title stringlengths 1 31 | text stringlengths 5 1.96k | embedding listlengths 1.79k 1.79k |
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電腦視覺 | * Introduction to computer vision（464KB pdf file）
* CMU's Computer Vision Homepage（頁面存檔備份，存於網際網路檔案館）
* Fudan University's Computer Vision Lab（頁面存檔備份，存於網際網路檔案館）
* Keith Price's Annotated Computer Vision Bibliography（頁面存檔備份，存於網際網路檔案館） and the Official Mirror Site Keith Price's Annotated Computer Vision Bibliography（頁面存檔備份，存於網際網路檔案館）
* HIPR2 image processing teaching package（頁面存檔備份，存於網際網路檔案館）
* USC Iris computer vision conference list（頁面存檔備份，存於網際網路檔案館）
* How to come up with new research ideas in computer vision? (in Chinese)（頁面存檔備份，存於網際網路檔案館）
* People in Computer Vision（頁面存檔備份，存於網際網路檔案館）
* The Computer Vision Genealogy Project（頁面存檔備份，存於網際網路檔案館）
## 參見
* 機器視覺
* 人工智慧與圖型識別
* 圖像處理
* 自動光學檢查
* 開放原始碼電腦視覺庫：OpenCV | [
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希腊字母Psi | **Psi**（大寫 Ψ，小寫 ψ，發音：/ˈpsaɪ/ 或 /ˈsaɪ/，中文音譯：**普西**），是第二十三個希臘字母。西里爾字母的Ѱ (Psi) 是由 Psi 演變而成。
符號 Ψ 或 ψ 用於：
* 心理學，精神病學，有時超心理學（涉及超能力或超自然的有關科目）。
* 在數學中，斐波納契數倒數和。
* 植物細胞水分運動之間的水勢。
* 在物理電磁場上，Ψ 用於表示磁鏈。
* 在生物化學，它是指稀有核苷酸偽尿苷酸。
* 在確定流流體力學函數曲線的流速始終相切。
* 在一個蛋白質的二面角的骨幹。
* 海王星。
* 薛丁格方程和整個量子力學， $\psi (x)\,$ 代表波的函數。
* 印第安納大學（該校首字母縮寫 I 和 U 的疊加效果）。
* 藥理學，一般藥店。
* 在病毒學的 ψ 網站是一個病毒的包裝信號。
* 粒子物理學上的 J/ψ 介子。
* 在可計算性理論， $\psi _{\mathrm {P} }(x)\,$ 代表一個程序 $\mathrm {P} \,$ 返回值 $\Upsilon \,$ 。 | [
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棕矮星 | **棕矮星**是質量介於最重的氣態巨行星和最輕的恆星之間的一種次恆星，具體而言，質量介於 13 至 75 或 80 倍木星質量 ，或大約 2.5×1028 kg 至大約 1.5×1029 kg。低於這個範圍的是次棕矮星（有時被稱為流浪行星），質量在這之上的是最輕的恆星紅矮星。棕矮星可能只有對流，而沒有分層或化學分化深度。
不同於主序帶上的恆星，棕矮星的質量不足以維持核心中氫（1H）融合成氦的核融合反應。然而，它們介於 13 _M_J 和 65 _M_J 之間的質量，被認為可以進行氘（2H）和鋰（7Li）的核融合。人們還在爭議能否不以核融合反應來定義，而以形成的過程更好的定義棕矮星。
恆星都按照光譜來分類，棕矮星的光譜分類為 M、L、T 和 Y 型 。儘管它們都被稱為棕矮星，但仍然有著不同的顏色。以人眼能看見的許多棕矮星，它們可能是橙色或紅色。在可見光的波長範圍內，棕矮星都很黯淡。
已知有行星環繞的棕矮星有 2M1207b、MOA-2007-BLG-192Lb、和 J044144b。
距離地球最近的棕矮星是在 2013 年發現的盧曼 16，這是距離 6.5 光年的一對聯星。截至 2017 年 12 月，在 NASA 的檔案中質量最大的系外行星是 HR 2562 b，估計質量在 30±15 _M_J，已經超過行星和棕矮星質量分界（13 _M_J）的兩倍多。
## 歷史
在 1960 年代，從理論上就已經推測出棕矮星的存在。
### 理論建立初期
在希夫・庫馬爾於 1960 年代建立理論時，現在所謂的 "棕矮星" 被稱為黑矮星 ；在對太空中自由漂流的次恆星天體的分類，這是一種不足以維持氫融合反應的天體。然而，（a）黑矮星這個名詞已被用來標示冷卻後的白矮星；（b）紅矮星可以燃燒氫；（c）這些天體在其生命的早期可能會發出可見光。正因為如此，這種天體曾經被提出的替代名詞包括**子星**和**次恆星**。在 1975 年，吉兒・塔特以其顏色接近棕色，提出了**棕矮星**這個名詞。
「黑矮星」這個名詞現在是指已經冷卻到不再發出可見光程度的白矮星。然而經過計算，即使是質量最低的白矮星，其冷卻到這樣的溫度所需要的時間也比目前的宇宙年齡還要長許多，因此預期這種天體尚不存在。 | [
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棕矮星 | 早期的理論認為質量最低的恆星和氫燃燒的極限在第一星族星是 0.07 太陽質量（_M_☉），在第二星族星是 0.09_M_☉，而且不會經歷正常的恆星演化歷程，就成為一顆緻密星 。第一次自一致性的氫燃燒最低質量計算，證實第一星族星的質量在 0.07 至 0.08_M_☉ 。
### 氘燃燒
在 1980 年代後期，受到氘燃燒將質量下限縮減為 0.012 太陽質量，還有在棕矮星外層低溫的大氣層有塵埃形成的影響，引發對這些理論的質疑。然而，因為它們幾乎不發出可見光，所以很難找到這樣的天體。這些天體輻射最強的光譜主要在紅外線上，而地面上觀測紅外線的儀器當時還不夠精良，無法輕易地分辨出任何棕矮星。
從那時以來，通過各種方法進行的多次搜索都在尋找這種天體。這些方法包括環繞著星場的多色成像調查、調查主序星中矮星和白矮星的微弱伴星、年輕星團的調查、和監測徑向速度以尋找緊鄰的伴星。
### GD 165B 和 "L" 類
多年來，發現棕矮星的努力毫無結果。然而在 1988 年，在以紅外線搜索發搜尋白矮星的工作中發現一顆被稱為 "GD 165 的恆星有黯淡的伴侶。GD 165B 的光譜是非常的紅且神祕，但沒有顯示預期的低質量紅矮星特徵。很顯然的 GD 165B 需要歸類在比當時已知的小矮星更冷的物體。GD 165B 在將近 10 年的期間都是獨一無二的，直到 2 微米全天巡天（2MASS）的調查展開，才發現更多具有這一類色彩和光譜特徵的天體。
如今，GD 165B 被認為是 "**L** 矮星 " 這一類型天體的原型。
儘管當時發現最低溫矮星的意義重大，但爭論的是 GD 165B 是該歸類為棕矮星，還僅僅是一顆質量很低的恆星。因為從外觀上來講，很難區分這兩種天體。
在發現 GD 165B 之後，其它的棕矮星候選者報告很快的就被提出。但是，大多數的候選者因為缺乏鋰，表明它們只是一顆恆星，因而喪失了後者的資格。真正的恆星在一億年不到的時間就會燃燒掉鋰，但棕矮星不會；而令人困惑的是棕矮星的亮度和溫度都類似於真正的恆星。因此，在年齡超過一億歲的天體大氣中檢測到鋰，才能確認它是一顆棕矮星。
### 葛利澤 229B 和 "T" 型甲烷矮星 | [
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棕矮星 | 在 1995 年，棕矮星的研究有了極大的突破，發現了兩個無可置疑的次恆星天體（泰德 1 和葛利澤 229B），並且以 670.8 奈米的譜線確認了鋰的存在。其中最著名的是後者，它被發現溫度和亮度都遠低於恆星的範圍。引人注目的是，其紅外線光譜清楚的呈現 2 微米的甲烷吸收帶，這種特徵以前只在巨行星和土星的衛星泰坦的光譜中觀測到。甲烷的吸收不會在主序星的溫度下出現，此一發現協助建立了另一類溫度比 L 矮星更低的 "T 矮星"，葛利澤 229B 就是此一類型的原型。
第一次確認的棕矮星是由西班牙天文物理學家 Rafael Rebolo（領導者）、瑪麗亞・羅莎・薩派特羅・奧索裡奧和愛德華多・馬丁在 1994 年發現的。他們發現的這個天體被稱為泰德 1，位於疏散星團昴宿星團內。發現的本文袃 1995 年春天提交給自然雜誌，該雜誌在 1995 年 9 月 14 日發表。_自然_在頭版頭條，很顯目的標示 "棕矮星的發現，正式版"。
發現泰德 1 的影像是 IAC 的團隊在 1994 年 1 月 6 日使用泰德峰天文台口徑 80 公分的望遠鏡（IAC 80）蒐集影像，而根據紀載，它的光譜是在 1994 年 12 月使用穆查丘斯羅克天文台（拉帕爾馬島）4.2 米的威廉・赫歇耳望遠鏡採集的。因為泰德 1 是昴宿星團的成員，它的距離、化學組成、和年齡，才能被估計出來。使用最先進的恆星演化模型，團隊估計泰德 1 的質量是 55 木星質量，這顯然低於恆星質量的下限。這個天體隨後成為年輕棕矮星工作相關引用的參考。
在理論上，質量低於 65 木星質量的棕矮星，在其演化過程中的任何時間點上都不會引燃鋰燃燒的過程。事實上，檢測鋰的原則是調查低光度和低表面溫度天體的本質之一。
由凱克 I 望遠鏡所獲得的高品質光譜資料，顯示泰德 1 一直保持形成昴宿星團的分子雲原本的組成分，也證實在核心始終未進行鋰燃燒的核融合反應。這些觀測確認泰德 1 是一顆棕矮星，也驗證了鋰光譜測試的成效。
有一段時間，泰德 1 是太陽系之外，經由直接觀測能見到的最小的天體。迄今，確認的棕矮星已經超過 1,800 顆，甚至有一些非常靠近地球，像是被引力束縛在一起，距離太陽大約 12 光年的一對棕矮星印地安座 εBa 和 Bb，還有距離僅有 6.5 光年的棕矮星聯星 Luhman 16。
## 理論 | [
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棕矮星 | 恆星誕生的標準機制是通過冷星際雲的氣體和塵埃形成引力坍縮。當雲氣收縮時，因為釋放出重力位能而使它的溫度升高。在進行收縮的早期，氣體會迅速的放熱，將多餘的能量輻射出去，使坍縮繼續進行。最後，中心區域會變得足夠緻密而陷阱輻射。因此，坍縮的雲氣中心溫度和密度隨著時間的推移而急遽增加，這使得收縮減緩，直到原恆星核心的溫度和壓力大到足以引發熱核反應。大多數的恆星，經由熱核融合成的氣體和輻射壓力將支撐它阻止任何進一步的引力收縮。達到流體靜力平衡的恆星，會花費絕大部分的生命期在氫融合成氦的主序星時期。
然而，如果原恆星的質量大約低於 0.08 太陽質量，在核心不會點燃正常的核融合反應。重力收縮不足以讓這麼小的原恆星產生足夠的熱，而在核心的溫度達到可以引發核融合反應之前，密度已經達到使原子密集到足以創建量子狀態的電子簡併壓力。依據棕矮星內部的模型，典型的核心溫度、壓力和密度將如下式預期所示：
* $10\,\mathrm {g/cm^{3}} \,\lesssim \,\rho _{c}\,\lesssim \,10^{3}\,\mathrm {{g}/{cm^{3}}} $
* $T_{c}\lesssim 3\times 10^{6}\,\mathrm {K} $
* $P_{c}\sim 10^{5}\,\mathrm {Mbar} $ 。
這意味著這顆原恆星是質量不夠和密度不足以達到維持氫融合所需要的條件。由電子簡併壓力所達到的密度和壓力，阻止了物質繼續向核心掉落。
進一步阻止引力收縮的結果是 "失敗的恆星"，或是棕矮星只是經由冷卻釋放掉其內部的熱能。
### 高質量棕矮星 VS 低質量恆星 | [
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棕矮星 | * 鋰普遍存在於棕矮星，而不存在於低質量恆星。要成為恆星，必需進行氫融合反應，這會迅速消耗掉它們的鋰。這是因為鋰 - 7 和質子的碰撞會產生兩顆氦 - 4 的原子核，而這種反應所需要的溫度恰好低於氫融合所需要的溫度。低質量恆星內部的對流確保整體的鋰會被耗盡，因此在次恆星的光譜中存在鋰線，是棕矮星候選者的強力指標。使用鋰來區分棕矮星和低質量恆星的方法通常稱為**鋰測試**，最早是由 Rafael Rebolo、愛德華多・馬丁、和 Antonio Magazzu 提出的。然而，在非常年輕的恆星中，因為還沒有足夠的時間來燃燒掉，也還存在著鋰。更重的恆星，像太陽，可以將鋰保存在外層的大氣中，而永遠不會獲得足夠的溫度使鋰枯竭，不過從它們的大小已足以分辨出它們不是棕矮星。與此相反的，在質量範圍上限的棕矮星，在它們年輕的時候可以熱到足以耗盡鋰。矮星的質量超過 65 木星質量，在它們活到 5 億歲時就可以燒盡它們的鋰，因此這種測試是不完美的。
* 不同於恆星，垂老的棕矮星有時會非常冷，冷到有很長的一段時間它們的大氣層可以收集到足以檢測出的甲烷含量。用這種方式確認的棕矮星包括葛利澤 229B。
* 主序星可以很冷，但是有最低的熱光度，使它們可以穩定的維持核融合進行。這在每顆恆星都不一樣，但一般至少是太陽的 0.01% 。棕矮星既冷且黑暗，在它們的有生之年穩定的步入黑暗：足夠老的棕矮星將會為太暗，以至於偵測不到。
* 鐵雨很可能是棕矮星的大氣對流換熱過程的一部分對流層，而且只出現在棕矮星，而不會在小的恆星中。對鐵雨的光譜研究仍在進行中－－不是所有的棕矮星都有，也不是永遠都有這種異常的大氣。在 2013 年，在最靠近太陽系的 Luhman 16 伴星 B 的大氣層影像內發現異常的鐵含量。
### 低質量棕矮星 vs 高質量行星
棕矮星另一個顯著的特性是它們的半徑大約都與木星的半徑相同。在它們質量的上限範圍（60-90 木星質量），棕矮星的體積主要由電子簡併壓力維持，如同白矮星。在質量的下限範圍（10 木星質量），其體積是由庫倫壓力維持，如同行星。由此，棕矮星的半徑隨著質量變化的範圍僅有 10-15%，這使得它們很難與行星區別開來。
此外，有許多的棕矮星沒有經歷過核融合：在質量的下限範圍（低於 13 木星質量），它們的核心從未熱到可以進行融合氘，甚至在質量的上限範圍（超過 60 木星質量），快速的冷卻使它們不能持續進行核融合，能進行核融合的時間大約只在 1,000 萬年以內。 | [
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棕矮星 | X 射線和紅外線光譜的分析也有些跡象。有些棕矮星能發出 X 射線；並且所有較溫暖的棕矮星持續地在紅光和紅外線的光譜中發出引人注目的光輝，直到它們的溫度低到像行星那樣（低於 1,000K）。
氣體巨星有一些棕矮星的特性。例如，木星和土星兩者的主要成分都像太陽一樣，是氫和氦。而土星的質量只是木星的 30%，但卻幾乎和木星一樣大。太陽系中的三顆巨大行星（木星、土星、和海王星）排放出來的熱量比從太陽收到的更多。而且 4 顆巨大的行星都有自己的 "行星系統"— 它們的衛星。棕矮星像恆星一樣，是單獨形成的，只是缺乏足夠的質量，不能像恆星一樣的燃燒。像所有的恆星一樣，它們可以是單獨的天體，也可以很靠近其他的恆星。有些軌道也像行星一樣，是橢圓的軌道。
目前，國際天文學聯合會考慮質量在可以燃燒氘的質量下限（目前以太陽系的金屬量計算是 13 木星質量）以上的天體是棕矮星，而在此質量下限以下（並且環繞其他恆星或恆星殘骸的），則被考慮是一顆行星。
以 13 木星質量做分界只是一個經驗法則，而不具有精確的物理意義。較大的天體會燒掉較多的氘，而較小的只會燒掉一些，13 木星質量大約在它們中間點的位置。氘的數量在某種程度上也取決於天體本身的一些組成分，特別是氦和氘相較於其他重元素的比例，這些決定了大氣透明度、進而影響到該天體冷卻的速率。
太陽系外行星百科包含質量高達 25 木星質量的系外行星，系外行星資料瀏覽器也收錄 24 木星質量的行星。
次棕矮星或「行星質量棕矮星」是在與恆星和棕矮星相同模式下形成的另一種天體（也就是說，它們經由氣體雲的坍縮，但是質量低於氘燃燒（大約 13 木星質量）的熱融合所需要的質量）。一些研究人員稱它們是自由漂浮行星，而另一些人則稱它們是行星質量棕矮星。
次棕矮星的形成方式與恆星相同，經由氣體雲的坍縮（或許有光侵蝕的協助），但是對於此過程是否會形成行星，以及在分類上則尚未取得共識自由漂浮次棕矮星在觀測上與最初在恆星形成，然後被逐出軌道的流浪行星沒有甚麼區別；另一方面，在星團中形成的次棕矮星也可能被捕獲，進入環繞恆星的軌道。國際天文學聯合會的太陽系外行星工作小組（WGESP）提出了 "次棕矮星" 這個名詞，定義為：在年輕星團中發現，質量在棕矮星質量下限之下的自由漂浮天體。
#### 低質量極限
氣體雲坍縮可以形成的次棕矮星質量下限大約是 1 MJ。這是因為氣體的重力塌縮需要輻射能量，但受到透明度的限制會加熱天體。在一篇論文中描述了 3 MJ 的候選者。 | [
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棕矮星 | #### 可能的行星質量棕矮星列表
**環繞一顆或更多恆星**
這些恆星的伴星是次棕矮星還是行星，還沒有獲得共識。
* WD 0806-661 B
* 處女座 DT c
* 金牛座 FW b
* ROXs 42b b
**環繞著棕矮星**
這些恆星的伴星是次棕矮星還是行星，還沒有獲得共識。
* 2MASS J04414489+2301513 的伴星，質量為 5–10MJ
* 2M1207b
**自由漂浮**
* WISE 0855–0714：距離地球約 7 光年，質量在 3-10 MJ
* S Ori 52
* UGPS J072227.51-054031.2
* Cha 110913-773444
* CFBDSIR2149-0403
## 觀測
### 棕矮星的類型
#### 光譜類型 M
光譜類型在 M6.5 與之後的恆星是棕矮星，它們也稱為後 M 矮星。
#### 光譜類型 L
長期以來，M 是古典的恆星光譜類型中溫度最低的一類。光譜以一氧化鈦（TiO）和釩氧化物（VO）分子的可見光吸收譜線為主。然而，GD 165B，白矮星 GD 165 的伴星，沒有標示 M 矮星的 TiO 吸收譜線特徵。隨著越來越多如同 GD 165B 的天體現身，最終導致柯克派翠克和其他人定義新的光譜類型，**L 矮星**：在紅光區不是微弱的金屬氧帶（TiO、VO），而是強烈的金屬氫化物帶（FeH、CrH、MgH、CaH）和顯著的鹼金屬線（NaI、Kl、CsI、RbI）。截至 2013 年 (2013-Missing required parameter 1=_month_!)，已經確認的譜線超過 900 條，多數都是由廣域巡天：2 微米全天巡天（2MASS）、近紅外南天深度巡天（DENIS）、和史隆數位巡天（SDSS）等的成果。
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棕矮星 | 如同 GD 165B 是 L 矮星的原型，葛利澤 229B 是第二種光譜類型 T 矮星的原型。而近紅外線（NIR）光譜的 L 矮星顯示強烈的水（H2O）和一氧化碳（CO）的吸收線，葛利澤 229B 的近紅外線光譜主要的吸收線則來自甲烷（CH4），這種特徵只曾在太陽系巨大的土星和它的衛星泰坦發現過。CH4、H2O、和氫分子的碰撞誘導吸收（CIA，collision-induced absorption），使得葛利澤 229B 的近紅外影像呈現藍色。它陡峭的紅色光譜也缺乏 L 矮星的 FeH 和 CrH 吸收帶，反而有異常寬廣的鹼金屬鈉和鉀的吸收特徵。這些差異導致柯克・派翠克提出有 H - 和 K - 波段與 CH4 吸收特徵的 T 矮星光譜類型。截至 2013 年 (2013-Missing required parameter 1=_month_!)，已經知道的 T 矮星有 355 顆。最近，亞當和湯姆・格巴爾研擬了 T 矮星近紅外光譜分類的新方案。理論認為 L 矮星是混和了質量非常低的恆星和次恆星天體（棕矮星），而 T 矮星完全都由棕矮星組成。因為鈉和鉀的吸收在 T 矮星光譜中綠色的部分，在人類視覺感知的 T 矮星真實顏色應該是煤焦油染料的洋紅色，而不是棕色。光譜類型 T 的棕矮星，像是 WISE 0316+4307，已經檢測出其距離太陽超過 100 光年。
#### 光譜類型 Y
還有一些人懷疑，如果還有的話，應該屬於 **Y 型矮星**，並預期它們將比 T 型矮星冷得多。雖然沒有明確定義的譜序，也尚未有原型，但它們已經有了模型。
在 2009 年，已知最冷的棕矮星溫度估計在 500-600k 之間，並已分配光譜為 T9。有三顆棕矮星的例子，分別是 CFBDS J005910.90-011401.3、ULAS J133553.45+113005.2、和 ULAS J003402.77−005206.7。這些天體的光譜顯示在 1.55 微米附近的吸收譜。Delorme 等人認為這些是氨的吸收譜線，應該被看作是 T-Y 的過渡，這些天體應該標誌為 Y0。但是，這項特徵很難區分是否是水和甲烷的吸收，因此其它的作者認為歸類為 Y0 尚言之過早。
在 2010 年 4 月，發現了兩顆超級冷的次棕矮星（UGPS 0722-05 和 SDWFS 1433+35），被提出來做為光譜分類 Y0 的原型。 | [
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棕矮星 | 在 2011 年 2 月，Luhman 等人報告在一顆白矮星發現了鄰近的伴星，是溫度大約 300K，7 木星質量的棕矮星。儘管這是行星的質量，但羅德里格斯等人認為它不可能像行星那樣的方法形成。
不久後，劉等人出版了一份_非常冷_（〜370K）的環繞著非常低質量棕矮星的棕矮星報表，並且註明_鑒於它們的低光度、非典型的顏色和寒冷的溫度，CFBDS J1458+10B 被架設是 Y 型矮星最有希望的候選者_。
在 2011 年 8 月，科學家使用 NASA 的 WISE 資料發現 6 顆 Y 型矮星－有著像恆星的身體，但溫度與人體一樣冷。
從 WISE 的資料已經發現了上百顆新的棕矮星，其中有 14 顆歸類為極冷的 Y 型矮星。一顆被稱為 WISE 1828+2650 的 Y 型矮星是在 2011 年 8 月發現的，是紀錄中溫度最低的棕矮星－完全沒有發射可見光。這種天體被認為像是自由漂浮行星，更勝於是恆星。最初估計 WISE 1828+2650 有溫度大約低於 300K 的大氣層 — 將它和室溫的上限 298K（25 °C，80 °F）比較，它的溫度已經修正和更新為範圍在 250 至 400K（−23–127 °C，−10–260 °F）。
在 2014 年 4 月，WISE 0855–0714 被宣布其溫度輪廓估計在 225 至 260K，質量在 3-10 木星質量，也很不尋常地觀測到它的視差，意味著它距離太陽系很近，只有 7.2±0.7 光年的距離。
### 棕矮星的光譜和大氣特性
L 型和 T 型矮星的排放通量多數是在 1 至 2.5 微米的近紅外線範圍內。從後期 M 型、L 型、T 型序列的低溫和遞減的溫度，結果是在近紅外有著豐富的譜線和種類繁多的特點。從中性原子到廣泛的分子帶，所有的一切都在溫度、重力和金屬量上具有不同的依賴性和相對較窄的譜線。此外，在這些低溫條件下，有利於冷凝氣體的狀態和晶粒的形成。
棕矮星大氣的典型溫度範圍從 2,200 下降至 750K。相較於恆星，內部穩定的核融合使自身暖起來，棕矮很快就結束並變得冰冷。質量大的棕矮星冷卻得比質量小的緩慢。
### 觀測的技術
日冕儀最近被用於檢測環繞著明亮恆星的暗弱天體，包括葛利澤 229B。
配備了靈敏的電子耦合裝置（CCD）的望遠鏡已經用來尋遙遠星團中暗弱的天體，包括泰德 1。
廣視野搜尋也已經確認一些獨立的黯淡天體，像是距離 30 光年遠的克魯 1。 | [
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棕矮星 | 在搜尋系外行星的巡天調查中，經常會發現棕矮星。系外行星偵測法對棕矮星也一樣適用，而且還更為容易。
### 里程碑
* 1995: 第一顆棕矮星被確認。泰德 1 是位於昴宿星團的一顆光譜類型為 M8 的天體，是被 Instituto de Astrofísica de Canarias 位於西班牙加那利群島中的拉帕爾瑪島上的穆查丘斯羅克天文台使用 CCD 篩檢出來的。
: 確認第一顆甲烷棕矮星。葛利澤 229B 是使用位於南加州帕洛馬山帕洛馬天文台 60 英吋反射鏡的自適應光學配合日冕儀，從銳化的影像中發現環繞著葛利澤 229A（距離地球 20 光年遠）的這顆棕矮星。隨後，200 英吋（5 米）的海爾望遠鏡紅外線光譜顯示有大量的甲烷。
* 1998 年：發現第一顆輻射出 X 射線的棕矮星。蝘蜓座 α1 是在蝘蜓座 I 暗星雲內的一個 M8 天體，被測出是一個 X 射線源，類似於對流的晚期型星。
* 1999 年 12 月 15 日：第一次檢測到來自棕矮星的 X 射線閃焰。加州大學的一個團隊使用錢卓 X 射線天文台監測 LP 944-20（距離 16 光年，60 木星質量），觀察到 2 小時的閃焰。
* 2000 年 7 月 27 日：第一次檢測到棕矮星的無線電發射（在閃焰和靜止期）。一個使用甚大天線陣研究 LP 944-20 的學生團隊在 2001 年 3 月 15 日的_自然_雜誌上發表了他們的成果。
* 2014 年 4 月 25 日：發現已知最冷的棕矮星。WISE J085510.83-071442.5 是距離地球 7.2 光年，是距離太陽第 7 近的一顆恆星，表面溫度在 - 13℃至 - 48℃[NASA]。
### 棕矮星和 X 射線源
自從 1999 年檢測到棕矮星的 X 射線閃焰，就建議改變有磁場的天體，將類似的低質量恆星也納入其中。 | [
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棕矮星 | 中心沒有強大核能來源，棕矮星的內部是在快速沸騰，或對流的狀態。當結合了大多數的棕矮星都展現出快速的自轉，為對流建立了強而有力的發展條件，在表面附近糾結磁場。錢卓 X 射線天文台觀測到來自 LP 944-20 的閃焰有可能起源自棕矮星表面下的湍流磁化熱材料。部分表面的閃焰可以傳導熱進入大氣層，使電流流動，產生 X 射線閃焰，就像發生閃電。在非燃燒期間缺乏來自 LP 944-20 的 X 射線也是一個值得注意的結果。它設置了最低能量以觀測棕矮星產生的穩定 X 閃焰能量，並顯示當棕矮星的表面溫度低於 2800K 時，日冕不再出現，並成為電中性。
使用 NASA 的錢卓 X 射線天文台，科學家已經在一個多星系統中發現來自低質量棕矮星的 X 射線。這是第一次在靠近母恆星（類似太陽的恆星 TWA 5A）的棕矮星解析出 X 射線來自日東京中央大學的坪井（Yohko tsuboi）說："我們的錢卓資料顯示，X 射線起源於棕矮星日面的電漿，它們的溫度高達 300 萬℃"，"這顆棕矮星的 X 射線亮得如同今天的太陽，但是它比太陽小了 50 倍"。"因此，這種觀測提高了巨大行星可能會在它們的青春期發出 X 射線的可能性！"
### 最近的發展
位於蝘蜓座，距離地球 500 光年的棕矮星蝘蜓座 110913-773444，可能是一個形成過程中的微型行星系統。來自賓夕法尼亞州立大學的天文學家相信他們檢測到類似於太陽系形成假說的氣體和塵埃盤面。Cha 110913-773444 是迄今為止發現最小的棕矮星（8 倍木星質量），如果它形成行星系統，它將是已知系統中最小的一個。他們的研究結果發表在 2005 年 12 月 10 日的天文物理雜誌。
目前觀測已知的棕矮星候選者已經揭露一種模式，紅外線輻射的閃耀和模糊不清，顯示相對較低的溫度，還有不透明的雲遮蔽內部的熱，激起極端的風。在這些天體上的天氣是極端劇烈的，可以媲美並遠遠超過木星上著名的大紅斑。
在 2013 年 1 月 8 日，天文學家使用 NASA 的哈伯太空望遠鏡和史匹哲太空望遠鏡探測被命名為 2MASS J22282889-431026 的棕矮星，繪製出迄今最詳細的棕矮星天氣圖，它顯示風力驅動著行星大小的雲。新的研究不僅是對棕矮星有更進一步的理解，也是研究太陽系外行星大氣的踏腳石。 | [
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棕矮星 | NASA 的 WISE 任務已經檢測到 200 顆新的棕矮星。實際上，在宇宙中鄰近我們的棕矮星比過去認知的要少。並非每有一顆棕矮星就有一顆恆星，而是每有一顆棕矮星，可能就有 6 顆恆星。
## 環繞棕矮星的行星
從相對較大的質量與大軌道推斷，環繞著棕矮星的行星質量天體 2M1207b、豺狼座 GQ b 和 2MASS J044144，可能是由環繞著棕矮星的雲氣塌縮而成，而不是吸積形成的，所以可能是次棕矮星而不是行星。然而，在 2013 年，發現第一顆在一個相對較小的軌道上環繞著棕矮星的低質量伴星（OGLE-2012-BLG-0358L b）。
環繞著棕矮星的盤面已經發現有許多特徵與恆星相同；因此預期會有吸積形成的行星環繞著棕矮星。有鑑於棕矮星盤面的低質量，大多數的行星將會是類地行星，而不是氣體巨星。如果有一顆環繞棕矮星的氣體巨行星橫越過我們的視線，然後，因為它們有著大約相似的直徑，會產生很明顯的凌日的訊號。行星的吸積區很接近棕矮星本身時，潮汐力就會有很大的影響。
環繞著棕矮星的行星很可能是耗盡水的碳行星。
## 棕矮星之最
* WD 0137-349 B: 第一次確認倖存於紅矮星階段之後的棕矮星。
* 在 1984 年，有一些天文學家假設可能有尚未被發現，環繞著太陽的棕矮星（有時被稱為復仇者），可以像經過的恆星一樣，和歐特雲產生交互作用。但這個理論已經沒落了。
## 相關條目
* 棕矮星列表
* 次棕矮星
* 棕矮星沙漠
* 亞恆星
* 紅矮星
* 藍矮星 (紅矮星階段)
* 暗物質
* 系外行星
* 橙矮星 — K 型主序星
* 黃矮星 — G 型主序星
## 參考資料
## 外部連結
* HubbleSite newscenter – Weather patterns on a brown dwarf （頁面存檔備份，存於網際網路檔案館）
### 歷史
* S. S. Kumar, _Low-Luminosity Stars_. Gordon and Breach, London, 1969—an early overview paper on brown dwarfs
* The Columbia Encyclopedia
### 細目
* A current list of L and T dwarfs | [
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] |
棕矮星 | * A geological definition of brown dwarfs, contrasted with stars and planets (via Berkeley)
* Neill Reid's pages at the Space Telescope Science Institute:
* On spectral analysis of M dwarfs, L dwarfs, and T dwarfs
* Temperature and mass characteristics of low-temperature dwarfs
* First X-ray from brown dwarf observed, Spaceref.com, 2000
* Brown Dwarfs and ultracool dwarfs（late-M, L, T） （頁面存檔備份，存於網際網路檔案館）—D. Montes, UCM
* Wild Weather: Iron Rain on Failed Stars （頁面存檔備份，存於網際網路檔案館）—scientists are investigating astonishing weather patterns on brown dwarfs, Space.com, 2006
* NASA Brown dwarf detectives （頁面存檔備份，存於網際網路檔案館）—Detailed information in a simplified sense
* Brown Dwarfs （頁面存檔備份，存於網際網路檔案館）—Website with general information about brown dwarfs（has many detailed and colorful artist's impressions）
### 恆星
* Cha Halpha 1 stats and history
* A census of observed brown dwarfs（not all confirmed）, ca 1998
* Epsilon Indi Ba and Bb, a pair of brown dwarfs 12 ly away | [
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棕矮星 | * Luhman et al., Discovery of a Planetary-Mass Brown Dwarf with a Circumstellar Disk
* Discovery Narrows the Gap Between Planets and Brown Dwarfs, 2007
* Y-Spectral class for Ultra-Cool Dwarfs, N.R.Deacon and N.C.Hambly, 2006 | [
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連續介質力學 | **連續介質力學**（Continuum mechanics）又稱**連續體力學**，是物理學、特別的是力學當中的一個分支，是處理包括固體和流體在內的所謂「連續介質」（continuous medium）或「連續體」（continuum，台灣也簡稱連體）宏觀性質的力學，由法國數學家奧古斯丁・路易・柯西在 19 世紀提出。
## 基本假設
連續介質力學的最基本假設是「連續介質假設」：即認為真實的流體和固體，可以近似看作連續的、充滿全空間的介質組成，各部分間無空隙（pore or empty）存在，物質的宏觀性質依然受牛頓力學的支配。此外，描述此介質各物理量所引用的數學函數，也均為連續函數。
此基本假設忽略物質的具體微觀結構（對固體和液體微觀結構研究，屬於凝聚態物理學的範疇），而用一組偏微分方程來表達宏觀物理量（如質量，速度，壓力等）。這些方程包括：本構方程（constitutive equation，也稱物性方程，描述介質性質的方程），和基本的物理定律（如質量守恆定律，動量守恆定律等）。連續介質力學排除了微觀及宏觀宇宙，只適用於一般工程科學的中等尺寸材料或對象，並不適用於：分子碰撞、原子內部、星體間等之力學分析。
## 研究對象
* 固體：固體不受外力時，具有確定的形狀。固體包括不可變形的剛體和可變形固體。剛體在一般力學中的剛體力學研究；連續介質力學中的固體力學則研究可變形固體在應力，應變等外界因素作用下的變化規律，主要包括彈性和塑性問題。
* 彈性：應力作用後，可恢復到原來的形狀。
* 塑性：應力作用後，不能恢復到原來的形狀，發生永久形變。
* 流體：流體包括液體和氣體，無確定形狀，可流動。經典流體最重要的性質是黏性（viscosity，經典流體對由剪切力引起的形變的抵抗力，無黏性的理想氣體和量子流體，不屬於經典流體力學的研究範圍）。從理論研究的角度，經典流體常被分為牛頓流體和非牛頓流體。
* 牛頓流體：滿足牛頓黏性定律的流體，比如水和空氣。
* 非牛頓流體：不滿足牛頓黏性定律的流體，介乎於固體和牛頓流體之間的物質形態。
## 主要分支學科
基本分支學科：
* 固體力學
* 彈性力學
* 塑性力學
* 損傷力學
* 斷裂力學
* 接觸力學
* 流體力學
* 流體靜力學
* 流體運動學
* 流體動力學
應用分支學科和交叉學科：
* 材料力學
* 結構力學
* 爆炸力學
* 空氣動力學 | [
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] |
連續介質力學 | * 電漿體動力學
* 磁流體動力學
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藍矮星 | **白矮星**，也稱為**簡併矮星**，是一種主要由電子簡併物質構成的恆星核殘骸。白矮星具有非常高的密度：它的質量與太陽相當，但體積與地球相近。白矮星沒有核融合來產生能量，微弱的亮度來自儲存的能量的熱輻射。已知距離最近的白矮星是天狼星 B，是在 8.6 光年的聯星，天狼星 A 的伴星。目前，在距離太陽最近的百顆恆星系統中，有 8 顆白矮星。異常昏暗的白矮星在 1910 年首次被確認；**白矮星**這名稱是威廉・魯伊登在 1922 年命名的。
白矮星被認為是恆星演化的最終狀態之一，是初始質量大約在 10_M_☉以下（這包括銀河系中超過 97% 的恆星），質量不夠高，因而不足以演化成為中子星。在主序帶末端的中、低質量的恆星結束氫融合階段後，將膨脹成為紅巨星，在此期間將通過 3 氦過程將在核心的氦融合成碳和氧。如果一顆紅巨星的質量不足以產生融合碳所需要的核心溫度（約 1 億 K），其將成為惰性的碳和氧積聚的核心。這樣的恆星在脫落其外層，並形成行星狀星雲後，它會留下一個核心，即殘存的白矮星。通常，這種白矮星主要由碳和氧組成，稱為**碳氧白矮星**（**CO white dwarf**）。如果初始質量介於 8 至 10.5_M_☉（太陽質量) 之間，核心的溫度足以融合碳，但仍不足以融合氖，這可能會形成**氧氖鎂白矮星**（**ONeMg** 或 **ONe white dwarf** ）。質量非常低的恆星無法燃燒氦，因此氦白矮星可能是在聯星系統中經由質量流失形成的。
白矮星不再進行核融合反應，因此它已經徹底耗盡了全部的燃料。所以，它不能通過核融合產生熱來抗拒重力以阻止重力崩潰，而只能靠電子簡併壓力來支撐，這導致它有極高的密度。簡併物理學推導出無自轉的白矮星的最大質量是 1.44_M_☉，即錢德拉塞卡極限，超過此上限，電子簡併壓力即無法支撐。接近這個質量極限的碳氧白矮星，通常通過伴星的質量轉移，可能經由一種稱為碳引爆的過程，爆炸成為一顆 Ia 超新星；SN 1006 就被認為是個著名的例子。 | [
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藍矮星 | 白矮星在形成之初仍十分熾熱，但是由於後續沒有能量來源，它會因為不斷的輻射能量而逐漸冷卻。這意味著，白矮星會經由它的輻射，從最初的高色溫，隨著時間的推移而降溫和轉紅。在極為漫長的一段冷卻時間裡，白矮星內的物質將從核心開始結晶。這顆恆星殘骸的低溫意味著它將不再發出顯著的熱量或光，最終將成為冰冷的黑矮星。不過白矮星達到這種狀態所要經歷的時間，經由理論推算，比當前的宇宙年齡（約 138 億年）還要長，所以認為還沒有黑矮星存在。現存最古老的白矮星仍然在以幾千度 K 的溫度持續輻射能量。
## 發現
第一顆被發現的白矮星是三合星的波江座 40，它的成員是主序星的波江座 40A，和在一段距離外組成聯星的白矮星波江座 40B 和主序星的波江座 40C。波江座 40B 和波江座 40C 這一對聯星是威廉・赫歇爾在 1783 年 1 月 31 日發現的, p. 73，它在 1825 年再度被瓦西里・雅可夫列維奇・斯特魯維觀測，1851 年被奧托・威廉・馮・斯特魯維觀測。在 1910 年，亨利・諾利斯・羅素、愛德華・皮克林和威廉敏娜・佛萊明發現它有一顆黯淡不起眼的伴星，而波江座 40B 的光譜類型是 A 型或是白色。在 1939 年，羅素回顧此一發現, p. 1：
> 我前往拜訪我的朋友，也是慷慨的恩人艾德華・C・皮克林教授。他一如往常的慈祥，自願檢視和討論我和 Hinks 在劍橋為觀察恆星視差所做的所有恆星光譜 — 還包括相互比較。這一段例行公事證明非常有效 — 發現了許多絕對星等相當黯淡的 M 型光譜恆星。在談到這個主題時（就我記憶所及），我請教皮克林一些不在我的名單上的暗星，特別是波江座 40B。他送了便條到天文台的辦公室，不久結果（我想是來自佛蘭銘夫人的）就來了，它的光譜是 A 型。即使在那天文領域的蠻荒年代，我也對其有足夠的了解，足以意識到其中的不對勁，那就是表面亮度和密度不符當時的理論可能數值。我臉上一定展現了我的困惑以及沮喪，這個完美的恆星規律上似乎出了個例外。但是皮克林微笑的對著我說：「就是因為這些例外，我們的知識才得以增長」，於是白矮星進入了研究領域！
>
>
對波江座 40B 的光譜正式的描述是在 1914 年由沃爾特・亞當斯提出的。 | [
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藍矮星 | 天狼星的伴星，天狼星 B，隨後也被發現。在 19 世紀，對有些恆星已經能夠精確的測量出其位置上的微小變化。貝塞爾使用這些精確的測量確定天狼星（大犬座 α）、南河三（小犬座 α）的位置都有些變動，在 1844 年他預言這兩顆恆星都有看不見的伴星：
> 如果我們認為_天狼星_和_南河三_是雙星，它們位置的變動就不會使我們驚訝了；我們必須承認它們，並且只由觀測確定它們的數量。但光度並不能反映質量的真實性質，如此多可見的恆星並不能證明沒有許多看不見的恆星。
>
>
貝塞爾粗略的估計出天狼星伴星的軌道週期是半個世紀 C. H. F. 彼得在 1851 年也計算出一個週期。直到 1862 年 1 月 31 日，阿爾萬・格雷厄姆・克拉克才看見這顆緊挨著天狼星的伴星，然後就證實這顆預期中存在的伴星，沃爾特・亞當斯在 1915 年宣布天狼星 B 的光譜和天狼星相似。
在 1917 年，范・馬南發現了一顆孤獨的白矮星，現在被稱為范馬南星。這三顆白矮星，最早發現的，是所謂的_經典的白矮星_, p. 2。終於，有許多的黯淡的白色恆星被發現，它們都有高自行，表示都是緊鄰地球的低光度天體，因此都是白矮星。
威廉・魯伊登在 1922 年要說明這種天體時，似乎是第一個使用_白矮星_這個名詞的人，稍後這個名詞經亞瑟・愛丁頓而通俗化了。儘管有各種的懷疑，第一顆非經典的白矮星大約直到 1930 年代才被辨認出來。在 1939 年已經發現了 18 顆白矮星, p. 3，在 1940 年代，魯伊登和其他人繼續研究白矮星，到 1950 年發現已經超過一百顆的白矮星，到了 1999 年，這個數目已經超過 2,000 顆之後的史隆數位巡天發現的白矮星就超過 9,000 顆，而且絕大多數都是新發現的。
## 組成和結構
雖然在已知的白矮星中，質量估計最低是 0.17 ，最高是 1.33 太陽質量，但質量分布明顯的在 0.6 太陽質量處是個高峰，大多數的質量都在 0.5 至 0.7 太陽質量之間。被觀測過的白矮星半徑估計在 0.008 和 0.02 太陽半徑之間。相較於地球的半徑是太陽的 0.009，白矮星將相當於太陽的質量封裝在只有太陽的百萬分之一，與地球相似的體積內，因此白矮星的平均密度大約是太陽密度的百萬倍，幾乎是 106 公克（1 噸） / 立方公分。白矮星是密度最大的已知天體種類之一，只有其他的緻密天體，像是中子星、黑洞和假設可能存在的夸克星能超越它。 | [
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藍矮星 | 白矮星在被發現之後就被確認是密度極端高的天體。如果一顆在聯星系統的恆星，像是天狼星 B 和波江座 40B，是可以從聯星的軌道估計出它的質量的。在 1910 年對天狼星 B 這樣做過，得到的質量是 0.94 太陽質量（現代的估計是 1 太陽質量）。由於高溫恆星的輻射量大於低溫恆星，恆星的表面亮度可以從有效表面溫度，也可以從光譜來估計。如果知道恆星的距離，它的整體光度也能估計出來。從這兩種圖表可以比較出恆星的半徑，由推理排出來的順序讓當時的天文學家非常困惑，因為天狼星 B 和波江座 40B 必須有非常高的密度。例如，當恩斯特・奧皮克在 1916 年估計一些聯星的密度時，他就發現波江座 40B 的密度超過太陽 25,000 倍以上，使他認為是 "不可能的"。如同亞瑟・史坦利・愛丁頓在 1927 年之後寫道, p. 50：
> 我們透過星光之中的訊息來學習與了解星星。當我們解讀了天狼星伴星所傳來的光訊息之後，我們得到以下的解譯："組成我的材料的密度，是比你所見過任何材料的密度都要高 3000 倍；光是一塊小到可以放進火柴盒裡的這種材料，它的重量就可以超過一噸。" 看到此訊息我們能做何回應？在 1914 年，我們通常只會有一種回應 -"閉嘴，別盡說些荒唐話。"
>
>
正如愛丁頓於 1924 年指出的那樣，根據廣義相對論，天狼 B 的光線將發生引力紅移。1925 年，亞當斯的觀測證實了引力紅移存在。
## 質量 - 半徑關係和質量極限
根據能量最小化原理，能簡單的推導出關於白矮星質量和半徑之間的粗略關係。我們可以把白矮星的初始能量近似的設定為與太陽的重力勢能和動能相當。 | [
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藍矮星 | 我們把 1 單位質量（就是說重力勢能公式中的 _m_=1）的白矮星的重力勢能計為 _Eg_，根據勢能公式，_Eg_=− _GM_/_R_，其中 _G_ 是萬有引力常數， _M_ 是白矮星質量， _R_ 是其半徑。同樣的，1 單位質量的動能計為 _Ek_，主要決定於其中的電子動能，所以它近似於 _N_ _p_2/2_m_，其中 _p_ 是電子平均動量，_m_ 是電子的質量， _N_ 是單位質量內的電子數。 電子是簡併物質，根據測不準原理，我們可用電子動量的測不準量 Δ_p_ 近似的表示 _p_。也就是說，Δ_p_Δ_x_ 近似的等於簡化普朗克常數_ħ_。其中的 Δ_x_ 近似於電子間平均距離，大致等於 _n_−1/3，也就是單位電子密度的立方根，其中的 _n_ 是 1 單位體積的電子數。基於白矮星的電子總數為 **N**×**M**，而它們的總體積正比於 _R3_，因此 _n_ 近似於 **N**×**M**/**R**3. 根據動能的微分公式 _Ek_，我們有：
: : $E_{k}\approx {\frac {N(\Delta p)^{2}}{2m}}\approx {\frac {N\hbar ^{2}n^{2/3}}{2m}}\approx {\frac {M^{2/3}N^{5/3}\hbar ^{2}}{2mR^{2}}}$ 。當白矮星的總能量 _Eg_ + _Ek_ 最小時，它處於穩定平衡態。從這點來看，重力勢能和動能應該相等。於是，我們得到下式：
: $|E_{g}|\approx {\frac {GM}{R}}=E_{k}\approx {\frac {M^{2/3}N^{5/3}\hbar ^{2}}{2mR^{2}}}$ 。由上式求解半徑 _R_，就得到
: $R\approx {\frac {N^{5/3}\hbar ^{2}}{2mGM^{1/3}}}$ 。上式中，_**N**_取決於白矮星的元素組成比例，而_ħ_是個普適常數（恆量）。由此，我們獲得白矮星質量與半徑之間的比例關係為：
: $R\sim {\frac {1}{M^{1/3}}},\,$
就是說，白矮星的半徑與其質量的立方根成反比例關係。 | [
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藍矮星 | 上述計算中的勢能採用了牛頓公式，所以計算結果是非相對論性的。假如我們對計算中的白矮星內電子速度做相對論性修正，就是說當電子速度逼近光速 _c_ 時，我們應把電子動能 _p_2/2_m_ 用狹義相對論的近似值 _pc_ 代替。經過這個替換，我們就發現
: : $E_{k\ {\rm {relativistic}}}\approx {\frac {M^{1/3}N^{4/3}\hbar c}{R}}$ 。如果我們把此式與 _Eg_ 聯立取等，就可看到 _R_ 已經消去，而質量 _M_ 的極限值約為
: $M_{\rm {limit}}\approx N^{2}\left({\frac {\hbar c}{G}}\right)^{3/2}$ 。
對這個質量極限的解釋是：因為白矮星的質量與其體積成反向關係，當我們增加白矮星的質量時，它的半徑反而縮小。於是，根據測不準原理，電子的動量或者說它的速度將增加。當電子運動速度逼近光速 _c_ 時，相對論性計算的準確度迅速提高，意味著白矮星質量 _M_ 將收斂於 _Mlimit_。因光速不可逾越，白矮星的質量不可能大於質量極限 _M_limit。
要更精確的計算白矮星的質量 —— 半徑關係和質量極限，必須考慮描述白矮星物質密度與壓強關係的狀態方程式。
## 形成
白矮星被認為是質量介於 0.07 至 10_M_☉為主序星的生命終點，產生的白矮星組成取決於恆星的初始質量。目前的星系模型表明，銀河系現在大約包含 100 億顆白矮星。
### 非常低質量的恆星
如果主序星的初始質量低於 0.5_M_☉，它永遠不會熱到足夠讓核心的氦參與核融合反應。因此，其壽命遠遠超過宇宙的年齡（大約 138 億年）。這樣的恆星最終將燃燒它所有的氫，在一段時間內成為一顆藍矮星，並結束其進化，成為主要由氦 - 4 組成的白矮星。由於這過程需要很長的時間，因此現在還觀察不到起源於它們的氦白矮星。相反的，現存的氦白矮星被認為是聯星系統中質量損失的產物，由於大行星伴星造成的質量損失。
### 低至中等質量的恆星 | [
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藍矮星 | 如果主序星的初始質量介於 0.5 至 8_M_☉之間，例如我們的太陽，它的核心會變得足夠的熱，可以通過 3 氦過程將氦融合成碳和氧，但仍永遠不足以將碳融合成氖。在核融合反應接近尾聲時，這樣的恆星將有一個不再進行核融合反應的碳氧核，周圍是仍在反應的氦殼，和更外層還在燃燒氫的殼。在赫羅圖上，可以在漸近巨星分支上找到這一類的恆星。然後它將排出其大部分的外層物質，形成一個行星狀星雲，直到只剩下碳氧的核心。這個過程將生成碳氧白矮星，成為我們觀測到的絕大多數白矮星。
### 中等質量到大質量的恆星
如果一顆恆星的質量足夠巨大，它的核心最終會熱到足夠將碳融合成氖，然後將氖繼續融合成更重的元素，直到融合成鐵。這樣的恆星不會變成白矮星，因為其不再有核融合反應的核心物質最初是由電子簡併壓力支撐，而最終將超過簡併壓力所能支撐的最大質量。於是恆星的核心將會崩潰，成為核心坍縮超新星並爆炸，這將留下一個中子星、黑洞，或者可能是更奇特的一種奇特的緻密星。一些主序星，質量或許在 8 至 10_M_☉，雖然規模大到足夠將碳融合成氖和鎂，但仍不足以將氖融合成更重的元素。這樣的恆星在核心不坍縮，並且核融合不會猛烈到將核心擊散成超新星的前提下，可能會留下一顆主要由氧、氖和鎂組成的白矮星。雖然已經確定一些白矮星可能是這種類型的，但存在這種白矮星的證據大部分來自稱為_氧氖鎂新星_或_氖新星_。這些新星的光譜顯示大量的氖、鎂和其它原子量介於中間的其他元素，這些元素似乎只能通過將物質累積到氧氖鎂白矮星上才能解釋。
### Iax 型超新星
Iax 型超新星，涉及氦白矮星的氣體累積，被提議成為這種恆星殘骸轉化的通道。在這種情況下，Ia 型超新星中產生的碳閃太弱，不會摧毀白矮星，但會產生不對稱的爆炸，只會將其質量的一小部分彈射，成為通常稱為殭屍恆星的恆星，有著高速的恆星運動學 #超高速星。在失敗的引爆過程中被轟擊出的物質，被白矮星重新累積，最重的元素，例如鐵，會落在核心，而累積在那兒。這些有_鐵芯_的白矮星比質量相近的碳氧白矮星小，冷卻和結晶的速度也比較快些。
雙星或者多星系統中，由於恆星質量（物質）的交換，恆星的演化過程與單獨的恆星不同，例如天狼星的伴星就是一顆年老的大約一個太陽質量的白矮星，但是天狼星是一顆大約 2.3 個太陽質量的主序星。
## 歷史上的發現 | [
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藍矮星 | 1862 年，阿爾萬・格雷厄姆・克拉克發現天狼星的伴星。根據對恆星數據的分析，這個伴星的質量約一個太陽質量，表面溫度大約 25000K，但是其光度大約是天狼星的萬分之一，所以根據光度和表面積的關係，推斷出其大小與地球相當。這樣的密度是地球上的物質達不到的。1917 年，阿德里安・范馬納恩發現目前已知離太陽最近的白矮星范馬南星。
在二十世紀初由馬克斯・普朗克等人發展出量子理論之後，拉爾夫・福勒於 1926 年建立一個基於費米 - 狄拉克統計的解釋白矮星的密度的理論。
1930 年，蘇布拉馬尼揚・錢德拉塞卡發現白矮星的質量上限（錢德拉塞卡極限），並因此獲得 1983 年的諾貝爾物理學獎。
## 相關條目
## 參考資料
## 外部連結和延伸讀物
### 一般
* White Dwarf Stars, Steven D. Kawaler, in _Stellar remnants_, S. D. Kawaler, I. Novikov, and G. Srinivasan, edited by Georges Meynet and Daniel Schaerer, Berlin: Springer, 1997. Lecture notes for Saas-Fee advanced course number 25. ISBN 3-540-61520-2.
### 物理
* _Black holes, white dwarfs, and neutron stars: the physics of compact objects_, Stuart L. Shapiro and Saul A. Teukolsky, New York: Wiley, 1983. ISBN 0-471-87317-9.
* Physics of white dwarf stars （頁面存檔備份，存於網際網路檔案館）, D. Koester and G. Chanmugam, _Reports on Progress in Physics_ **53**（1990）, pp. 837–915.
* _White dwarf stars and the Chandrasekhar limit_ （頁面存檔備份，存於網際網路檔案館）, Dave Gentile, Master's thesis, 帝博大學，1995. | [
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] |
藍矮星 | * Estimating Stellar Parameters from Energy Equipartition（頁面存檔備份，存於網際網路檔案館）, sciencebits.com. Discusses how to find mass-radius relations and mass limits for white dwarfs using simple energy arguments.
### 變化性
* Asteroseismology of white dwarf stars, D. E. Winget, _Journal of Physics: Condensed Matter_ **10**, #49（1998-12-14）, pp. 11247–11261. DOI 10.1088/0953-8984/10/49/014.
### 磁場
* Magnetism in Isolated and Binary White Dwarfs （頁面存檔備份，存於網際網路檔案館）, D. T. Wickramasinghe and Lilia Ferrario, _Publications of the Astronomical Society of the Pacific_ **112**, #773 (July 2000), pp. 873–924.
### 頻率
* White Dwarfs and Dark Matter （頁面存檔備份，存於網際網路檔案館）, B. K. Gibson and C. Flynn, _Science_ **292**, #5525 (2001-06-22), p. 2211. DOI 10.1126/science.292.5525.2211a.
### 觀測的
* Testing the White Dwarf Mass-Radius Relation with HIPPARCOS （頁面存檔備份，存於網際網路檔案館）, J. L. Provencal, H. L. Shipman, Erik Hog, P. Thejll, _The Astrophysical Journal_ **494**（1998-02-20）, pp. 759–767. | [
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藍矮星 | * Discovery of New Ultracool White Dwarfs in the Sloan Digital Sky Survey （頁面存檔備份，存於網際網路檔案館）, Evalyn Gates, Geza Gyuk, Hugh C. Harris, Mark Subbarao, Scott Anderson, S. J. Kleinman, James Liebert, Howard Brewington, J. Brinkmann, Michael Harvanek, Jurek Krzesinski, Don Q. Lamb, Dan Long, Eric H. Neilsen, Jr., Peter R. Newman, Atsuko Nitta, and Stephanie A. Snedden, _The Astrophysical Journal_ **612**, #2（September 2004）, pp. L129–L132.
* Villanova University White Dwarf Catalogue WD （頁面存檔備份，存於網際網路檔案館）, G. P.McCook and E. M. Sion.
* Dufour, P.; Liebert, James; Fontaine, G.; Behara, N. Rare White dwarf stars with carbon atmospheres. Nature. 2007, **450**: 522–524 [2008-01-02]. （原始內容存檔於 2021-04-18）. | [
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化學鍵 | **化學鍵**（英語：Chemical Bond）是一種粒子間的結合模式，其中粒子可以是原子或離子。透過化學鍵，粒子可組成多原子的化學物質。鍵由兩相反電荷間的電磁力引起，電荷可能來自電子和原子核，或由偶極子造成。化學鍵種類繁多，其能量大小、鍵長亦有所不同。
在原子中，帶負電、繞原子核運行的電子與核內帶正電的質子互相吸引，而位於兩原子核之間的電子則皆受兩方吸引。因此，原子核和電子間最穩定的組態，是當電子位處兩原子核間之時。這些電子使原子核能夠彼此相吸，形成所謂的化學鍵。然而，化學鍵並不能減少個別粒子所構成的體積。由於電子的質量較小且具有物質波性質，它們相較於原子核而言佔據了極大部分的體積，使原子核之間距離較遠。
一般而言，強化學鍵的形成伴隨著原子間電子的共用或轉移。分子、晶體、金屬和雙原子氣體。
## 化學鍵的類型
化學鍵有強與弱之分。一般較強的化學鍵有離子鍵及共價鍵。分子內部共價鍵可以很強，而多原子分子之間共價鍵強度則與各原子的相互角度有關。
化合物的化學鍵類型，也會影響其物理性質，例如：熔點、沸點等。在高分子中它作為分子內部的力出現。
### 離子鍵
正離子、負離子通過靜電作用形成的化學鍵稱作**離子鍵**。兩個原子間的電負性相差極大時，一般是金屬與非金屬，例如：氯與鈉，若他們要結合，電負性大的氯會從電負性小的鈉搶走一個電子，以符合八隅體。之後氯會以 - 1 價的方式存在，而鈉則以 + 1 價的方式存在，兩者再以庫侖靜電力因正負相吸而結合在一起。離子鍵可以延伸，理想的離子化合物中並無分子結構。然而實際上，由於離子間總有極化作用的發生，所以離子之間的電子雲並不可能完全無重疊，因此離子化合物總是帶有一部份共價性。
離子鍵亦有強弱之分。其強弱影響該離子化合物的熔點、沸點和溶解性等性質。離子鍵越強，其熔點越高。離子半徑越小或所帶電荷越多，負、正離子間的作用就越強。例如鈉離子 ${\ce {Na+}}$ 的微粒半徑比鉀離子 ${\ce {K+}}$ 的微粒半徑小，電荷密度大，則氯化鈉 ${\ce {NaCl}}$ 中的離子鍵較氯化鉀 ${\ce {KCl}}$ 中的離子鍵強，而氯化鈉的熔點比氯化鉀的高。
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化學鍵 | 根據化合物中所含化學鍵類型的不同，把含有離子鍵的化合物稱爲**離子化合物**（**ionic compound**），鹼類（如 ${\ce {KOH}}$ ）、大多數鹽類（如 ${\ce {MgCl2}}$ ）（其中 BeCl2，AlCl3 為共價）、大多數金屬氧化物（可理解為活潑金屬氧化物）（如 ${\ce {CaO}}$ ）都是離子化合物。離子化合物中可能存在共價鍵，這與其定義並不矛盾（參看下文對**共價化合物**的定義），如 ${\ce {NH4Cl}}$ 、 ${\ce {NaOH}}$ 便是既具有共價鍵又具有離子鍵的離子化合物。
### 共價鍵
原子間通過共用電子形成的化學鍵，叫做**共價鍵**。它通過兩個電負度相近的原子，例如兩個氧，互相共用其外圍電子以符合八隅體的鍵結方式結合，因此也有人說這是非金屬原子間的結合方式。而共價鍵有鍵角及方向的限制，因此不能隨意延伸，也就是有分子結構。共價鍵廣泛存在於氣體之中，例如氫氣、氯氣、二氧化碳。有些物質如金剛石，則是由碳原子通過共價鍵（巨型共價結構）形成的。共價鍵又可分爲非極性共價鍵與極性共價鍵，以及配位鍵。
#### 共價化合物
**只**含有**共價鍵**的化合物稱爲**共價化合物**（**covalent compound**），如 ${\ce {HCl}}$ （在溶液中會成為 ${\ce {H+}}$ 及 ${\ce {Cl-}}$ ）、 ${\ce {H2O}}$ 、 ${\ce {CO2}}$ 、 ${\ce {CH4}}$ 、 ${\ce {NH3}}$ 等。因此根據其定義，共價化合物中肯定不存在離子鍵。共價鍵鍵能強，但由於分子間作用力較弱，通常具有低熔沸點特性。
#### 極性共價鍵
**極性共價鍵**是在化合物中，不同種的原子形成的共價鍵，由於兩個原子吸引電子的能力不同，共用電子對必然偏向吸引電子能力較強的原子一方，因而吸引電子能力較弱的原子一方相對的呈正電性。叫做極性共價鍵 (polar covalent)，簡稱極性鍵。
#### 非極性共價鍵
**非極性共價鍵**是在單質分子中，同種原子形成共價鍵，兩個原子吸引電子的能力相同，共用電子對不偏向任何一個原子，因此成鍵的原子都不顯電性。叫做非極性共價鍵 (nonpolar covalent)，簡稱非極性鍵。
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化學鍵 | **金屬鍵**則是金屬原子間的鍵結方式，金屬陽離子通過與帶負電的電子間的庫侖靜電力，金屬原子間共用遊走於空價軌域的電子雲，而結合成穩定態，因此金屬有很高的延展性，且有很高的熔點（汞除外），並無分子結構。
## 鍵斷裂
在一般情況下，有鍵裂解分兩種情況：均等分裂和不均等分裂，並因為過程的特性而有所不同。
均等分裂中，兩產物斷開共價鍵後兩者之間的電子平分。
不均等分裂中，斷裂後電子會偏好留在某一產物上。
「鍵解離能」指的是能量的裂解化學鍵所需的能量。
在生物化學打破大分子的過程（通過拆解其內部能量）的分解代謝。
酶，其催化鍵裂解被稱為裂合酶，而它們通過水解或氧化 / 還原，所以亦可稱為水解酶和氧化還原酶。
## 分子軌道理論
科學家認為「電子」是以一種稱為費米子（Fermions）的型態存在於物質空間之中，電子在原子之中時是以不同能階的原子軌域（Atomic Orbital）存在，氫原子光譜的實驗就是電子在不同能量原子軌域間存在的證明，由此可更進一步的推廣多電子原子的原子軌域，以及原子和原子形成分子的分子軌域理論（Molecular Orbital Theory）之理論模型，最後再利用量子化學中數學形式的條件限制，探討與解釋電子存在空間中的情況。
由微觀世界的眼光看來，分子所存在的空間即是其電子密度分佈所佔據的空間，化學家定義連結原子和原子成為分子的作用為「化學鍵」，以量子化學的觀念看來，化學鍵的形成主因是由於原子軌域的相互重疊，使得原有的軌域能階分佈改變，因而形成能量較穩定的分子軌域，也因此最直接影響化學鍵的形成因素就是電子密度的重新分佈。
利用電子密度之分佈可以找出兩個相鄰原子間電子密度極大值，其電子密度極大值的連線便是原子間化學鍵鍵結之路徑，進一步的利用原子間電子密度極大值的等位面加以分別原子所佔有的區域，可以界定出原子位於分子中所佔據的空間大小。
## 參見
## 外部連結
* 簡單化學鍵結構概念 （頁面存檔備份，存於網際網路檔案館），賴亭吟著寫、陳藹然責任編輯，國科會高瞻計畫資源平台。
* http://ispccss.mysinablog.com/index.php?op=ViewArticle&articleId=1945407 原子和化學鍵 (普照科學 Po （頁面存檔備份，存於網際網路檔案館） Chiu Integraded Science) | [
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芳香性 | **芳香性**是一種化學性質，有芳香性的分子中，由不飽和鍵、孤對電子和空軌道組成的共軛系統具有特別的、僅考慮共軛時無法解釋的穩定作用。可以將芳香性看作是環狀離域和環共振的體現。一般認為在這些體系中的電子，可以自由在由原子組成的環形結構上運動（離域），這些環形結構含有單鍵和雙鍵相間，離域的結果是環鍵的鍵級趨於均化，給予體系穩定作用。
芳香性的理論最初由凱庫勒發展出來，是以六元的苯分子為原型建立起來。理論中的苯有兩個共振形態，並有單鍵和雙鍵相間，但理論上的兩種苯（環己三烯）衍生物的這類異構體在實際上無法檢測或分離出來，苯事實上是這兩個異構體的「雜化體」，並且具有不考慮電子離域時無法解釋的穩定性。
## 理論
右上圖中的雙箭頭於化學中代表共振的意思，但不代表該分子會變成那兩個形態，反而是指該分子處於該兩個形態的混合狀態。事實上，那兩個形態都不能代表分子實際的形態，但指出了電子大概的分佈。由此可見，分子真實的形態就是那兩個形態的融合，而它所發出的電磁波波長就是該兩個形態所發出的平均長度。
用組成 π 鍵的電子密度分佈的位置，來更準確地解釋就是電子分佈在分子的環的上、下。
## 歷史
凱庫勒最初在苯之中發現芳香性這特性，其後於 1931 年才有人以量子力學來解釋此現象。當年亦證實芳香性之中的 π 電子的數量必定是 4n+2 的定律，並肯定芳香族分子的環必定是平面的。而 4n+2 的概念亦和伍德沃德 - 霍夫曼規則有所相關。
## 芳香性化合物的特徵
被分類為芳香性的化合物通常有以下的條件：
1. 有一些離域電子組成一些 π 鍵，並且令整個環系統可以當成單與雙鍵的組合；
2. 給出離域電子形成 π 鍵的原子需處於同一個平面；
3. 原子需組成一個環；
4. 組成 π 鍵的電子總數需為 4n+2，即不是 4 的倍數的雙數（休克爾規則）；
5. 可進行親電芳香取代反應和親核芳香取代反應。
苯就是一個較好的例子，它適合以上所有條件，並且有 6 個離域電子（即 n=1）。有 4n+2 個 π 電子的化合物通常都是芳香性的。環丁二烯只有 4 個離域電子，所以不屬於芳香性化合物。這些只有 4n 個 π 電子而又具近似平面結構的環狀化合物稱為反芳香性化合物。
非芳香族的有機物就叫做脂肪族，脂肪族沒有芳香族具有的特殊的共振穩定作用。在受到與平面垂直的外磁場作用時，離域的 π 電子環電流產生的感應磁場可將整個空間劃分為屏蔽區與去屏蔽區，可在核磁共振結果上顯示出來。 | [
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芳香性 | 在商業中最重要的芳香化合物就是苯和甲苯，每年產量極高。從石油中得到的苯和甲苯可用來做其他極有用的日用品材料，包括苯乙烯、苯酚、苯胺及尼龍。
## 芳香性化合物的分類
芳香性化合物大致可分為：簡單的芳香化合物、多環芳香化合物和雜環化合物。
### 雜環化合物
雜環化合物之中組成環的原子不僅包括碳，還包括氮、氧或硫等原子，例如五元的吡咯、呋喃和噻吩、六元的吡啶、多環的吲哚、喹啉等。
### 多環芳香化合物
多環芳香化合物中有一類多環芳香烴，其分子由超過一個不包含雜環或取代基的芳香環融合在一起並同時分享兩個碳原子所組成，當中有一些是致癌物質。例子有萘、蒽、菲。
### 簡單的芳香化合物
芳香烴有大量的有機化合物的結構當中都包含簡單的芳香環，例如 DNA（包含嘌呤、嘧啶等）、三硝基甲苯（含苯環）、乙醯水楊酸（含苯環）和對乙醯氨基酚（含苯環）等。
## 參考文獻
## 參見
* 反芳香性
* 同芳香性
* 芳香烴
* 多環芳香烴
* 芳香環
* 休克爾規則
* 非芳香性 | [
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布萊特-拉比方程 | **塞曼效應**（英語：**Zeeman effect**），在原子物理學和化學中的光譜分析裡是指原子的光譜線在外磁場中出現分裂的現象，是 1896 年由荷蘭物理學家彼得・塞曼譯註發現的，隨後荷蘭物理學家亨德里克・勞侖茲在理論上解釋了譜線分裂成 3 條的原因。這種現象稱為「塞曼效應」。進一步的研究發現，很多原子的光譜在磁場中的分裂情況非常複雜，稱為**反常塞曼效應**（anomalous Zeeman effect）譯註。完整解釋塞曼效應需要用到量子力學，電子的軌道磁矩和自旋磁矩耦合成總磁矩，並且空間取向是量子化的，磁場作用下的附加能量不同，引起能階分裂。在外磁場中，總自旋為零的原子表現出正常塞曼效應，總自旋不為零的原子表現出反常塞曼效應。塞曼效應是繼 1845 年法拉第效應和 1875 年克爾效應之後發現的第三個磁場對光有影響的實例。塞曼效應證實了原子磁矩的空間量子化，為研究原子結構提供了重要途徑，被認為是 19 世紀末 20 世紀初物理學最重要的發現之一。利用塞曼效應可以測量電子的電荷質量比。在天體物理中，塞曼效應可以用來測量天體的磁場。塞曼效應也在核磁共振頻譜學、電子自旋共振頻譜學、磁振造影以及梅斯堡譜學方面有重要的應用。
## 塞曼效應的歷史
1896 年，荷蘭物理學家塞曼使用半徑 10 英尺的凹形羅蘭光柵觀察磁場中的鈉火焰的光譜，他發現鈉的 D 譜線似乎出現了加寬的現象。這種加寬現象實際是譜線發生了分裂。隨後不久，塞曼的老師、荷蘭物理學家勞侖茲應用古典電磁理論對這種現象進行了解釋。他認為，由於電子存在軌道磁矩，並且磁矩方向在空間的取向是量子化的，因此在磁場作用下能階發生分裂，譜線分裂成間隔相等的 3 條譜線。塞曼和勞侖茲因為這一發現共同獲得了 1902 年的諾貝爾物理學獎。
1897 年 12 月，普雷斯頓（T.Preston）報告稱，在很多實驗中觀察到光譜線有時並非分裂成 3 條，間隔也不盡相同，人們把這種現象叫做為**反常塞曼效應**，將塞曼原來發現的現象叫做**正常塞曼效應**。反常塞曼效應的機制在其後二十餘年時間裡一直沒能得到很好的解釋，困擾了一大批物理學家。1925 年，兩名荷蘭學生喬治・烏倫貝克（G.E.Uhlenbeck,1900-1974）和撒姆爾・高斯密特（S.A.Goudsmit,1902-1978）提出了電子自旋假設，很好地解釋了反常塞曼效應。 | [
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布萊特-拉比方程 | 應用正常塞曼效應測量譜線分裂的頻率間隔可以測出電子的電荷質量比。由此計算得到的電荷質量比數值與約瑟夫・湯姆森在陰極射線偏轉實驗中測得的電子電荷質量比數量級是相同的，二者互相印證，進一步證實了電子的存在。
塞曼效應也可以用來測量天體的磁場。1908 年美國天文學家喬治・海爾等人在威爾遜山天文台利用塞曼效應，首次測量到了太陽黑子的磁場。
## 正常塞曼效應的理論解釋
不加外磁場時，原子在兩個能階 E1 和 E2（E1<E2）之間躍遷的能量差為
: $\Delta E=h\nu =E_{2}-E_{1}\,$
原子核的磁矩比電子磁矩小大約三個數量級。如果只考慮電子的磁矩對原子總磁矩的貢獻，那麼磁場引起的附加能量為
: $\Delta U=-\mathbf {\mu } \cdot \mathbf {B} =-\mu _{z}B=m_{J}g_{J}\mu _{B}B$
這裡將磁感應強度 **B** 的方向取為 z 軸方向，μZ 是磁矩在 z 方向上的投影。mJ 是電子總角動量 J 在 z 方向投影的量子數，可以取 - J,-J+1,…J-1,J 共 2J+1 個值，gJ 是電子總角動量的朗德因子，μB 是波耳磁元。
這樣，原子的每一個能階分裂成若干分立的能階，兩個能階之間躍遷的能量差為：
: $\Delta E'=h\nu '=E'_{2}-E'_{1}=E_{2}-E_{1}+(m_{2J}g_{2J}-m_{1J}g_{1J})\mu _{B}B$
對於自旋為零的體系有 $g_{1J}=g_{2J}=1$ 。由於躍遷的選擇定則 $\Delta m_{J}=m_{2J}-m_{1J}=0,\pm 1$ ，頻率 ν 只有三個數值：
: $h\nu '=h\nu +\left\{{\begin{matrix}\mu _{B}B\\0\\-\mu _{B}B\end{matrix}}\right\}$ | [
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布萊特-拉比方程 | 因此一條頻率為 ν 的譜線在外磁場中分裂成三條譜線，相互之間頻率間隔相等，為 ${\frac {\mu _{B}B}{h}}$ 。勞侖茲應用古典電磁理論解釋了正常塞曼效應，計算出了這個頻率間隔。通常把這個能量差的波數間隔 $\Delta ({\frac {1}{\lambda }})={\frac {\mu _{B}B}{hc}}={\frac {e\hbar B}{2m_{e}hc}}={\frac {eB}{4\pi m_{e}c}}\approx 46.7Bm^{-1}T^{-1}$ 稱為**勞侖茲單位**，符號 ${\hat {L}}$ 。
鎘的 643.847nm（1D2 態向 1P1 態的躍遷）譜線在磁場不太強時就是表現出正常塞曼效應。這兩個態的 g 都等於 1，在外磁場中，1D2 分裂成 5 個子能階，1P1 分裂成 3 個子能階，由於選擇定則，這些子能階之間有 9 種可能的躍遷，有 3 種可能的能量差值，所以譜線分裂成 3 條。
## 塞曼效應的偏振特性
對於 Δm=+1，原子在磁場方向的角動量減少了一個 $\hbar $ ，由於原子和光子的角動量之和守恆，光子具有與磁場方向相同的角動量 $\hbar $ ，方向與電矢量旋轉方向構成右手螺旋，稱為 σ+ 偏振，是左旋偏振光。反之，對於 Δm=-1，原子在磁場方向的角動量增加了一個 $\hbar $ ，光子具有與磁場方向相反的角動量 $\hbar $ ，方向與電矢量旋轉方向構成左手螺旋，稱為 σ-偏振，是右旋偏振光。對於 Δm=0，原子在磁場方向的角動量不變，稱為 π 偏振。如果沿磁場方向觀察，只能觀察到 σ+ 和 σ-譜線的左旋偏振光和右旋偏振光，觀察不到 π 偏振的譜線。如果在垂直於磁場方向觀察，能夠觀察到原譜線分裂成 3 條：中間一條是 π 譜線，是線偏振光，偏振方向與磁場方向平行，σ+ 和 σ-線分居兩側，同樣是線偏振光，偏振方向與磁場方向垂直。
## 反常塞曼效應
只有自旋為單態，即總自旋為 0 的譜線才表現出正常塞曼效應。非單態的譜線在磁場中表現出反常塞曼效應，譜線分裂條數不一定是 3 條，間隔也不一定是一個勞侖茲單位。 | [
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布萊特-拉比方程 | 例如鈉原子的 589.6nm 和 589.0nm 的譜線，在外磁場中的分裂就是反常塞曼效應。589.6nm 的譜線是 2P1/2 態向 2S1/2 態躍遷產生的譜線。當外磁場不太強時，在外磁場作用下，2S1/2 態能階分裂成兩個子能階，2P1/2 態也分裂成兩個子能階，但由於兩個態的朗德因子不同，譜線分裂成 4 條，中間兩條是 π 線，外側兩條分別是 σ+ 線和 σ-線。589.0nm 的譜線是 2P3/2 態向 2S1/2 態躍遷產生的，2P3/2 態能階在外磁場不太強時分裂成四個子能階，因此 589.0nm 的譜線分裂成 6 條。中間兩條 π 線，外側兩邊各兩條 σ 線。
## 逆塞曼效應
實驗中不僅可以觀察到光譜發射線的塞曼效應，吸收線也會發生塞曼效應，這被稱為逆塞曼效應。
## 塞曼效應的破壞
只有當外磁場的強度比較弱，不足以破壞自旋 - 軌道耦合時才會出現反常塞曼效應，這時自旋角動量和軌道角動量分別圍繞總角動量作快速進動，總角動量繞外磁場作慢速進動。當磁場很強時，自旋角動量和軌道角動量不再合成總角動量，而是分別圍繞外磁場進動。這時反常塞曼效應被帕邢 - 巴克效應所取代，其效果是恢復到正常塞曼效應，即譜線分裂成 3 條，相互之間間隔一個勞侖茲單位。這裡磁場的「強」與「弱」是相對的，例如 3T 的磁場對於鈉 589.6nm 和 589.0nm 的雙重線是弱磁場，不會引起帕邢 - 巴克效應，但對於鋰的 670.785nm 和 670.800nm 的雙重線是強磁場，足夠觀察到帕邢 - 巴克效應。
## 參閱
* 明顯對稱性破缺
## 參考文獻
## 延伸閱讀
## 譯註
* **塞曼**在一些文獻中也譯作**季曼**、**齊曼**。
* **反常**塞曼效應也譯作**異常**塞曼效應，**複雜**塞曼效應。 | [
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蛋白质-蛋白质对接 | **分子對接**（macromolecular docking ）是分子模擬的重要方法之一，其本質是兩個或多個分子之間的識別過程，其過程涉及分子之間的空間匹配和能量匹配。分子對接方法在藥物設計、材料設計等領域有廣泛的應用。
## 歷史
分子對接這一想法的歷史可以追溯到 19 世紀提出的受體學說，Fisher 提出的受體學說認為，藥物與體內的蛋白質大分子即受體會發生類似鑰匙與鎖的識別關係，這種識別關係主要依賴兩者的空間匹配。隨著受體學說的發展，人們對生理活性分子與生物分子的相互作用有了更加深刻的認識，從基於空間匹配的剛性模型逐漸發展成為基於空間匹配和能量匹配的柔性模型。模型的優化使通過計算模擬分子間相互作用的設想更容易實現。另一方面，計算機和計算科學的迅速發展又使得人們能夠處理大量數據，這兩方面的因素共同促成了分子對接方法的出現。
早期的分子對接方法用分子力學方法或者量子化學方法計算小分子之間分子識別，在一些分子模擬軟體包中也含有分子對接的模塊。但是由於算法和計算機處理能力的限制，早期的對接方法較難處理含有大分子的分子對接過程。
1995 年由 Accelrys 公司開發的計算化學軟體 Affinity 上市，這是第一個可以進行有大分子參與的商業化分子對接軟體，此後，商業化和免費的分子對接軟體層出不窮。現在應用中的分子對接軟體涵蓋了剛性對接、半柔性對接、柔性對接等各種對接方法，在能量優化方面則使用了人工神經網絡、遺傳算法、模擬退火、禁忌搜索、局部搜索等各種方法，目前的分子對接方法是研究小分子與大分子相互作用模式、生物大分子間識別、分子自組裝、超分子結構等課題的常用方法之一。
## 原理與方法
分子對接方法的兩大課題是分子之間的空間識別和能量識別。空間匹配是分子間發生相互作用的基礎，能量匹配是分子間保持穩定結合的基礎。對於幾何匹配的計算，通常採用格點計算、片斷生長等方法，能量計算則使用模擬退火、遺傳算法等方法。
各種分子對接方法對體系均有一定的簡化，根據簡化的程度和方式，可以將分子對接方法分為三類。
**剛性對接**：剛性對接方法在計算過程中，參與對接的分子構像不發生變化，僅改變分子的空間位置與姿態，剛性對接方法的簡化程度最高，計算量相對較小，適合於處理大分子之間的對接。 | [
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蛋白质-蛋白质对接 | **半柔性對接**：半柔性對接方法允許對接過程中小分子構像發生一定程度的變化，但通常會固定大分子的構像，另外小分子構像的調整也可能受到一定程度的限制，如固定某些非關鍵部位的鍵長、鍵角等，半柔性對接方法兼顧計算量與模型的預測能力，是應用比較廣泛的對接方法之一。
**柔性對接**：柔性對接方法在對接過程中允許研究體系的構像發生自由變化，由於變量隨著體系的原子數呈幾何級數增長，因此柔性對接方法的計算量非常大，消耗計算機時很多，適合精確考察分子間識別情況。
## 主要分子對接軟體
### DOCK
Dock 是目前應用最廣泛的分子對接軟體之一，由 Kuntz 課題組開發。Dock 應用半柔性對接方法，固定小分子的鍵長和鍵角，將小分子配體拆分成若干剛性片斷，根據受體表面的幾何性質，將小分子的剛性片斷重新組合，進行構像搜索。在能量計算方面，Dock 考慮了靜電相互作用、范德華力等非鍵相互作用，在進行構像搜索的過程中搜索體系勢能面。最終軟體以能量評分和原子接觸罰分之和作為對接結果的評價依據。
### AutoDock
Autodock 是另外一個應用廣泛的分子對接程序，由 Olson 科研組開發。AutoDock 應用半柔性對接方法，允許小分子的構像發生變化，以結合自由能作為評價對接結果的依據。自從 AutoDock3.0 版本以後，對能量的優化採用拉馬克遺傳算法 (LGA)，LGA 將遺傳算法與局部搜索方法相結合，以遺傳算法迅速搜索勢能面，用局部搜索方法對勢能面進行精細的優化。
### FlexX
FlexX 是德國國家信息技術研究中心生物信息學算法和科學計算研究室開發的分子對接軟體，目前已經作為分子設計軟體包 BioSolveIT LeadIT 的一個模塊實現商業化。FlexX 使用碎片生長的方法尋找最佳構像，根據對接自由能的數值選擇最佳構像。FlexX 程序對接速度快效率高，可以用於小分子資料庫的虛擬篩選。
## 參考文獻
## 參見
* 分子力學
* 分子模擬
* 計算化學 | [
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濱松醫科大學 | **濱松醫科大學**（はままついかだいがく、Hamamatsu University School of Medicine）位於靜岡縣濱松市半田山的日本國立大學。
## 沿革
* 1973 年 9 月 1 日 靜岡大學國立醫科大學創設準備室設立
* 1974 年 6 月 7 日 濱松醫科大學成立。醫學部醫學科設立
* 1995 年 4 月 1 日 醫學部護理學科設立
* 1999 年 在第 93 次醫生國家考試中取得全國最高合格率（98.1％）
* 2003 年 參加 21 世紀 COE 項目
* 2004 年 4 月 1 日 成為國立大學法人濱松醫科大學
## 大學組織
* 學部
* 醫學部
* 醫學科
* 基礎講座（解剖學第一、解剖學第二、生理學第一、生理學第二、生化學第一、生化學第二、病理學第一、病理學第二、藥理學、微生物學、衛生學、公眾衛生學、寄生蟲學、法醫學）
* 臨床講座（內科學第一、內科學第二、內科學第三、精神神經醫學、外科學第一、外科學第二、腦神經外科學、整形外科學、皮膚科學、放射線醫學、麻醉・復甦學、婦產科學、小兒科學、齒科口腔外科學、臨床檢査醫學、臨床藥理學、急救醫學）
* 護理學科
* 基礎護理學講座（基礎護理學、人類科學、環境科學、疾病科學）
* 臨床護理學講座（小兒護理學、成人護理學、老人護理學、母性護理學、精神護理學）
* 地域護理學講座
* 大學院
* 醫學研究科博士課程
* 形態系專業、生理系專業、生化系專業、生態系專業
## 附屬設施
* 附屬圖書館
* 保健管理中心
* 光量子醫學研究中心
* 醫療廢棄物處理中心
* 信息處理中心
* 醫學部附屬設施
* 動物實驗設施
* 實驗實習機器中心
* 附屬醫院
## 交通
在 JR 濱松站北出口汽車站 13 號上車口乘坐「磐田山手線醫大方向」巴士，在「醫科大學」站下車（行程約 35 分鐘）
## 外部連結
* 濱松醫科大學 （頁面存檔備份，存於網際網路檔案館） | [
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周培源 | **周培源**（1902 年 8 月 28 日 —1993 年 11 月 24 日），男，江蘇宜興人，中國理論物理學家、力學家，中國科學院院士，在流體力學領域的研究中有重要貢獻，最重要的工作是導出了雷諾應力的輸運方程。曾任國家科協主席、北京大學校長。
## 生平
1924 年畢業於清華學校。1926 年春獲美國芝加哥大學學士學位，同年底獲碩士學位。1928 年獲美國加州理工學院博士學位。
1936 年在美國普林斯頓高等研究院跟從愛因斯坦研究廣義相對論。其後，在德國萊比錫大學和瑞士蘇黎世聯邦理工學院繼續進行量子力學的博士後研究。
歷任清華大學物理系教授、清華大學教務長，北京大學數學力學系教授、北京大學副校長，北京大學校長。
1980 年 3 月 15 日，中國科學技術協會第二次全國代表大會在北京召開，會議選出第二屆全國委員會，其被選為主席。1986 年 6 月，中國科學技術協會第三次全國代表大會在北京召開，選舉產生第三屆全國委員會。6 月 28 日，第三屆全國委員會第一次會議召開，推舉其擔任名譽主席。
## 家庭
妻子王蒂澂（1910 年出生），北平女子師範大學畢業。1932 年，周培源王蒂澂在北京結婚，由時任清華校長梅貽琦主婚。
周王共育有四個女兒：
* 周如枚，1980 年去世。
* 第一任丈夫梁從誡是梁思成和林徽因的兒子。兩人在文革期間離婚。兩人的兒子梁鑒又名周志兵。梁鑒的女兒梁周洋，又名周洋，美籍華人模特。
* 第二任丈夫麻醉學家謝榮，1968 年生女謝蘭。謝蘭現名 Cecilia Xie Lan Birge，1989 年六四事件之後移民美國，曾任美國新澤西州蒙哥馬利鎮區議員、區長。
* 周如雁
* 周如玲
* 周如苹 **周培源基金會**（2020.7.7 民政部將其列入**社會組織嚴重違法失信名單**）理事長、法人代表。周如苹已於幾年前去世。
## 紀念
1993 年，周培源基金會（http://www.cpyf.org.cn）正式成立。
2022-07-07 民政部對**周培源基金會**作出警告的行政處罰（周培源基金會 2020 年度公益事業支出比例為 2.57%）民政部將其列入**社會組織嚴重違法失信名單**（https://chinanpo.mca.gov.cn/xwxq?newsType=2351&id=19973）。
2003 年，周培源銅像在北京大學落成。
以周培源命名的獎項：
* 周培源力學獎 | [
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] |
周培源 | * 周培源物理獎
以周培源命名的研究單位：
* 清華大學周培源應用數學研究中心
## 參考文獻
## 參見
* 物理學家列表
## 外部連結
* 周培源先生簡介
* （簡體中文）周培源基金會網站（頁面存檔備份，存於網際網路檔案館） | [
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短剑号 | **劍標**（†），HTML 字符編碼為「†」，U+2020，是在星標之後的第二種腳註標記。原來用於天主教典禮中，指示何時應該劃十字聖號。
**雙劍標**（‡），HTML 字符編碼為「‡」，U+2021，則是在劍標之後的第三種腳註標記，用來刪除文本中的無關內容。
由於劍標與雙劍標也是基督教十字架的另一種造型，在德語環境中這兩個符號標在人名前後是象徵那人已經過世。所以在當作腳註標記使用時，不可把它們標在一個在世之人的名字前後。歷史上主要用於標註出譯文中重複贅餘的成分或錯漏之處。
在量子力學中，劍標常用以表示共軛轉置。在生物學中，該符號常用來表示已滅絕的分類單元。
## 編碼
* U+2020 † DAGGER ，HTML：`†` `†`，Windows Alt 碼:`Alt`+`0``1``3``4`
* U+2021 ‡ DOUBLE DAGGER ，HTML：`‡` `‡`，Windows：`Alt+0135`，macOS：`option-shift-7`
* U+2E36 ⸶ DAGGER WITH LEFT GUARD - used in Alexander John Ellis's "palaeotype" transliteration to indicate retracted pronunciation
* U+2E37 ⸷ DAGGER WITH RIGHT GUARD - used in Alexander John Ellis's "palaeotype" transliteration to indicate advanced pronunciation
* U+2E38 ⸸ TURNED DAGGER - used in Alexander John Ellis's "palaeotype" transliteration to indicate retroflex pronunciation
* U+2E4B ⹋ TRIPLE DAGGER - A variant with three handles.
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量子 | 在物理學中，**量子**（quantum）是參與基本相互作用的任何物理實體（物理性質）的最小量。量子的概念，皆涉及一個不可分割的基本個體。例如，「光的量子」是光的單位。而延伸出的量子力學、量子光學等成為不同的專業研究領域。
某物理性質可以被「量化」的基本概念，稱為**量化假設**；這意味著物理性質的「量值」（magnitude），只能採用由一個量子的整數倍組成的離散值。
量子的基本概念為所有的有形性質是「可量子化的」。「量子化」指其物理量的數值是特定的，而不是任意值。例如，在（休息狀態的）原子中，電子的能量是可量子化的。這決定原子的穩定和一般問題。在 20 世紀的前半期，出現了新的概念：許多物理學家將量子力學視為瞭解和描述自然的的基本理論。在量子被發現的 100 多年間，經過普朗克，愛因斯坦，波爾等科學家的不懈努力，已初步建立量子力學理論。
英語量子一詞來自拉丁語 _quantum_，意為「有多少」，代表「相當數量的某物質」。
## 歷史
量子物理是研究量子化的物理分支，在 1900 年根據熱輻射理論延伸建立量子理論。由於馬克斯・普朗克（M. Planck）試圖解決黑體輻射問題，所以他大膽提出量子假設，並得出了普朗克輻射定律，沿用至今。
當時德國物理界聚焦於黑體輻射問題的研究。馬克斯・普朗克在 1900 年 12 月 14 日的德國物理學學會會議中第一次發表能量量子化數值、Avogadro-Loschmidt 數的數值、一個分子摩爾（mole）的數值及基本電荷。其數值比以前的更準確，提出的理論也成功解決了黑體輻射的問題，標誌著量子力學的誕生。
量子假設的提出有力地衝擊了古典物理學，促進物理學進入微觀層面，奠基現代物理學。但直到現在，物理學家關於量子力學的一些假設仍然不能被充分地證明，仍有很多需要研究的地方。
## 相關方程
### 黑體輻射量子方程
黑體輻射量子方程是量子力學的第一部分。在 1900 年 10 月 7 日面世。
當物體被加熱，它以電磁波的形式散發紅外線輻射。物體變得熾熱時，紅色波長部分開始變得可見。大多數熱輻射是紅外線，除非物體變得像太陽的表面一樣熱，但當時的實驗室內不能夠達成這種條件而且只可以量度部分黑體光譜。
能量 $E$ 、輻射頻率 $f$ 及溫度 $T$ 可以被寫成：
$$
E={\frac {hf}{e^{\frac {hf}{kT}}-1}}
$$ | [
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] |
量子 | _h_ 是普朗克常數及 _k_ 是玻爾茲曼常數。兩者都是物理學中的基礎。基礎能量的量子是 _hf_。可是這個單位正常之下不存在並不需要量子化。
## 參考書籍
* M. Planck，_A Survey of Physical Theory_，transl. by R. Jones and D.H. Williams，Methuen & Co.，Ltd.，London 1925(Dover editions 1960 and 1993)including the Nobel lecture.
* J. Mehra and H. Rechenberg，_The Historical Development of Quantum Theory_，Vol.1，Part 1，Springer-Verlag New York Inc.，New York 1982.
* Lucretius，"On the Nature of the Universe"，transl. from the Latin by R.E. Latham，Penguin Books Ltd.，Harmondsworth 1951. There are，of course，many translations，and the translation's title varies. Some put emphasis on how things work，others on what things are found in nature.
## 參看
* 量子力學
* 量子狀態
* 量子數
* 量子場論
* 量子電腦
* 量子密碼學
* 量子演算
* 磁束量子
* 量子化
* 次原子粒子
* 基本粒子 | [
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原子能階 | **能階**（英語：energy level）又稱**能級**，是描述微觀粒子體系（原子、電子、分子等）可能存在的相對穩定狀態下，所對應一系列不連續的、分立的且確定的「內在」能量值或狀態。
**能階理論**則是一種解釋原子核外電子運動軌道的理論。它認為電子只能在特定的、分立的軌道上運動，各個軌道上的電子具有分立的能量，這些能量值即為能階。由於距原子核越遠的電子越不受束縛，因此其能階越高，該電子具有越多的電子能量。
電子可以在不同的軌道間發生躍遷，電子吸收能量可以從低能階躍遷到高能階或者從高能階躍遷到低能階從而輻射出光子。氫原子的能階可以由它的光譜顯示出來。
## 背景
19 世紀末 20 世紀初，人類開始走進微觀世界，物理學家提出許多關於原子機構的模型，這裡就包括拉塞福的核式模型。核式模型能很好地解釋實驗現象，因而得到許多人的支持；但是該模型與古典的電磁理論有著深刻的矛盾。
## 古典理論的局限
實驗事實表明：原子具有高度的穩定性，即使受到外界干擾，也很不易改變原子的屬性；且氫原子所發出的光譜為線狀光譜，與古典電磁理論得出的結論完全不同。
然而，按古典電磁理論，電子繞核轉動具有加速度，加速運動著的電荷（電子）要向周圍空間輻射電磁波，電磁波頻率等於電子繞核旋轉的頻率，隨著不斷地向外輻射能量，原子系統的能量逐漸減少，電子運動的軌道半徑也越來越小，繞核旋轉的頻率連續增大，電子輻射的電磁波頻率也在連續地變化，因而所呈現的光譜應為連續光譜。
由於電子繞核運動時不斷向外輻無線電磁波，電子能量不斷減少，電子將逐漸接近原子核，最後落於核上，這樣，原子應是一個不穩定系統。
## 新理論的提出
丹麥物理學家尼爾斯・波耳於 1913 年提出了自己的原子結構假說，認為圍繞原子核運動的電子軌道半徑只能取某些分立的數值，這種現象叫軌道的量子化，不同的軌道對應著不同的狀態，在這些狀態中，儘管電子在做高速運動，但不向外輻射能量，因而這些狀態是穩定的。原子在不同的狀態下有著不同的能量，所以原子的能量也是量子化的。 | [
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因数分解 | 在數學中，**整數分解**（英語：integer factorization）又稱**質因數分解**（prime factorization），是將一個正整數寫成幾個因數的乘積。例如，給出 45 這個數，它可以分解成 $$ $3^{2}\times 5$ 。根據算術基本定理，這樣的分解結果應該是獨一無二的。這個問題在代數學、密碼學、計算複雜性理論和量子計算機等領域中有重要意義。
## 因子分解
完整的因子列表可以根據因數分解推導出，將冪從零不斷增加直到等於這個數，算出可以整除這個數的所有整數。
例如，因為 $45=\,$ $3^{2}\times 5$ ，由此可知，
45 可以被以下數字因子分解:
* 30 ×50 =1
* 30×51 =5
* 31×50 =3
* 31×51 =15
* 32×50 =9
* 32×51 =45
相對應的，因數分解只包括因數因子。參見因數分解算法。
## 實際應用
給出兩個整數，很容易就能將它們兩個相乘。
但是，給出一個大整數 (100 位數以上的整數)，找出它們的因數就顯得不是那麼容易了。
這就是許多現代密碼系統的關鍵所在。如果能夠找到解決整數分解問題的快速方法，幾個重要的密碼系統將會被攻破，
包括 RSA 加密演算法公鑰算法和 Blum Blum Shub 隨機數發生器。
儘管快速分解是攻破這些系統的方法之一，仍然會有其它的不涉及到分解的其它方法。
所以情形完全可能變成這樣：整數分解問題仍然是非常困難，這些密碼系統卻是能夠很快攻破。
有的密碼系統則能提供更強的保證：如果這些密碼系統被快速破解（即能夠以**多項式時間複雜度破解**），
則可以利用破解這些系統的算法來快速地以**多項式時間複雜度**分解整數。
換句話說，破解這樣的密碼系統不會比整數分解更容易。
這樣的密碼系統包括 Rabin cryptosystem(RSA 的一個變體）以及 Blum Blum Shub 隨機數發生器。
## 當今的新進展
2005 年，作為公共研究一部分的，有 663 個二進制數位之長的 RSA200 已經被一種一般用途的方法所分解。
如果一個大的，有 _n_ 個二進制數位長度的數，是兩個差不多大小相等的因數的乘積，
現在還沒有很好的演算法來以多項式時間複雜度分解它。
這就意味著沒有已知算法可以在 O（_nk_）（_k_ 為常數）的時間內分解它。 | [
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因数分解 | 但是現在的算法也是比 O (en) 快的。
換句話說，現在我們**已知最好的算法比指數數量級時間要快，比多項式數量級時間要慢**。
已知最好的漸近線運行時間是普通數體篩選法（GNFS）。
時間是：
: $\mathrm {O} \left(\exp \left(\left({\frac {64}{9}}n\right)^{\frac {1}{3}}(\log n)^{\frac {2}{3}}\right)\right)$
對於平常的計算機，GNFS 是我們已知最好的對付 _n_ 個二進制數位大因數的方法。
不過，對於量子計算機， 彼得・秀爾在 1994 年發現了一種可以用多項式時間來解決這個問題的算法。
如果大的量子計算機建立起來，這將對密碼學有很重要的意義。
這個算法在時間上只需要 O (_n_3)，空間只要 O (_n_) 就可以了。
構造出這樣一個算法只需要 2_n_ 個量子位。
2001 年，第一個 7 量子位的量子計算機第一個運行這個算法，它分解的數是 15。
### 難度與複雜度
現在還不確切知道整數分解屬於哪個複雜度類。
我們知道這個問題的判定問題形式（「請問 _N_ 是否有一個比 _M_ 小的因數？」）是在 NP 與反 NP 之中。
因為不管是答案為是或不是，我們都可以用一個質因數以及該質因數的質數證明來驗證這個答案。
由秀爾演算法可知，這個問題在 BQP 中。
大部份的人則懷疑這個問題不在 P、NP 完全、以及反 NP 完全這三個複雜性類別中。
如果這個問題可以被證明為 NP 完全或反 NP 完全，則我們便可推得 NP = 反 NP。
這將會是個很震撼的結果，也因此大多數人猜想整數分解這個問題不在上述的複雜性類別中。
也有許多人嘗試去找出多項式時間的演算法來解決這個問題，但是都尚未成功，因此這個問題也被多數人懷疑不在 P 中。
有趣的是，判定一個整數是否是質數則比分解該整數簡單許多。
AKS 算法証明前者可以在多項式時間中解決。
測試一個數是否為質數是 RSA 演算法中非常重要的一環，因為它在一開始的時候需要找很大的質數。
## 整數分解算法
### 特殊用途算法
一個特別的因數分解算法的運行時間依賴它本身的未知因子：大小，類型等等。在不同的算法之間運行時間也是不同的。
* 試除法
* 輪式因數分解法
* 波拉德 RHO 算法 | [
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0.030476979911327362,
-0.10149238258600235,
0.0654795914888382,
0.10294006764888763,
0.3045123517513275,
-0.32341820001602173,
-0.5081366300582886,
-0.0658753514289856,
-0.11266987025737762,
-0.08433480560779572,
-0.12149835377931595,
0.2434309720993042,
-0.22129511833190918,
-0.391809344291687,
-0.34510326385498047,
0.27644118666648865,
-0.269740492105484,
-0.1061532199382782,
-0.07481953501701355,
-0.05856335908174515,
-0.047058332711458206,
0.5159715414047241,
-0.07270713895559311,
-0.2394399493932724,
0.2447170615196228,
0.16290132701396942,
-0.09911300241947174,
0.1010061576962471,
-0.265273779630661,
0.06203430891036987,
0.5468114614486694,
-0.3069773316383362,
-0.1492525190114975,
0.1350448876619339,
0.07396293431520462,
0.02249263785779476,
0.33984649181365967,
-0.25821495056152344,
0.4441993534564972,
-0.24344737827777863,
0.45528656244277954,
-0.05311269313097,
-0.11250495165586472
] |
因数分解 | * 代數群因式分解算法，其中包括 Pollard's _p_− 1 算法、Williams' _p_+ 1 算法和 Lenstra 橢圓曲線分解法
* 費馬質數判定法
* 歐拉因式分解法
* 特殊數體篩選法
### 一般用途算法
一般用途算法的運行時間僅僅依賴要分解的整數的長度。這種算法可以用來分解 RSA 數。大部分一般用途算法基於平方同餘方法。
* Dixon 算法
* 連分數分解法（CFRAC）
* 二次篩選法
* 有理篩選法
* 普通數體篩選法
* Shanks' square forms factorization（SQUFOF）
### 其他算法
* 秀爾算法
## 參見
* 質因數表 | [
-0.24531124532222748,
0.1976342648267746,
0.0032675722613930702,
-0.039009351283311844,
-0.05686512216925621,
0.4516872465610504,
0.5261110067367554,
-0.0072819823399186134,
-0.035665951669216156,
-0.14840583503246307,
-0.17290061712265015,
0.2675771117210388,
0.036621443927288055,
0.026821983978152275,
0.07372763007879257,
-0.13629457354545593,
-0.10766589641571045,
-0.10539492964744568,
-0.876043975353241,
-0.23087620735168457,
0.07617012411355972,
0.004877543542534113,
-0.2955154776573181,
0.02001330256462097,
-0.40823113918304443,
-0.13388805091381073,
-0.05424230545759201,
0.2611965537071228,
-0.0595928430557251,
-0.21887728571891785,
0.034258052706718445,
-0.32779964804649353,
-0.044462110847234726,
-0.29747751355171204,
-0.071495421230793,
-0.07976298779249191,
0.004462534561753273,
0.15932410955429077,
-0.030590210109949112,
0.18411210179328918,
0.31527093052864075,
-0.057772956788539886,
-0.17222106456756592,
-0.29922184348106384,
0.1762276291847229,
0.3236073851585388,
0.02852702885866165,
0.19944049417972565,
0.10257145762443542,
-0.0308589655905962,
0.20837347209453583,
0.1261776089668274,
0.014026930555701256,
0.03153945133090019,
-0.006577253341674805,
0.024203229695558548,
-0.20983098447322845,
-0.6612885594367981,
0.24298283457756042,
-0.2544800341129303,
-0.07525532692670822,
-0.1016288697719574,
-0.18479321897029877,
-0.5358970165252686,
0.21627232432365417,
-0.1488686352968216,
0.2622755169868469,
0.09085599333047867,
-0.30601102113723755,
-0.045117899775505066,
0.40919411182403564,
0.06908093392848969,
0.08352154493331909,
-0.05779963731765747,
-0.259234756231308,
-0.4761793613433838,
-0.14857183396816254,
-0.6772712469100952,
-0.4247991442680359,
0.16427500545978546,
0.08877386897802353,
-0.030802730470895767,
-0.0691584125161171,
-0.250270277261734,
0.06819893419742584,
0.10726330429315567,
0.029665421694517136,
0.28822094202041626,
-0.30115213990211487,
0.039764635264873505,
-0.059461504220962524,
-0.00018101907335221767,
0.5004414916038513,
-0.21217221021652222,
-0.09025125205516815,
0.05566781759262085,
0.23246629536151886,
0.08400756865739822,
-0.0831589475274086,
0.2319122850894928,
-0.26010817289352417,
0.16446159780025482,
0.14464807510375977,
0.2580850124359131,
-0.08485159277915955,
0.3463171422481537,
0.03714427724480629,
0.1185218021273613,
0.16071882843971252,
0.20460034906864166,
0.1746920496225357,
0.04477008432149887,
-0.1356552690267563,
-0.11253537237644196,
0.20839251577854156,
-0.1809377819299698,
-0.05603804066777229,
0.05432223901152611,
-0.21534517407417297,
0.21291734278202057,
0.017437437549233437,
0.21034009754657745,
0.33087101578712463,
-0.07720550894737244,
0.14722077548503876,
0.15845322608947754,
0.20174731314182281,
-0.07710802555084229,
0.3940087854862213,
-0.03357553109526634,
0.10382665693759918,
-0.37764978408813477,
0.024826018139719963,
0.1740085780620575,
-0.33122020959854126,
-0.26184841990470886,
0.14081238210201263,
0.40018558502197266,
0.027113934978842735,
0.4450293481349945,
0.05356451869010925,
0.2503565549850464,
0.09699896723031998,
-0.06008515506982803,
-0.3362637162208557,
-0.15275585651397705,
0.1635632961988449,
-0.34628480672836304,
0.1635400652885437,
-0.006970379501581192,
-0.6789370775222778,
-0.18003512918949127,
-0.28171539306640625,
-0.05015439912676811,
-0.08930029720067978,
-0.12190009653568268,
-0.0008104973239824176,
-0.11795485019683838,
0.43700745701789856,
0.07189486175775528,
0.14324522018432617,
0.22547093033790588,
0.12383536249399185,
-0.21181485056877136,
0.04919955134391785,
0.20361769199371338,
0.2429272085428238,
0.0044756000861525536,
0.28670984506607056,
-0.00720785278826952,
0.2011253833770752,
-0.23565858602523804,
-0.20157107710838318,
-0.11634790152311325,
-0.049335580319166183,
0.04770255833864212,
-0.13826024532318115,
0.07019689679145813,
0.49797824025154114,
0.3885725736618042,
0.05131406709551811,
-0.017348451539874077,
0.015395302325487137,
0.26771342754364014,
0.45890867710113525,
-0.4049347937107086,
-0.026311015710234642,
0.4806148409843445,
0.07014357298612595,
0.11414873600006104,
-0.14109739661216736,
-0.07192754745483398,
0.24474592506885529,
-0.3373822867870331,
-0.3800765872001648,
-0.13780713081359863,
-0.2585640251636505,
-0.2188822329044342,
0.21684576570987701,
0.16879789531230927,
-0.22921669483184814,
-0.7721529603004456,
0.40808799862861633,
-0.09190234541893005,
0.19435881078243256,
0.021914813667535782,
0.059230294078588486,
0.27841004729270935,
0.22925207018852234,
0.0785125121474266,
-0.1411498636007309,
-0.2763957977294922,
0.15247546136379242,
0.08988892287015915,
-0.015089143998920918,
0.41797536611557007,
0.042259715497493744,
0.4417618215084076,
0.1716800481081009,
0.1456845998764038,
0.27080562710762024,
-0.4583373963832855,
0.005830288864672184,
0.09252462536096573,
-0.05861499905586243,
0.02782955765724182,
-0.16690003871917725,
-0.20500557124614716,
-0.18829938769340515,
0.1007438525557518,
-0.14910395443439484,
0.40275701880455017,
0.15357014536857605,
0.022434763610363007,
-0.06669824570417404,
0.6212202906608582,
0.1594392955303192,
0.22449979186058044,
0.44568106532096863,
-0.09553807228803635,
-0.17806562781333923,
0.5965871214866638,
0.26524484157562256,
-0.3114829957485199,
0.0399257130920887,
0.4448837637901306,
0.29147911071777344,
-0.15874803066253662,
0.27201560139656067,
-0.23508599400520325,
-0.10908787697553635,
0.6684730648994446,
0.3710094094276428,
0.5356057286262512,
0.10270188003778458,
0.37886297702789307,
0.08941982686519623,
0.01882793940603733,
0.3912612497806549,
0.6458674669265747,
-0.19098350405693054,
-0.45017239451408386,
-0.38526976108551025,
0.08400768041610718,
-0.014451326802372932,
0.40034082531929016,
-0.0983843058347702,
0.6357110142707825,
-0.07582362741231918,
-0.009847309440374374,
0.09009639918804169,
0.16689079999923706,
0.584576427936554,
0.0801611840724945,
0.10482171177864075,
-0.3434692323207855,
0.26371702551841736,
-0.056181397289037704,
0.021726571023464203,
-0.05510799214243889,
-0.03447108715772629,
0.027957353740930557,
0.020775089040398598,
-0.12862537801265717,
-0.8025004863739014,
-0.13099989295005798,
-0.5666882991790771,
0.2730730175971985,
0.620488703250885,
-0.09913758933544159,
0.09386671334505081,
0.27990394830703735,
0.009444938972592354,
0.14271043241024017,
0.15599524974822998,
0.13706102967262268,
0.18144828081130981,
0.1575276404619217,
-0.20279256999492645,
0.42087846994400024,
-0.47225385904312134,
-0.21960528194904327,
0.0014882709365338087,
0.23416906595230103,
-0.3477051258087158,
-0.1269538700580597,
0.06878623366355896,
0.4046201705932617,
-0.23461377620697021,
0.011326019652187824,
0.12092973291873932,
-0.1566179394721985,
0.15622487664222717,
0.18210367858409882,
-0.3038271367549896,
0.23447555303573608,
-0.3940889835357666,
-0.23974423110485077,
-0.12587763369083405,
-0.20760288834571838,
0.011614937335252762,
0.06286381930112839,
0.2550124526023865,
0.07401568442583084,
0.0946757048368454,
-0.2542143762111664,
-0.23187443614006042,
-0.5312632322311401,
0.2960195541381836,
-0.04872540012001991,
0.2882729768753052,
-0.16909103095531464,
0.4832160472869873,
0.257231205701828,
-0.020027082413434982,
-0.08229490369558334,
0.36099109053611755,
0.36681511998176575,
-0.7709561586380005,
-0.21667355298995972,
0.4031525254249573,
0.09461910277605057,
-0.48055949807167053,
-0.3951604664325714,
0.05627266690135002,
-0.19713321328163147,
0.46089085936546326,
0.07684970647096634,
-0.4327441453933716,
-0.547864556312561,
-0.01974458061158657,
0.229122593998909,
0.07032030075788498,
-0.4190448820590973,
-0.22026970982551575,
-0.07815384864807129,
0.09253349155187607,
0.01460964884608984,
0.06215180084109306,
-0.07792746275663376,
-0.4574800133705139,
-0.05933384597301483,
0.6043879985809326,
0.2231547236442566,
-0.05381496995687485,
-0.39731860160827637,
-0.2644253969192505,
0.1642015427350998,
-0.12457526475191116,
-0.1974402219057083,
0.02957119047641754,
-0.10262718796730042,
-0.51838219165802,
0.20480570197105408,
-0.37980929017066956,
0.15628251433372498,
-0.2624925374984741,
-0.04919158294796944,
-0.2896481454372406,
-0.14099471271038055,
0.3954029679298401,
0.3445951044559479,
-0.416204571723938,
0.33963143825531006,
-0.17882788181304932,
-0.3377525806427002,
0.23284371197223663,
0.2336486279964447,
-0.03785209730267525,
-0.05544621869921684,
0.23593094944953918,
0.16870823502540588,
-0.3347373306751251,
0.060419030487537384,
-0.1838475912809372,
0.12380794435739517,
0.13354413211345673,
0.2442537397146225,
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0.17281073331832886,
-0.03664340078830719,
-0.5611833930015564,
-0.2042037397623062,
-0.30627110600471497,
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0.288708359003067,
0.06109127029776573,
-0.6209875345230103,
-0.2219633311033249,
-0.2106265425682068,
0.2943178415298462,
0.19014951586723328,
0.33930835127830505,
0.18179626762866974,
-0.1846996396780014,
0.13717220723628998,
0.04942655563354492,
0.2046232968568802,
0.007752600125968456,
-0.40374094247817993,
0.180438831448555,
-0.04715992882847786,
0.1271693855524063,
-0.23261769115924835,
-0.3815448582172394,
0.28783345222473145,
0.26220083236694336,
0.14351703226566315,
-0.09165647625923157,
0.29991185665130615,
-0.10049332678318024,
-0.5657872557640076,
0.19146177172660828,
-0.002940217964351177,
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0.08030025660991669,
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0.10028038173913956,
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基组 | **量子化學中的基組**是用於描述體系波函數的若干具有一定性質的函數。基組是量子化學從頭計算的基礎，在量子化學中有著非常重要的意義。基組的概念最早脫胎於原子軌道，隨著量子化學的發展，基組的概念已經大大擴展，現已不局限於原子軌道的原始概念。在量子化學計算中，根據體系的不同，需要選擇不同的基組，構成基組的函數越多，基組便越大，計算的精度也越高，計算量也隨之增大。
## 斯萊特型基組
斯萊特型基組形式為：
斯萊特型基組是比較原始的基組，函數形式滿足接近原子核的 cusp 條件，但難以計算多中心雙電子積分。隨著高斯型基組的引進，斯萊特型基組在以波函數為基礎的計算方法 (HF 及 Post-HF) 中很少使用。在密度泛函理論中，因為計算方式的不同，仍然有一定的用途，如 ADF 程序包支持斯萊特型基組。
## 高斯型基組
高斯型基組用高斯函數替代了原來的斯萊特函數，其形式如下：
由於高斯型函數屬於二次方函數，因此可以將三中心和四中心的雙電子積分輕易轉化為二中心的雙電子積分，因而可以藉由中心之轉換減少計算的次數以加快計算的速度，但是高斯型函數並無法正確表示靠近原子核附近波函數的波函數狀況，因此直接使用高斯型函數構成基組的精度不及斯萊特型基組。
例如：可以氫原子的 STO-3G 之高斯型基組為例子，STO-3G 代表利用 3 個高斯型基組做線性組合，來近似 1 個斯萊特型基組。
## 壓縮高斯型基組
壓縮高斯基組是用壓縮高斯型函數構成的量子化學基組。為了彌補高斯型函數在 ${\vec {r}}=0$ 處行為的巨大差異，量子化學家使用多個高斯型函數進行線性組合，以組合獲得的新函數作為基函數參與量子化學計算，這樣獲得的基組一方面可以較好地模擬原子軌道波函數的形態，另一方面可以利用高斯型函數在數學上的良好性質，簡化計算。壓縮高斯型基組是目前應用最多的基組，根據研究體系的不同性質，量子化學家會選擇不同形式的的壓縮高斯型基組進行計算。
### 最小基組
最小基組又叫 STO-NG 基組，STO 是斯萊特型原子軌道的縮寫，NG 表示每個斯萊特型原子軌道是由 N 個高斯型函數線性組合獲得。STO-2G 基組是規模最小的壓縮高斯型基組。其中高斯型函數的指數和線性組合係數通過對原子進行 HF 方程進行自洽場計算，得到最低能量。
STO-NG 基組規模小，計算精度差，雖然計算量最小，但隨著計算機速度的提高，已經很少被使用。
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基组 | 要提高量子化學計算精度，必須加大基組的規模，即增加基組中基函數的數量，增大基組規模的一個方法是劈裂原子軌道，也就是使用多個基函數來表示一個原子軌道。
劈裂價鍵基組就是應用上述方法構造的較大型基組，所謂劈裂價鍵就是將價層電子的原子軌道用兩個或以上基函數來表示。常見的劈裂價鍵基組有 3-21G、4-21G、4-31G、6-31G、6-311G 等，在這些表示中前一個數字用來表示構成內層電子原子軌道的高斯型函數數目，「-」以後的數字表示構成價層電子原子軌道的高斯型函數數目。如 6-31G 所代表的基組，每個內層電子軌道是由 6 個高斯型函數線性組合而成，每個價層電子軌道則會被劈裂成兩個基函數，分別由 3 個和 1 個高斯型函數線性組合而成。
劈裂價鍵基組能夠比 STO-NG 基組更好地描述體系波函數，同時計算量也比最小基組有顯著的上升需要根據研究的體系不同而選擇相應的基組進行計算。
例如：6-31G (H 軌道 1s,2s;C,N,O,F 軌道 1s,2sp,3sp)，6-311G (H 軌道 1s,2s,3s;C,N,O,F 軌道 1s,2sp,3sp,4sp)。
### 極化基組
劈裂價鍵基組不能較好地描述電子云的變型等性質，為了解決這一問題，方便強共軛體系的計算，量子化學家在劈裂價鍵基組的基礎上引入高角動量函數，構成了極化基組。
所謂極化基組就是在劈裂價鍵基組的基礎上添加更高角動量所對應的基函數，如在第一周期的氫原子上添加 p 軌道波函數，在第二周期的 C 原子上添加 d 軌道波函數，在過渡金屬原子上添加 f 軌道波函數等等。這些新引入的基函數雖然經過計算沒有電子分布，但是實際上會對內層電子構成影響，因而考慮了極化基函數的極化基組能夠比劈裂價鍵基組更好地描述體系。
極化基組的表示方法基本沿用劈裂價鍵基組，所不同的是需要在劈裂價鍵基組符號的後面添加 \* 號以示區別，如 6-31G\*\* 就是在 6-31G 基組基礎上擴大而形成的極化基組，兩個 \* 符號表示基組中不僅對重原子添加了極化基函數，而且對氫等輕原子也添加了極化基函數
也可以用 (ho,lo) 的型式來表示基組的極化函數，其中 ho 表示重原子的極化函數選取，lo 表示輕原子極化函數選取。如，(3df, 3pd) 比表示對重原子添加 3 個 d 型的和 1 個 f 型基函數，而對輕原子添加 3 個 p 型和 1 個 d 型的基函數。 | [
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] |
基组 | 例如：6-31G\*\*(H 軌道 1s,2s,2p;C,N,O,F 軌道 1s,2sp,3sp,3d)，6-311G\*\*(H 軌道 1s,2s,2p,3s;C,N,O,F 軌道 1s,2sp,3sp,3d,4sp)。
### 彌散基組
彌散基組是對劈裂價鍵基組的另一種擴大。在高斯函數 $\phi _{1s}^{GF}(\alpha ,{\vec {r}}-{\vec {R_{A}}})=\left({\frac {2\alpha }{\pi }}\right)^{\frac {3}{4}}exp\left(-\alpha |{\vec {r}}-{\vec {R_{A}}}|^{2}\right)$ 中，變量 α 對函數形態有極大的作用，當 α 的取值很大時，函數圖像會向原點附近聚集，而當 α 取值很小的時候，函數的圖像會向著遠離原點的方向彌散，這種 α 很小的高斯函數被稱為彌散函數。所謂彌散基組就是在劈裂價鍵基組的基礎上添加了彌散函數的基組，這樣的基組可以用於非鍵相互作用體系的計算。
### 高角動量基組
高角動量基組是對極化基組的進一步擴展，它在極化基組的基礎上進一步添加高能級原子軌道所對應的基函數，這一基組通常用於在電子相關方法中描述電子間相互作用。
## 參見
* 量子化學 | [
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哈特里－福克方程 | **哈特里－福克**方程式（英語：Hartree–Fock equation），又稱為 **HF 方程式**，是一個應用變分法計算多電子系統波函數的方程式，是量子物理、凝聚態物理學、量子化學中最重要的方程式之一。HF 方程式形式上是單電子本徵方程式，求得的本徵態是單電子波函數，即分子軌域。以 HF 方程式為核心的數值計算方法稱為「哈特里－福克方法」（Hartree–Fock method）。
基於分子軌域理論的所有量子化學計算方法都是以 HF 方法為基礎的。鑑於分子軌域理論在現代量子化學中的廣泛應用，HF 方程式被視為現代量子化學的基石。
## 哈特里－福克近似
**哈特里－福克近似**也稱為**分子軌域近似**或**單行列式近似** ，認為多電子體系波函數可以由體系分子軌域波函數構造的單個斯萊特行列式表示：
: : $\Psi ={\frac {1}{\sqrt {N!}}}{\begin{vmatrix}\chi _{1}(1)&\chi _{2}(1)&\cdots &\chi _{N}(1)\\\chi _{1}(2)&\chi _{2}(2)&\cdots &\chi _{N}(2)\\\vdots &\vdots &\ddots &\vdots \\\chi _{1}(N)&\chi _{2}(N)&\cdots &\chi _{N}(N)\\\end{vmatrix}}=|\chi _{1}\chi _{2}\cdots \chi _{N}\rangle $
這裡 $N$ 為電子數， $\chi _{i}(x)$ 是能量最低的 $N$ 個分子軌域波函數。而分子軌域波函數是由體系中所有原子軌域波函數經過線性組合構成的，
: : $\chi _{a}(x)=\sum _{i}C_{ai}\phi _{i}(x)$
通常原子軌域波函數 $\phi _{i}(x)$ 由計算時指定的基組給出。
在此近似下，由變分法，體系的多電子薛丁格方程式可化為單電子方程式
: : $f_{(1)}\chi _{a}(1)=\varepsilon _{a}\chi _{a}(1)$ | [
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哈特里－福克方程 | 其中 $f_{(1)}$ 為福克算符， $\chi _{a(1)}$ 為體系的分子軌域， $\varepsilon _{a}$ 是分子軌域 $\chi _{a(1)}$ 的軌域能。這就是**哈特里－福克方程式**。求解該方程式，即求解使方程式成立的一組分子軌域；這組分子軌域使得在 HF 近似下的體系總能量最低。
## 發展簡史
1927 年，物理學家瓦爾特・海特勒和弗里茨・倫敦完成了氫氣分子的量子力學計算之後，開啟了量子化學的時代。從此，人們便開始嘗試使用量子力學理論來解釋化學物質結構和化學現象。
為了解決多電子體系薛丁格方程式近似求解的問題，量子化學家道格拉斯・哈特里在 1928 年提出了哈特里假設，將每個電子看做是在其他所有電子構成的平均勢場中運動的粒子，並且首先提出了迭代法的思路。哈特里根據他的假設，將體系電子哈密頓算子分解為若干個單電子哈密頓算子的簡單代數和，每個單電子哈密頓算子中只包含一個電子的坐標，因而體系多電子波函數可以表示為單電子波函數的簡單乘積，這就是**哈特里方程式**。但是由於哈特里沒有考慮電子波函數的反對稱要求，他的哈特里方程式實際上是非常不成功的。
1930 年，哈特里的學生弗拉基米爾・福克和約翰・斯萊特分別提出了考慮包立原理的自洽場迭代方程式和單行列式型多電子體系波函數，這就是今天的**哈特里 - 福克方程式**。但是由於計算上的困難，HF 方程式誕生後整整沉寂了二十年，在 1950 年，量子化學家克萊門斯・羅特漢想到將分子軌域用原子軌域的線性組合來近似展開，而得到了閉殼層結構的羅特漢方程式。
1953 年，美國的 R. 帕里瑟、R. 帕爾和英國的約翰・波普爾花費兩年時間使用手搖計算器分別獨立地實現了對氮氣分子的 RHF 自洽場計算，這是人類首次通過求解 HF 方程式獲得對化學結構的量子力學解釋，也是量子化學計算方法第一次實際完成。在第一次成功之後，伴隨著電腦技術的迅猛發展，HF 方程式與量子化學一道獲得長足發展，在 HF 方程式的基礎上，人們發展出了高級量子化學計算方法，使得計算精度進一步提高，通過對 HF 方程式電子積分的簡化和參數化，人們大大縮減了量子化學的計算量，使得對超過 1000 個原子的中等大小分子的計算成為可能。
## 方程式的導出 | [
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哈特里－福克方程 | 哈特里－福克方程式源於對多電子體系電子波函數的變分法處理。在波恩 - 歐本海默近似法條件下，一個多電子體系的電子運動與能量可以與原子核的運動和能量相互分離，這樣利用電子哈密頓算符和多電子波函數便可以計算體系的電子能量。其能量的表達式為:
: : $E_{0}=\langle \Psi _{0}|H_{ele}|\Psi _{0}\rangle $
式中 $E_{0}$ 表示體系基態電子能量； $H_{ele}$ 表示體系的電子哈密頓算符，
: : $H_{ele}=\sum _{i}^{N}{\frac {-1}{2}}\nabla _{i}^{2}-\sum _{i=1}^{N}\sum _{a=1}^{M}{\frac {Z_{A}}{R_{iA}}}+\sum _{i=1}^{N}\sum _{j>i}^{N}{\frac {1}{r_{ij}}}$
根據作用方式，可以將 $H_{ele}$ 分解為兩部分
: : $H_{ele}=O_{1}+O_{2}$
其中 $O_{1}$ 為單電子算符
: : $O_{1}=\sum _{i}^{N}{\frac {-1}{2}}\nabla _{i}^{2}-\sum _{i=1}^{N}\sum _{a=1}^{M}{\frac {Z_{A}}{R_{iA}}}=\sum _{i}^{N}-\left({\frac {1}{2}}\nabla _{i}^{2}+\sum _{a=1}^{M}{\frac {Z_{A}}{R_{iA}}}\right)=\sum _{i}^{N}h_{i}$
描述單個電子的動能和原子核吸引位能；而 $O_{2}$ 為二電子算符
: : $O_{2}=\sum _{i=1}^{N}\sum _{j>i}^{N}{\frac {1}{r_{ij}}}$
描述電子間交互作用。
$\Psi _{0}$ 代表基態多電子波函數，是體系單電子分子軌域波函數為基函數組建的斯萊特行列式。構建 $\Psi _{0}$ 的各個分子軌域相互之間是正交歸一的，即約束條件為
: : $\langle \chi _{a}|\chi _{b}\rangle =\delta _{ab}$ | [
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哈特里－福克方程 | 考慮此約束條件，應用拉格朗日乘數法對函數 $L=E_{0}[\{\chi _{i}\}]-\sum _{a,b}^{N}\varepsilon _{ab}\left(\langle a|b\rangle -\delta _{ab}\right)$ 變分求極值。式中 $\varepsilon _{ab}$ 是拉格朗日待定因子， $\langle a|b\rangle $ 是 $\langle \chi _{a}|\chi _{b}\rangle $ 的簡寫。
變分法的處理過程如下：令
: : $\delta L=\delta E_{0}[\{\chi _{i}\}]-\sum _{a,b}^{N}\varepsilon _{ab}\delta [\chi _{a}|\chi _{b}]=0$
其中
: : ${\begin{aligned}\delta E_{0}=&\sum _{a}^{N}[\delta \chi _{a}|h|\chi _{a}]+\sum _{a}^{N}[\chi _{a}|h|\delta \chi _{a}]\\&+{\frac {1}{2}}\sum _{a,b}^{N}\left([\delta \chi _{a}\chi _{a}|\chi _{b}\chi _{b}]+[\chi _{a}\delta \chi _{a}|\chi _{b}\chi _{b}]+[\chi _{a}\chi _{a}|\delta \chi _{b}\chi _{b}]+[\chi _{a}\chi _{a}|\chi _{b}\delta \chi _{b}]\right)\\&+{\frac {1}{2}}\sum _{a,b}^{N}\left([\delta \chi _{a}\chi _{b}|\chi _{b}\chi _{a}]+[\chi _{a}\delta \chi _{b}|\chi _{b}\chi _{a}]+[\chi _{a}\chi _{b}|\delta \chi _{b}\chi _{a}]+([\chi _{a}\chi _{b}|\chi _{b}\delta \chi _{a}]\right)\end{aligned}}$
這裡定義單電子積分記號 | [
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哈特里－福克方程 | : : $[\chi _{a}|\chi _{b}]=\langle \chi _{a}|\chi _{b}\rangle =\int dx\chi _{a}^{*}(x)\chi _{b}(x)$
以及二電子積分記號
: : $[\chi _{a}\chi _{b}|\chi _{c}\chi _{d}]=\int dx_{1}dx_{2}\chi _{a}^{*}(x_{1})\chi _{b}(x_{1})r_{12}^{-1}\chi _{c}^{*}(x_{2})\chi _{d}(x_{2})$
考慮到流動坐標的不可區分性，可以簡化為：
: : ${\begin{aligned}\delta E_{0}=&\sum _{a}^{N}[\delta \chi _{a}|h|\chi _{a}]+\sum _{a,b}^{N}\left([\delta \chi _{a}\chi _{a}|\chi _{b}\chi _{b}]-[\delta \chi _{a}\chi _{b}|\chi _{b}\chi _{a}]\right)\\&+\left(\sum _{a}^{N}[\delta \chi _{a}|h|\chi _{a}]+\sum _{a,b}^{N}\left([\delta \chi _{a}\chi _{a}|\chi _{b}\chi _{b}]-[\delta \chi _{a}\chi _{b}|\chi _{b}\chi _{a}]\right)\right)^{*}\end{aligned}}$
同理， $\delta L$ 中的 $\sum _{a,b}^{N}\varepsilon _{ba}\delta [a|b]$ 項有：
: : $\sum _{a,b}^{N}\varepsilon _{ba}\delta [a|b]=\sum _{a,b}^{N}\varepsilon _{ba}[\delta \chi _{a}|\chi _{b}]+\left(\sum _{a,b}^{N}\varepsilon _{ba}[\delta \chi _{a}|\chi _{b}]\right)^{*}$
將兩項相加， $\delta L$ 表示為： | [
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] |
哈特里－福克方程 | : : ${\begin{aligned}\delta L=&\sum _{a}^{N}[\delta \chi _{a}|h|\chi _{a}]+\sum _{a,b}^{N}\left([\delta \chi _{a}\chi _{a}|\chi _{b}\chi _{b}]-[\delta \chi _{a}\chi _{b}|\chi _{b}\chi _{a}]\right)+\sum _{a,b}^{N}\varepsilon _{ba}[\delta \chi _{a}|\chi _{b}]\\&+\left(\sum _{a}^{N}[\delta \chi _{a}|h|\chi _{a}]+\sum _{a,b}^{N}\left([\delta \chi _{a}\chi _{a}|\chi _{b}\chi _{b}]-[\delta \chi _{a}\chi _{b}|\chi _{b}\chi _{a}]\right)+\sum _{a,b}^{N}\varepsilon _{ba}[\delta \chi _{a}|\chi _{b}]\right)^{*}\\=&0\end{aligned}}$
若 $L$ 函數處於極值點，則變數 $\left\{\chi _{i}\right\}$ 向各個方向的微小變化都應該有 $\delta L=0$ 。可以取 $\left\{\delta \chi _{i}\right\}=\left\{i\delta \chi _{i}\right\}$ 沿虛數軸變分，則在 $\delta L$ 表達式中，第一項前會產生一個 $i$ 的係數，第二項複共軛會產生一個 $-i$ 係數： | [
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哈特里－福克方程 | : : ${\begin{aligned}\delta L=&i\sum _{a}^{N}[\delta \chi _{a}|h|\chi _{a}]+\sum _{a,b}^{N}\left([\delta \chi _{a}\chi _{a}|\chi _{b}\chi _{b}]-[\delta \chi _{a}\chi _{b}|\chi _{b}\chi _{a}]\right)+\sum _{a,b}^{N}\varepsilon _{ba}[\delta \chi _{a}|\chi _{b}]\\&-i\left(\sum _{a}^{N}[\delta \chi _{a}|h|\chi _{a}]+\sum _{a,b}^{N}\left([\delta \chi _{a}\chi _{a}|\chi _{b}\chi _{b}]-[\delta \chi _{a}\chi _{b}|\chi _{b}\chi _{a}]\right)+\sum _{a,b}^{N}\varepsilon _{ba}[\delta \chi _{a}|\chi _{b}]\right)^{*}\\=&0\end{aligned}}$
消去虛數單位，兩式相加，可以消去表達式中的複共軛項：
: : $\delta L=\sum _{a}^{N}[\delta \chi _{a}|h|\chi _{a}]+\sum _{a,b}^{N}\left([\delta \chi _{a}\chi _{a}|\chi _{b}\chi _{b}]-[\delta \chi _{a}\chi _{b}|\chi _{b}\chi _{a}]\right)+\sum _{a,b}^{N}\varepsilon _{ba}[\delta \chi _{a}|\chi _{b}]=0$
在引入庫侖算符和交換算符的概念之後，上述表達式可以改寫為：
: : $\sum _{a}^{N}\int dx_{1}\delta \chi _{a}^{*}(1)\left[h_{1}\chi _{a}(1)+\sum _{b}^{N}\left(J_{b(1)}-K_{b(1)}\right)\chi _{a}(1)-\sum _{b}^{n}\varepsilon _{ba}\chi _{b}(1)\right]=0$ | [
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哈特里－福克方程 | 對任意 $\left\{\delta \chi _{i}\right\}$ 上述等式均應成立，因而必須有:
: : $h_{1}\chi _{a}(1)+\sum _{b}^{N}\left(J_{b(1)}-K_{b(1)}\right)\chi _{a}(1)-\sum _{b}^{n}\varepsilon _{ba}\chi _{b}(1)=0$
整理得到：
: : $\left[h_{(1)}+\sum _{b}^{N}\left(J_{b(1)}-K_{b(1)}\right)\right]\chi _{a}(1)=\sum _{b}^{N}\varepsilon _{ba}\chi _{b}(1)$
定義福克算符
: : $f_{(1)}=h_{(1)}+\sum _{b}^{N}\left(J_{b(1)}-K_{b(1)}\right)$
方程式可以表達為
: : $f_{(1)}\chi _{a}(1)=\sum _{b}^{N}\varepsilon _{ba}\chi _{b}(1)$
即哈特里－福克方程式。為了求解，通過對分子軌域波函數進行酉轉換處理，使得軌域能矩陣 $\varepsilon _{ba}$ 對角化，將一般的不可解的 HF 方程式轉化為**正則哈特里－福克方程式**：
: : $f_{(1)}\chi _{a}(1)=\epsilon _{a}\chi _{a}(1)$
此方程式形式上為本徵方程式，但是福克算符中的庫侖算符和交換算符都與分子軌域有關，因此只能夠通過自洽迭代的方法近似求解，即哈特里 - 福克自洽場（HF-SCF）方法。HF-SCF 方法是組態交互作用方法、多體微擾理論、半經驗量子化學計算等現代量子化學計算方法的基礎。
## 方程式的解與自洽場方法
正則哈特里－福克方程式雖然具有簡單的本徵方程式形式，但福克算子中的庫侖算子和交換算子中含有所有 $\chi _{a}$ 的表達式，因而方程式的實際形式非常複雜，無法求得精確的解析解，只能使用迭代法求解，即量子化學中的自洽場方法。 | [
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哈特里－福克方程 | 在實際操作中，人們會首先給定基組，將本徵方程式轉化為矩陣方程式。這一轉變使分子軌域可以表示為 $\chi _{a}(x)=\sum _{i}C_{ai}\phi _{i}(x)$ ，可以用一個變化的 $K$ 維向量 $\mathbb {C_{a}} $ 來代表分子軌域， $K$ 為基組中基函數的數量。同理，可以將福克算符轉化為福克矩陣。然而由於基函數不一定正交，存在重疊積分
: : $S_{ij}=\int dx\phi _{i}^{*}(x)\phi _{j}(x)$
最終 HF 方程式的形式轉化為廣義本徵方程式
: : $\mathbb {F} \mathbb {C} =\mathbb {\varepsilon } \mathbb {S} \mathbb {C} $
此方程式稱為**哈特里 - 福克 - 羅特漢**方程式，或**羅特漢方程式**。這樣，相對複雜的本徵方程式就轉化為只需要進行簡單代數計算就可以求解的矩陣本徵方程式，而原方程式中複雜的積分則在上述轉化過程中一次性完成了。
求解時首先需要給定一個分子軌域的初始猜測 $\mathbb {C} $ ，用這套分子軌域計算福克矩陣 $\mathbb {F} $ ，再求解哈特里 - 福克 - 羅特漢方程式，得到一組新的分子軌域 $\mathbb {C} $ 和相應的本徵值矩陣 $\mathbb {\varepsilon } $ ，再使用新軌域 $\mathbb {C} $ 計算新的福克矩陣 $\mathbb {F} $ …… 重複上述過程，直到 $\mathbb {C} $ 的各個矩陣元數值不再有明顯的變化，稱作自洽迭代收斂，就得到了 HF 方程式的解。
得到收斂的 $\mathbb {C} $ 矩陣後，將這些係數代入 $\chi _{a}=\sum _{i}C_{ia}\phi _{i}$ 與基函數結合，便獲得了最終的分子軌域波函數以及體系電子總能量等各種性質。
## 方程式的應用
HF 方程式在量子化學中有著廣泛的應用，所有分子軌域理論的量子化學計算都是以 HF 方程式為基礎的。
* **組態交互作用方法**（CI）：在 CI 方法中，通過 HF 方程式解得的一系列分子軌域用於構建多電子基函數集，在構建了多電子基函數集後再通過變分法處理獲得 CI 能量的最低點，因而進行 CI 計算必須首先完成 HF 方程式的求解。 | [
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哈特里－福克方程 | * **多體微擾理論方法**（MPn）：MPn 計算將體系所有福克算符的代數和定義為哈特里－福克哈密頓算符，將電子間交互作用所產生的能量項看作是對哈特里－福克哈密頓算符的微擾，經過微擾處理後可以獲得體系能量的近似值。進行多體微擾計算也需要首先進行 HF 方程式的求解，以獲得需要的分子軌域波函數形式和分子軌域能量。
* **半經驗量子化學計算**：半經驗量子化學計算是對 HF 方程式求解過程的簡化。在 HF 方程式的求解中，絕大部分計算量都分布在由正則 HF 方程式向矩陣本徵方程式形式轉變的過程中，如果將這一過程中大量的電子積分用經驗數值代替，便可以極大地縮短 HF 方法的計算時間。為此，針對不同的研究體系，量子化學家開發了不同的積分經驗常數，與之相應地產生了各種半經驗量子化學計算方法。本質上講，半經驗計算仍然是通過自洽場方法求解 HF 方程式的過程。
## 參考文獻
## 外部連結
* A Numerical Hartree-Fock Program（頁面存檔備份，存於網際網路檔案館）
## 參見
* 量子化學
* 羅特漢方程式
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Slater行列式 | **斯萊特行列式**是多電子體系波函數的一種表達方式，他以量子物理學家斯萊特的名字命名。這種形式的波函數可以滿足對多電子波函數的反對稱要求（即所謂泡利原理）：交換體系中任意兩個電子，則波函數的符號將會反轉。在量子化學中，所有基於分子軌道理論的計算方法都用斯萊特行列式的形式來表示多電子體系的波函數。
## 形式
### 基本形式
斯萊特行列式最原初的形態是一個由單電子波函數即分子軌道波函數構成的行列式：
$$
\Psi _{(x_{1},x_{2},\cdots ,x_{n})}={\frac {1}{\sqrt {N!}}}{\begin{vmatrix}\chi _{i(x_{1})}&\chi _{j(x_{1})}&\cdots &\chi _{k(x_{1})}\\\chi _{i(x_{2})}&\chi _{j(x_{2})}&\cdots &\chi _{k(x_{2})}\\\vdots &\vdots &\ddots &\vdots \\\chi _{i(x_{n})}&\chi _{j(x_{n})}&\cdots &\chi _{k(x_{n})}\end{vmatrix}}
$$
行列式中每一行是由同一電子的不同可能波函數組成，每一列是由不同電子的相同可能波函數組成，行列式前的係數 $\left(N!\right)^{-{\frac {1}{2}}}$ 是保證波函數歸一性的歸一係數。
根據行列式的性質，互換行列式中的兩行行列式的符號會反轉：
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Slater行列式 | {\begin{vmatrix}\chi _{i(x_{1})}&\chi _{j(x_{1})}&\cdots &\chi _{k(x_{1})}\\\chi _{i(x_{2})}&\chi _{j(x_{2})}&\cdots &\chi _{k(x_{2})}\\\vdots &\vdots &\ddots &\vdots \\\chi _{i(x_{n})}&\chi _{j(x_{n})}&\cdots &\chi _{k(x_{n})}\end{vmatrix}}=-{\begin{vmatrix}\chi _{i(x_{2})}&\chi _{j(x_{2})}&\cdots &\chi _{k(x_{2})}\\\chi _{i(x_{1})}&\chi _{j(x_{1})}&\cdots &\chi _{k(x_{1})}\\\vdots &\vdots &\ddots &\vdots \\\chi _{i(x_{n})}&\chi _{j(x_{n})}&\cdots &\chi _{k(x_{n})}\end{vmatrix}}
$$
這一性質正符合多電子體系的泡利原理
### 其他形式
* 考慮到行列式在書寫過程中的不便，通常人們用右矢的形式代表斯萊特行列式：
$$
\Psi _{(x_{1},x_{2},\cdots ,x_{n})}=\mid \chi _{i},\chi _{j},\cdots ,\chi _{k}\rangle
$$
需要注意的是，這種右矢形式僅僅用來代表行列式，並非數學上的相等關係。
* 將行列式展開後，可以用置換算子形式來表示斯萊特行列式:
$$
\Psi _{(x_{1},x_{2},\cdots ,x_{n})}={\frac {1}{\sqrt {N!}}}\sum _{n=1}^{N!}(-1)^{p_{n}}P_{n}\left[\chi _{i(x_{1})}\chi _{j(x_{2})}\cdots \chi _{k(x_{n})}\right]
$$ | [
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] |
Slater行列式 | 其中算子 $P_{n}$ 叫做置換算子，其作用是將各分子軌道波函數中的電子序號進行交換，根據排列的原理，在由 N 個電子組成的體系中，這樣的算子一共有 N! 個。 $p_{n}$ 是置換算子的奇偶性，即任何置換算子可以轉化為若干兩兩對換的置換算子的乘積，所謂奇偶性就是一個置換算子所分解成的對換算子的個數的奇偶性。與上面提到的右矢形式不同，這種由置換算子來表達的形式與行列式表達式在數學上是嚴格相等的。
* 對斯萊特行列式的置換算子形式進一步簡化可以用反對稱化算子形式來表示：
$$
\Psi _{(x_{1},x_{2},\cdots ,x_{n})}=A\left[\chi _{i(x_{1})}\chi _{j(x_{2})}\cdots \chi _{k(x_{n})}\right]
$$
其中算子 $A={\frac {1}{\sqrt {N!}}}\sum _{n=1}^{N!}(-1)^{p_{n}}P_{n}$ 叫做反對稱化算子。
## 應用
斯萊特行列式在量子化學中應用廣泛，經過自洽場方法解 HF 方程獲得的最終解便是一個斯萊特行列式型多電子波函數，高級的量子化學計算方法也應用到斯萊特行列式，組態相互作用方法得到的多電子體系波函數是若干個斯萊特行列式的線性組合:
$$
\Phi =\sum _{i}C_{i}\Psi _{i}
$$
經過對這個由許多行列式組成的巨大波函數的變分法處理，可以獲得比 HF 方程更加精確的量子化學計算結果
## 參見
* 量子化學
* 分子軌道
* 波函數 | [
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电传导率 | **電導率**（electrical conductivity）或**比電導**（specific conductance），又譯電傳導性、電導係數，是表示物質傳輸電流能力強弱的一種測量值。當施加電壓於導體的兩端時，其電荷載子會呈現朝某方向流動的行為，因而形成電流。電導率 $\sigma \,\!$ 是以歐姆定律定義為電流密度 $\mathbf {J} \,\!$ 和電場強度 $\mathbf {E} \,\!$ 的比率：
: $\mathbf {J} =\sigma \mathbf {E} \,\!$ 。
有些物質會有異向性 (anisotropic) 的電導率，必需用 3 X 3 矩陣來表達（使用數學術語，第二階張量，通常是對稱的）。
電導率是電阻率 $\rho \,\!$ 的倒數。在國際單位制中的單位是西門子 / 公尺 (S・m-1)：
: $\sigma ={1 \over \rho }\,\!$ 。
電導率儀 (electrical conductivity meter) 是一種是用來測量溶液電導率的儀器。
## 電傳導性
**電傳導性** (electric conductivity) 是物質可以傳導電子的性質。按物質是否具有電傳導性，可把物質分為導體、半導體和絕緣體。
* 導體：金屬、電解質溶液，一般有很高的電導率，很低的電阻率。
* 絕緣體：像玻璃、乾燥的木材、塑料、橡膠或真空這類物質的電導率很低，電阻率很高。
* 半導體：電導率在導體和絕緣體之間。在不同的狀況下，電導率會有很大的變化。例如，暴露於電場或某種頻率的光波，最重要地，溫度和半導體材料的成份。
固態半導體的摻雜程度會造成電導率很大的變化。增加摻雜程度會造成高電導率。水溶液的電導率高低跟其內含溶質鹽的濃度有關，或其它會分解為電解質的化學雜質。水樣本的電導率是測量水的含鹽成分、含離子成分、含雜質成分等等的重要指標。水越純淨，電導率越低（電阻率越高）。水的電導率時常以**電導係數**來紀錄；電導係數是水在 25°C 溫度的電導率。
## 一些物質的電導率
## 複電導率
假若物質處於隨時間而變的電場，則電導率為複數（對於各向異性物質，電導率為複矩陣），稱為導納率 (admittivity) 。這方法可以應用於電阻抗斷層成像 (electrical impedance tomography) ，一種工業或醫學成像。 | [
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电传导率 | 另外一種計算交流的方法使用實值的電導率（跟時間有關）和實值的電容率。電導率越大，物質會越快吸收交流訊號，物質越不透明。
## 電導率與溫度的關係
電導率與溫度緊密相關。金屬的電導率隨著溫度的增高而降低。半導體的電導率隨著溫度的增高而增高。在一段溫度值域內，電導率可以被近似為與溫度成正比。為了要比較物質在不同溫度狀況的電導率，必須設定一個共同的參考溫度。電導率與溫度的相關性，時常可以表達為，電導率對上溫度線圖的斜率，用方程式寫為
: $\sigma (T)={\sigma _{0} \over 1+\alpha (T-T_{0})}\,\!$ ；
其中， $T_{0}\,\!$ 是參考溫度， $T\,\!$ 是測量溫度， $\sigma \,\!$ 是物質的電導率， $\sigma _{0}\,\!$ 是物質在參考溫度的固定參考電導率（通常在室溫）， $\alpha \,\!$ 是物質的**溫度補償斜率**。
最常見的水的溫度補償斜率大約為 2 %/°C 。但是，它可以在 (1 to 3) %/°C 值域之間。這數值與地球化學有關，可以很容易地在實驗室裡測量出來。
在非常低溫的狀況（在絕對零度附近），有些物質的電導率會變得非常高。稱這現象為超導現象。
## 參考文獻
## 參閱
* 導電性：講述電導率的物理本質。
* 電阻
* 熱導率
* 古典與量子電導率 (classical and quantum conductivity)
* 莫耳電導率 (molar conductivity) 講述在溶液裡因離子產生的電導率。
* 傳輸現象 (transport phenomena) | [
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伪随机性 | **偽隨機性**（英語：Pseudorandomness）是一個過程似乎是隨機的，但實際上並不是。例如**偽亂數**是使用一個確定性的演算法計算出來的似乎是隨機的數序，因此偽亂數實際上並不隨機。在計算偽亂數時假如使用的開始值不變的話，那麼偽亂數的數序也不變。偽亂數的隨機性可以用它的統計特性來衡量，其主要特徵是每個數出現的可能性和它出現時與數序中其它數的關係。偽亂數的優點是它的計算比較簡單，而且只使用少數數值很難推算出計算它的演算法。一般人們使用一個假的亂數，比如電腦上的時間作為計算偽亂數的開始值。
## 電腦偽亂數函式
用來計算偽亂數的函式被稱為隨機函式，使用隨機函式產生隨機數的演算法稱為亂數生成器。一些隨機函式是周期性的，雖然一般來說使用非周期性的函式要好得多，但周期性的隨機函式往往快得多。有些周期函式的係數可以調整，之後它們的周期非常大，基本上與非周期的函式效果一樣。
### C 語言
可見，偽亂數是由一套產生亂數的演算法實現的。
## 使用
在電腦類比中偽亂數用來類比產生隨機的過程，背景雜訊產生器中也可應用偽亂數。由於偽亂數不是真的亂數，在有些方面它們不能被使用，例如在密碼學中使用偽亂數要小心，因為其可計算性是一個可以攻擊的地方。統計學、蒙特・卡羅方法上使用的偽隨機數也必須挑選週期極長、隨機性夠高的隨機函式，以確保計算結果有足夠的隨機性。
偽亂數的一個特別大的優點是它們的計算不需要外部的特殊硬體的支援，因此在計算機科學中偽亂數依然被使用。真正的亂數必須使用專門的裝置，比如熱噪訊號、量子力學的效應、放射性元素的衰退輻射，或使用無法預測的現象，譬如使用者按鍵盤的位置與速度、使用者運動滑鼠的路徑坐標等來產生。對於移動式計算，採用加速度感測器協助亂數生成亦是一種普遍做法。
## 參見
* 線性同餘方法
* 平方取中法
* M-sequence
* 梅森旋轉演算法
* 偽亂數二進位數列
## 延伸閱讀
## 外部連結
* HotBits: Genuine random numbers, generated by radioactive decay （頁面存檔備份，存於網際網路檔案館）
* Using and Creating Cryptographic-Quality Random Numbers | [
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伪随机性 | * In RFC 1750, the use of pseudorandom number sequences in cryptography is discussed at length. | [
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多体微扰理论 | **多體微擾理論**是一種基於分子軌域理論的高級量子化學計算方法。這種方法以 Hartree-Fock 方程的自洽場解為基礎，應用微擾理論，獲得考慮了相關能的多電子體系近似解，其計算精度與組態相互作用方法的 DCI 接近，但計算量遠小於 DCI，是應用比較廣泛的高級量子化學計算方法。
## 歷史
多體微擾理論是由量子化學家 Møller 和 Plesset 在 1934 年提出的，所以這一方法也經常以二人的名字所寫 MP 表示，MPn 表示的是多體微擾 n 級近似。
## 基本假設
多體微擾理論是以 Hartree-Fock 方程為基礎的，應用微擾法處理的計算方法。微擾法要求將複雜體系的哈密頓算子分解為可精確求解項和微擾項兩部分，在多體微擾理論中，引入 Hartree-Fock 哈密頓算子的概念：
$$
H_{0}=\sum _{i}^{N}f_{i}
$$
可以證明，由 Hartree-Fock 方程解得的單電子分子軌道波函數所構成的斯萊特行列式波函數是 $H_{0}$ 的本徵函數，構成斯萊特行列式的各分子軌道軌道能的代數和是 $H_{0}$ 的本徵值：
$$
H_{0}|\Psi _{0}>=\sum _{i}^{N}\epsilon _{i}|\Psi _{0}>
$$
將多電子體系哈密頓算子分解為 Hartree-Fock 哈密頓算子和微擾項的代數和：
$$
H_{ele}=H_{0}+V
$$
在這個假設下，多電子體系電子哈密頓算子被分解為可精確求解的 Hartree-Fock 哈密頓算子和微擾算子，應用微擾方法進行近似處理。
## 能量和波函數的各級近似
### 零級能量和波函數
在多體微擾理論下，基態零級能量就是構成基態斯萊特行列式的各分子軌道軌道能的代數和，零級波函數就是基態斯萊特行列式波函數。可以看出，多體微擾理論的零級能量精度甚至不如 Hartree-Fock 方程所得的能量。
### 一級校正
根據微擾理論，能量的一級校正 $E_{0}^{(1)}$ 為：
$$
E_{0}^{(1)}=<\Psi _{0}^{(0)}|V|\Psi _{0}^{(0)}>
$$
將微擾算子 V 的表達式代入得到：
可以看到，經過能量的一級校正後體系能量為：
### 二級校正
根據微擾理論，體系基態能量的二級校正 $E_{0}^{(2)}$ 為：
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多体微扰理论 | E_{0}^{(2)}=\sum _{n\neq \;0}{\frac {{\begin{vmatrix}<\Psi _{0}^{(0)}|V|\Psi _{n}^{(0)}>\end{vmatrix}}^{2}}{E_{0}^{(0)}-E_{n}^{(0)}}}
$$
其中 $\Psi _{n}^{(0)}$ 為 Hartree-Fock 哈密頓算子本徵能量為 $E_{n}^{(0)}$ 的波函數，其本質是體系激發態的斯萊特行列式
可以證明，只有對雙激發的斯萊特行列式才有 ${\begin{vmatrix}<\Psi _{0}|V|\Psi _{a,b}^{r,s}>\end{vmatrix}}\neq \;0$ 所以體系能量的二級校正為：
$$
E_{0}^{(2)}=\sum _{a<b,r<s}{\frac {{\begin{vmatrix}<\Psi _{0}|V|\Psi _{a,b}^{r,s}>\end{vmatrix}}^{2}}{\epsilon _{a}+\epsilon _{b}-\epsilon _{r}-\epsilon _{s}}}
$$
將分子項展開，得到：
$$
E_{0}^{(2)}=\sum _{a<b,r<s}{\frac {{\begin{vmatrix}<\chi _{a}\chi _{b}||\chi _{r}\chi _{s}>\end{vmatrix}}^{2}}{\epsilon _{a}+\epsilon _{b}-\epsilon _{r}-\epsilon _{s}}}
$$
最終體系經過二級校正的基態能量為：
$$
E=E_{0}^{HF}+\sum _{a<b,r<s}{\frac {{\begin{vmatrix}<\chi _{a}\chi _{b}||\chi _{r}\chi _{s}>\end{vmatrix}}^{2}}{\epsilon _{a}+\epsilon _{b}-\epsilon _{r}-\epsilon _{s}}}
$$
由於式中 $\epsilon _{r}$ $\epsilon _{s}$ 是體系未占據分子軌道的軌道能，在基態，其能量衡高於 $\epsilon _{a}$ 和 $\epsilon _{b}$ 所以能量的二級微擾是一個負值，因而考慮二級微擾的體系能量低於 Hartree-Fock 方程得到的體系能量，這一差異來自電子相互作用。 | [
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] |
多体微扰理论 | 考慮二級校正的多體微擾計算簡稱 MP2
### 高級校正
更高級的校正是以較低級校正為計算基礎的，隨著校正級別的提高，計算量也急劇增加，理論上講，隨著校正級別的提高最終的體系能量會逐漸逼近真實值。目前的計算方法最高可以進行 MP5 計算，即體系能量的五級校正。
## 方法評價
多體微擾理論方法是一種量子化學高級計算方法，在考慮相關能的計算方法中，多體微擾理論方法是計算量相對最小的。MP1 可以達到 HF 方程的計算精度，MP2 一般可以達到 60% 的相關能，與 DCI 方法相當，但計算過程僅需要計算少量雙電子積分，遠遠小於 DCI；MP4 一般可以達到 85% 的相關能。
MPn 方法是一個大小一致的方法，即對電子數不同的體系，使用 MPn 計算的精度是相同的，這一特性使得 MPn 方法特別適合進行化學反應的模擬計算。但是由於 MPn 方法以 HF 方程為基礎，因而受到 HF 方程的局限，對於那些應用 HF 方程不能很好處理的體系，如非限制性開殼層體系，MPn 方法也不能很好處理。
## 參考文獻
## 參見
* 量子化學
* Hartree-Fock 方程 | [
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量子化学中的双电子积分 | **雙電子積分**就是涉及兩個電子坐標的積分，是量子化學計算中出現頻率最高的一類積分，也是進行 Hartree-Fock 方程自洽場計算和其他高級量子化學計算過程中計算量最大的一個部分。構成一個雙電子積分的，是二至四個不同的軌道波函數、一個涉及兩個電子坐標的算子（即雙電子算子）和兩套電子坐標。在量子化學計算中，出現頻率最高的雙電子算子是 ${\frac {1}{r_{12}}}$ ，即在原子單位下表徵兩電子間庫侖排斥力的算子。
## 雙電子積分的基本形式
雙電子積分的基本形式是這樣的：
$$
\int dx_{1}dx_{2}\chi _{1}^{*}(x_{1})\chi _{2}^{*}(x_{2}){\frac {1}{r_{12}}}\chi _{1}(x_{1})\chi _{2}(x_{2})
$$
其中的 $\chi _{1}(x_{1})$ 、 $\chi _{2}(x_{2})$ 表示參與積分的單電子波函數； $x_{1}$ 、 $x_{2}$ 表示電子坐標，其中包含三個方向的笛卡兒坐標和一個自旋坐標； ${\frac {1}{r_{12}}}$ 即上面提到的雙電子算子。
也可以用狄拉克符號來簡寫上述雙電子積分：
$$
\langle \chi _{1}(x_{1})\chi _{2}(x_{2})|{\frac {1}{r_{12}}}|\chi _{1}(x_{1})\chi _{2}(x_{2})\rangle
$$
兩者在數學上是完全一樣的。
## 物理符號與化學符號
對於使用 ${\frac {1}{r_{12}}}$ 算子的雙電子積分，由於在量子化學中出現的頻率極高，因而使用專門的符號來表示，即所謂物理符號和化學符號
### 物理符號
物理符號的形式是 $\langle \chi _{i}\chi _{j}|\chi _{k}\chi _{l}\rangle $ ，有時也簡單表示為 $\langle ij|kl\rangle $ ，這一表示等價於：
$$
\int dx_{1}dx_{2}\chi _{i}^{*}(x_{1})\chi _{j}^{*}(x_{2}){\frac {1}{r_{12}}}\chi _{k}(x_{1})\chi _{l}(x_{2})
$$ | [
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量子化学中的双电子积分 | 分布在表示中單豎線之前的是取復共軛的軌道波函數，分布在單豎線之後的是不取復共軛的軌道波函數，在豎線同一側的兩個波函數中，位於左則的一個波函數其變量為 $x_{1}$ ；位於右則的波函數變量為 $x_{2}$ ，也就是 $\langle 12|12\rangle $ 。
我們注意到電子座標 $x_{1}$ 及 $x_{2}$ 是可交換的，所以
$$
\langle ij|kl\rangle $ = $\langle kl|ij\rangle
$$
在此基礎上可以進一步定義更加複雜的物理符號：
$$
\langle ij||kl\rangle =\langle ij|kl\rangle -\langle ij|lk\rangle
$$
這一表示也可改寫如下:
$$
\int dx_{1}dx_{2}\chi _{i}^{*}(x_{1})\chi _{j}^{*}(x_{2})*{\frac {1-P_{12}}{r_{12}}}\chi _{k}(x_{1})\chi _{l}(x_{2})
$$
其中 $P_{12}$ 為交換電子 1 及電子 2 的算子。
考慮到電子坐標的等價性和符號本身的數學意義，物理符號有如下性質：
$$
\langle ij||kl\rangle =\langle ji||lk\rangle
$$
$$
\langle ij||kl\rangle =-\langle ij||lk\rangle
$$
### 化學符號
化學符號的形式是 $[\chi _{i}\chi _{k}|\chi _{j}\chi _{l}]$ ，有時候也簡單地表示為 $[ik|jl]$ ，這一表示等價於：
$$
\int dx_{1}dx_{2}\chi _{i}^{*}(x_{1})\chi _{j}^{*}(x_{2}){\frac {1}{r_{12}}}\chi _{k}(x_{1})\chi _{l}(x_{2})
$$ | [
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量子化学中的双电子积分 | 分布在表示中單豎線之前的是電子坐標為 $x_{1}$ 的軌道波函數，分布在單豎線之後的是電子坐標為 $x_{2}$ ；的軌道波函數，在豎線同一側的兩個波函數中，位於左則的一個波函數須取復共軛，並在積分中位於算子的左側位於右則的波函數不取復共軛，並在積分中位於算子的右側。也就是 $[11|22]$ ，與物理符號 $\langle 12|12\rangle $ 相同的是，兩個同樣數字的（同樣電子座標）的軌域中，靠左邊的是取複共軛，靠右邊的是沒取的。
由於電子 1 與電子 2 的交換不影響積分結果。所以我們有 $[ij|kl]=[kl|ij]$ 。而在量子化學計算裡，波函數通常是實數，因此有 $[ij|kl]=[ji|kl]=[ij|lk]$ 。
組合以上關係，共有八種交換對稱：
$[ij|kl]=[ji|kl]=[ij|lk]=[ji|lk]$
$=[kl|ij]=[kl|ji]=[lk|ij]=[lk|ji]$
與物理符號一樣，化學符號也有更進一步的形式：
$$
[ik||jl]=[ik|jl]-[il|jk]
$$
由於將相同變量的波函數集中在符號的一側，因而化學符號在使用中比物理符號更方便，在量子化學計算中，出現的頻率更高。
在實際應用中還有約化掉自旋函數的化學符號：
$$
(ik|jl)
$$
在這個積分中，參與積分的軌道波函數僅僅含有空間部分，積分的變量也僅僅含有空間笛卡兒坐標，自旋函數以及自旋坐標被分離後單獨積分了，而空間函數的積分規則與化學符號 $[ik|jl]$ 完全一致。
### 物理符號與化學符號的關係
由兩者的表示規則可以得出兩者之間的關係為：
$$
\langle ij|kl\rangle =[ik|jl]
$$
$$
\langle ij||kl\rangle =[ik||jl]
$$
## 參見
量子化學
Hartree-Fock 方程 | [
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高能理論物理 | **粒子物理學**是研究組成物質和射線的基本粒子以及它們之間交互作用的一個物理學分支。由於許多基本粒子在大自然的一般條件下不存在或不單獨出現，物理學家只有使用粒子加速器在高能相撞的條件下才能生產和研究它們，因此粒子物理學也被稱為**高能物理學**。
## 次原子粒子
現代粒子物理學的研究集中在次原子粒子上。這些粒子的結構比原子要小，其中包括原子的組成部分如電子、質子和中子（質子和中子本身又是由夸克所組成的粒子）和放射和散射所造成的粒子如光子、微中子和緲子，以及許多其它奇特的粒子。
嚴格地說「粒子」這個稱呼不精確，粒子物理學中研究的所有的物體都遵守量子力學的規則，它們都顯示波粒二象性，根據不同的實驗條件它們顯示粒子的特性或波的特性。在物理理論中，它們既非粒子也非波，理論學家用希爾伯特空間中的狀態向量來描寫它們，詳細的理論基礎為量子場論。但按照粒子物理學的常規在這篇文章中這些物體依然被稱為「粒子」，雖然這些粒子也具有波的特性。
今天所知的所有基本粒子都可以用一個叫做粒子物理標準模型的量子場論來描寫。標準模型是目前粒子物理學中最好的理論，它包含 37 種基本粒子，這些基本粒子相互結合可以形成更加複雜的粒子。從 1960 年代以來實驗物理學家已經發現和觀察到了上百種複合粒子了。標準模型理論幾乎與至今為止觀察到的所有的實驗數據相符合。雖然如此，大多數粒子物理學家相信標準模型理論依然是一個不完善的理論，而一個更加基本的理論尚有待發現。最近發現的微中子靜質量不為零是第一個與標準模型出現偏差的實驗觀測。
## 歷史
前 6 世紀古希臘的哲學家就提出物質是由基本粒子組成的猜測。流西普斯、德謨克里特斯和伊比鳩魯是「原子論」的代表人物。17 世紀時艾薩克・牛頓也有過物質是由粒子組成的想法。1802 年約翰・道爾頓正式提出所有物質是由原子組成的理論。
1869 年季米特里・門捷列夫發表的元素週期表加深了原子論的設想。約瑟夫・湯姆森發現了原子中存在帶有負電荷、質量非常小的電子，認為原子是由質子和被束縛的電子組成的。歐尼斯特・拉塞福證明質子集中在非常緊密的原子核中。1932 年英國物理學家查兌克發現了中子，至此，人們認識到原子核是由質子和中子組成的，電子在原子核外運動。
20 世紀原子物理學和量子物理學的研究導致了裂變和聚變的發現和實驗成功。人類能夠將一個元素的原子轉換成另一個元素的原子。 | [
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高能理論物理 | 1950 年代和 60 年代中許多新的粒子被發現，它們被統稱為「粒子動物園」。直到 1970 年代粒子物理的標準模型建立，將大多數這些粒子看作是少數基本粒子的複合粒子後這個混亂才減輕。
## 標準模型理論
目前描寫基本粒子的最成功的理論是標準模型理論，它使用規範玻色子來描寫強交互作用、弱交互作用和電磁交互作用。光子、W 及 Z 玻色子和膠子都屬於規範玻色子。標準模型包含了 24 種基本費米子 (12 種基本粒子與對應之反粒子)，這些組成了現有的物質。最後這個理論還預言了希格斯玻色子。彼得・希格斯與弗朗索瓦・恩格勒爾後榮獲 2013 年諾貝爾物理學獎。
## 實驗粒子物理學
實驗粒子物理學通常藉由大量產生對撞物理事件，進而分析物理現象。早於 CERN 中期（1980 年代），此領域便需要集結大量不同國籍之研究員，全球資訊網便是為了因應研究員溝通之便利而誕生。因為需要處理大量數據，如今此領域之研究發展也與電腦科學息息相關。由於各個高能實驗經費正節節高升，以求更高能量、更高亮度或是更高的時空解析度 (對於時間和空間的最小辨識力)，通常經費來自於政府，大型強子對撞機已成為經費最高之高能物理實驗。因為嚴重排擠了政府其他項目之預算，高昂經費之高能物理實驗往後可能漸漸減少。
大的實驗粒子物理學國際合作有：
* 歐洲核子研究組織：位於法國和瑞士邊境日內瓦附近，其主要儀器如下：
* 大型正負電子對撞機（2001 年停用，現已拆除）
* 超級質子同步加速器
* 大型強子對撞機
* 德國電子加速器：位於德國漢堡，其主要設備是強子電子環設備（HERA），可用電子和正電子與質子相撞。
* SLAC 國家加速器實驗室：位於美國帕洛阿圖附近，其主要設備是 PEP-II，用來碰撞電子和正電子。
* 費米國立加速器實驗室：位於美國芝加哥附近，其主要設備是兆電子伏特加速器（Tevatron），碰撞質子與反質子。
* 布魯克黑文國家實驗室：位於美國長島，其主要設備是相對論性重離子對撞機，用來使重離子如金離子與質子相撞。
* 布德克核子物理研究所（BINP）：位於俄羅斯新西伯利亞。 | [
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高能理論物理 | * 超級神岡探測器: 1998 年，超級神岡探測器的領導者、日本科學家小柴昌俊發表了測量結果，給出微中子振盪的首個確切證據，認為微中子在三種不同「味」之間是可以相互轉換的，這也表明微中子是有質量的，而不是粒子物理標準模型中預言的零質量粒子。2002 年，超級神岡探測器證實反應爐中產生的微中子發生了振盪。這個探測結果在微中子天文學和粒子物理學中具有里程碑式的意義，小柴昌俊因此獲得 2002 年的諾貝爾物理學獎。
* 高能加速器研究機構：位於日本筑波，擁有一個測試微中子振盪的 K2K 和測試正反 B 介子違反電荷宇稱守恆性的 Belle 實驗。如今 (2018 年底) SuperKEK 加速器 (預期亮度為舊型 Belle 實驗 KEK 電子正子加速器之 40 倍) 正在進行建造與測試，Belle II 實驗相關偵測器與軟體也正在開發，以接收日後大量的物理事件數據。
* 大亞灣核反應爐微中子實驗：位於中國大亞灣核電站北側，主要物理目標是利用核反應爐產生的反微中子來測量微中子混合角 $\theta _{13}$ ，該項目 （頁面存檔備份，存於網際網路檔案館）由中科院高能物理研究所主持。
此外世界各地還有許多其它粒子加速器，比如中國大陸的北京正負電子對撞機與台灣新竹科學工業園區的國家同步輻射研究中心。
## 理論粒子物理學
理論粒子物理學試圖描述自然界的一切交互作用，研究能解釋今天實驗結果並能預言未來實驗結果的模型、理論構架和數學工具。今天在這方面有許多不同的努力。
一個重要的工作點是更好地理解標準模型理論和其實驗結果，從試驗中獲得更精確的參數，這個工作點測試標準模型理論的極限來擴大我們對自然的理解。這個工作最大的困難在於量子色動力學中對多個物體計算時的困難。一些理論家將他們的精力集中在有效場論。
另一個重要的工作點是建立超出標準模型理論的模型。由於今天的實驗數據還不夠，這個工作非常困難。新的理論結構有超對稱、阮桑模型、前子理論等等。
第三個重要的工作點是弦理論，其目的在於建立一種基於微小弦與膜而不是基於粒子的理論來統一描述量子力學和廣義相對論。如果這一理論取得成功，可以被看作一種「萬有理論」。
此外還有一些其它的理論工作如迴圈量子引力理論等。
## 還原論
還原論是將世界上的事物的解釋簡化到一些基礎的理論的哲學觀點。在粒子物理學這個觀點是提出一個可以解釋世界上的一切的一種最基礎的物理理論，或者用一個比較大眾化的語言來說，來尋找一個概括宇宙一切的公式。 | [
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高能理論物理 | 但在粒子物理學的發展過程中也一直有人批評這種極端的還原論。這些批評者中有粒子物理學家、化學家、生物學家、固態物理學家和整體論者。他們並不向標準模型理論本身挑戰，但他們認為基本粒子的特性並不一定也是它們所組成的原子、分子或更大的結構的特性，尤其是它們並不能表達很多粒子組成的系統的特性。基於混沌理論，一些批評者認為即使物理學家完全認識基本粒子的所有的特性的話，人們以此不能完全理解所有的自然的過程；另一些批評者懷疑人們能夠完全理解基本粒子的特性。
## 公共政策
粒子物理學的實驗結果需要使用巨大的粒子加速器才能取得。這些加速器非常昂貴（往往需要上十億美元）因此需要大量政府資助。因此粒子物理學的研究也關係到公共政策的決定。
許多人認為花這麼多的錢不值得，而且粒子物理學消耗了許多可以用到更重要的研究和教育方面的錢。20 世紀 80 年代，在美國德克薩斯州開始建造一台超級超導對撞機，這是個宏偉的計劃，費用高達 80 億美元，美國國會為此花掉了 20 億美元，在建成一條 22 公里的隧道後取消了這項工程。許多科學家（包括超導超大型加速器的支持者和反對者）相信這個決定的原因之一是因為冷戰結束後美國沒有必要花這麼多錢在這方面與蘇聯競爭了。
此外許多反對者懷疑單一國家是否還有能力運行如此昂貴的對撞機。
粒子物理學的支持者認為為最基本的理論值得花這麼多錢，這些錢對科學的其它方面也有好處。例如加速器與其副產品同步輻射在生物與醫學上的應用，以及最早由 CERN 研究員所創立的全球資訊網。他們指出今天所有的加速器都是國際合作建立和運行的，他們懷疑取消製造加速器所節省下來的預算仍會使用在其它科學和教育的方面上。
## 展望
世界各地的粒子物理學家對粒子物理學近期和中期最重要的目標的見解是一致的。近期的目標是於 2007 年完成大型強子對撞機並用它來尋找希格斯玻色子和超對稱粒子。中期的目標是建造國際直線對撞機（International Linear Collider, ILC）。這個對撞機的技術實現方法已於 2004 年 8 月決定，但其地址還沒有決定。國際直線對撞機與大型強子對撞機是互相補充的實驗設備，大型強子對撞機更適合用來尋找新的粒子，而國際直線對撞機則更適合用來精確地測量這些粒子的特性。
粒子物理學的其它重要目標包括測量微中子的靜質量和澄清質子的雙重 β 衰變是否存在。這些實驗不一定需要使用對撞機。
## 參考文獻
## 參見
* 基本粒子
* 原子物理學 | [
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] |
高能理論物理 | * 原子核物理學
* 量子物理學
## 外部連結
* 美國高能量物理諮詢小組的文章 Quantum Universe: The Revolution in 21st Century Particle Physics 列出粒子物理學與天體物理學的九個基礎問題。 | [
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] |
波恩－奥本海默近似 | **波恩–歐本海默近似**（Born-Oppenheimer approximation，簡稱 **BO 近似**，又稱**絕熱近似**）是一種普遍使用的解包含電子與原子核的體系的量子力學方程式的近似方法。
在用量子力學處理分子或其他體系時，需要通過解薛定鍔方程式或其他類似的偏微分方程式獲得體系波函數。這個過程往往由於體系自由度過多而非常困難，甚至無法進行。據波恩－歐本海默近似中，考慮到原子核的質量要比電子大很多，一般要大 3-4 個數量級，因而在同樣的交互作用下，電子的移動速度會較原子核快很多，這一速度的差異的結果是使得電子在每一時刻仿佛運動在靜止原子核構成的勢場中，而原子核則感受不到電子的具體位置，而只能受到平均作用力。由此，可以實現原子核坐標與電子坐標的近似變量分離，將求解整個體系的波函數的複雜過程分解為求解電子波函數和求解原子核波函數兩個相對簡單得多的過程。
在波恩－歐本海默近似下，體系波函數可以被寫為電子波函數與原子核波函數的乘積
: : $\Psi _{\mathrm {total} }=\chi _{\mathrm {electronic} }\times \phi _{\mathrm {nuclear} }$
波恩－歐本海默近似由於在大多數情況下非常精確，又極大地降低了量子力學處理的難度，被廣泛應用於分子結構研究、凝聚體物理學、量子化學、化學反應動力學等領域。
波恩–歐本海默近似是由物理學家歐本海默與其導師波恩共同提出的。
## 利用波恩－歐本海默求解薛定鍔方程式的步驟
首先將體系的哈密頓算符分解為原子核動能算符與電子哈密頓算符兩項
: ${\hat {H}}={\hat {T}}_{n}+{\hat {H}}_{e}$
其中電子哈密頓算符包含所有靜電交互作用 (電子－電子，電子－核及核－核交互作用)，以及電子的動能算符。
接下來，解以下的電子薛丁格方程式以獲得電子波函數和絕熱位能面：
: ${\hat {H}}_{\mathrm {e} }\;\chi (\mathbf {r,R} )=E_{\mathrm {e} }(\mathbf {R} )\;\chi (\mathbf {r,R} )$ | [
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波恩－奥本海默近似 | 這一步一般被稱為電子結構計算，由於電子的數量往往遠大於原子核，這一步往往是整個計算最昂貴的一步。根據計算能力，體系複雜程度和精度的需要，可以採用各種從頭計算法直接近似求解，也可以採用密度泛函理論，半經驗方法，分子力學勢場等一系列方法獲得位能面。
獲得位能面之後，可以用其求解原子核的定態薛丁格方程式得到原子核波的定態函數和體系能量
: $\left[{\hat {T}}_{\mathrm {n} }+E_{\mathrm {e} }(\mathbf {R} )\right]\phi (\mathbf {R} )=E\phi (\mathbf {R} )$
或通過求解含時薛丁格方程式模擬原子核波函數隨時間的演化
: $\left[{\hat {T}}_{\mathrm {n} }+E_{\mathrm {e} }(\mathbf {R} )\right]\phi (\mathbf {R;t} )=i\hbar {\frac {\partial }{\partial t}}\phi (\mathbf {R;t} )$
這兩種方法獲得的結果可以互相轉化。原子核定態波函數描述了分子的振動，一般用來獲得振動零振動點能，進行紅外、微波光譜或電子光譜振動與轉動結構的計算；而波函數隨時間的演變則描述了化學反應的動力學過程。
注意到第二步原子核薛丁格方程式的求解中，需要電子態能量以原子核位置為坐標的函數，因而往往需進行多次，乃至數千萬次電子結構的計算，造成計算嚴重的困難。即使在小分子體系中，往往也無法採用完全量子力學的方法求解波函數。由於原子核的質量比電子大得多，量子效應不如電子明顯，因此這一步往往採用比電子結構步驟更高程度的近似方法，以減輕計算負擔。例如常採用古典力學或者半古典近似處理原子核運動；在振動幅度較小的情況下，可將位能面近似為二次曲面，而原子核的運動則轉化為量子諧振子問題而可解析解出。某些情況下亦可採用平衡態的近似，運用統計力學或過渡態理論等統計方法則，只需要在少量關鍵位置進行電子結構和振動計算就得到關於分子的分布或反應速率等資訊。
## 適用性
波恩 - 歐本海默近似只有在所在電子態與其他電子態能量都足夠分離的情況下才有效。當電子態出現交叉或者接近時，波恩－歐本海默近似即失效。
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] |
波恩－奥本海默近似 | 對於基態，不加其他近似的情況下通過波恩－歐本海默近似得出的體系總能量一定小於體系真實能量，因而給出了真實能量的下限。與此相對，另一絕熱近似方法波恩－黃近似則給出體系真實能量的上限。
## 相關條目
* 非絕熱耦合
* 薛丁格方程式
* 波恩－黃近似
* 波恩定則
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稳定岛理论 | **穩定島理論**是核子物理中的一個理論推測，核物理學家推測原子核的質子數和中子數為「幻數」的超重元素會特別穩定。假如這個猜測正確的話，那麼某些特定的超重元素的同位素將比其他同位素更穩定，這些同位素的放射性衰變過程可能會相對非常慢。
穩定島理論最初是格倫・西奧多・西博格提出的。他認為原子核中的核子與原子核外的電子一樣是分布在不同的「殼層」中的，實際上每個殼層就是一組相近的量子能階。不同殼層之間的能量差則比較大。假如一個原子核中的質子和中子正好填滿一個殼層的話，那麼每個核子之間的結合能就會達到最大，因此這個原子核就最穩定，比其附近核子沒有填滿或超出一個殼層的同位素要穩定。
飽和的殼層的中子和質子數被稱為「幻數」（也稱為「魔數」），中子和質子數同時為幻數的原子核被稱為「雙幻核」。一個可能的中子魔數是 184，可能的質子魔數是 114、120 和 126，也就是說，298Fl、304Ubn 和 310Ubh 可能比較穩定。這些同位素至今為止未能被合成。但 Fl 的帶有 114 個質子和少於 184 個中子的同位素比元素週期表中鄰近的元素的同位素的衰變要慢得多。
處於穩定島的元素非常可能依然是放射性元素，只是它們相對於其附近的同位素「比較穩定」，雖然有人懷疑有些同位素的半衰期可能大於一日或甚至更長，但也可能小於一秒，而在目前發現的元素中，289Fl 的半衰期為 2.6 秒，285Cn 的半衰期為 11 分鐘，293Lv 的半衰期有 0.05 秒，跟附近的元素比起來是很驚人的。
#### 註解
## 參見
* 鈇
* 鎶
* 鉝
* 幻數 (物理學)
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對偶向量空間 | 在數學裡，任何向量空間 _V_ 都有其對應的對偶向量空間（或簡稱為對偶空間），由 _V_ 的線性泛函組成。此對偶空間俱有一般向量空間的結構，像是向量加法及純量乘法。由此定義的對偶空間也可稱之為代數對偶空間。在拓撲向量空間的情況下，由連續的線性泛函組成的對偶空間則稱之為連續對偶空間。
**對偶空間**是 行向量（ $1\times n$ ）與列向量（ $n\times 1$ ）的關係的抽象化。這個結構能夠在無限維度空間進行並為測度，分佈及希爾伯特空間提供重要的觀點。**對偶空間**的應用是泛函分析理論的特徵。傅立葉變換亦內蘊對偶空間的概念。
## 代數對偶空間
設 $V$ 為 在域 $F$ 上的向量空間，定義其**對偶空間** $V^{*}$ 為由 _$V$_ 到 _$F$_ 的所有線性函數的集合。
即是 _$V$_ 的標量線性變換。 $V^{*}$ 本身是 $F$ 的向量空間，並且對所有 $V^{*}$ 中的 $\phi $ 及 $\varphi $ 、所有 _$F$_ 中的 $a$ 、所有 _$V$_ 中的 $x$ 滿足以下加法及標量乘法：
: $(\phi +\varphi )(x)=\phi (x)+\varphi (x)\,$
: $(a\phi )(x)=a\phi (x)\,$
在張量的語言中， _$V$_ 的元素被稱為反變或逆變（contravariant）向量，而 $V^{*}$ 的元素被稱為共變或協變（covariant）向量、「餘向量」或「同向量」（co-vectors），「線性型」或「1 - 形式」（one-form）。
### 例子
如果 _$V$_ 是有限維的， $V^{*}$ 的維度和 V 的維度便相等；
如果 $\left\{e_{1},...,e_{n}\right\}$ 是 _$V$_ 的基， $V^{*}$ 便應該有相對基 $\left\{e^{1},...,e^{n}\right\}$ ，記作：
: $e^{i}(e_{j})=\left\{{\begin{matrix}1,&{\mbox{if }}i=j\\0,&{\mbox{if }}i\neq j\end{matrix}}\right.$
如果 _$V$_ 是平面幾何向量的空間， $V^{*}$ 便是一組組的平行線。我們能從平行線應用到任何向量產生一個標量。 | [
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對偶向量空間 | 如果 _$V$_ 是無限維度， $e^{i}$ 不能產生 $V^{*}$ 的基；而 $V^{*}$ 的維度比 _$V$_ 的大。
例如空間 $R^{(\omega )}$ 的元素是實數列，其擁有很多非零數字。 $R^{\omega }$ 的雙對空間是所有實數數列的空間。這些數列 $(a_{n})$ 被用於元素 $(x_{n})$ 而產生 $\sum _{n}a_{n}x_{n}$ 。
### 線性映射的轉置
設 $f:V\rightarrow W$ 是線性映射。 $f$ 的_轉置_ ${}^{t}\!f:W^{*}\rightarrow V^{*}$ 定義為
:
對任何向量空間 $V,W$ ，定義 $L(V,W)$ 為所有從 $V$ 到 $W$ 的線性映射組成的向量空間。 $f|\rightarrow {}^{t}\!f$ 產生從 $L(V,W)$ 至 $L(W^{*},V^{*})$ 的單射；這是個同構若且唯若 $W$ 是有限維的。
若 線性映射 _f_ 表示作其對 $V,W$ 的基之矩陣 $A$ , 則 ${}^{t}\!f$ 表示作其對 $V^{*},W^{*}$ 的對偶基之轉置矩陣。
若 $g:W\rightarrow X$ 是另一線性映射，則 ${}^{t}\!\left(g\circ f\right)={}^{t}\!f\circ {}^{t}\!g$ 。
在範疇論的語言裡，_為任何向量空間取對偶_及_為任何線性映射取轉置_都是向量空間範疇的逆變函子。
### 雙線性乘積及對偶空間
正如所見，如果 $V$ 擁有有限維度， $V$ 跟 $V^{*}$ 是同構的，但是該同構並不自然；它是依賴於我們開始所用的 $V$ 的基。事實上，任意同構 $\phi =\left(V\rightarrow V^{*}\right)$ 在 $V$ 上定義了一個唯一的非退化的雙線性型：
: $\langle v,w\rangle =(\Phi (v))(w)\,$
相反地從每個在有限維空間中的非退化的雙線性積可以產生由 $V$ 映射到 $V^{*}$ 的同構。
### 到雙對偶空間內的單射
存在一個由 $V$ 到其雙對偶 $V^{**}$ 的自然映射 $\psi $ ，定義為
$$
\left(\psi (v)\right)(\varphi )=\varphi (v)\forall v\in V,\varphi \in V^{*}
$$ | [
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對偶向量空間 | $\psi $ 常是單射；若且唯若 $V$ 的維數有限時， $\psi $ 是個同構。
## 連續對偶空間
處理拓撲向量空間時，我們一般僅對該空間射到其基域的連續線性泛函感興趣。由此導致**連續對偶空間**之概念，此乃其代數對偶空間之一子空間。向量空間 _$V$_ 之連續對偶記作 _$V$_ ′。此脈絡下可逕稱連續對偶為_對偶_。
線性賦範向量空間 _$V$_ （如一巴拿赫空間或一希爾伯特空間）之連續對偶 _$V$_ 產生一線性賦範向量空間。對一 _$V$_ 上之連續線性泛函，其範數 $\left\Vert \varphi \right\Vert $ 定義為
: $\|\phi \|=\sup\{|\phi (x)|:\|x\|\leq 1\}$
此法變一連續對偶為一線性賦範向量空間，實為巴拿赫空間。
### 例子
對任意有限維之線性賦範向量空間或拓撲向量空間，正如歐幾里得空間，其連續與代數對偶不二。
令 $1<p<\infty $ 為實數，並考慮所有序列 $a=(a_{n})$ 構成之巴拿赫空間 _l__p_，使其範數
: $\|\mathbf {a} \|_{p}=\left(\sum _{n=0}^{\infty }|a_{n}|^{p}\right)^{1/p}$
有限。以 ${\frac {1}{p}}+{\frac {1}{q}}=1$ 定義 $q$ ， $I^{p}$ 其連續對偶遂自然等同於 $I^{q}$ ：給定一元素 $\varphi \in (I^{p})$ ， $I^{q}$ 中相應元素為序列 $\left(\varphi (e_{n})\right)$ ，其中 $e_{n}$ 謂第 $n$ 項為 1 且餘項皆 0 之序列。反之，給定一元素 $a=(a_{n})\in I^{q}$ ， $I^{p}$ 上相應之連續線性泛函 $\varphi $ 定為 $\varphi (a)=\sum _{n}a_{n}x_{n}$ （對一切 $a=(a_{n})\in I^{p}$ ，見 Hölder 不等式）。
準此， $I^{1}$ 之連續對偶亦自然同構於 $I^{\infty }$ 。再者，巴拿赫空間 $c$ （賦以上確界範數之全體收斂序列）及 $c_{0}$ （ _$c$_ 中收斂至零者）之連續對偶皆自然同構於 $I^{1}$ 。
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對偶向量空間 | 若 _$V$_ 為希爾伯特空間，則其連續對偶亦然，並反同構於 _$V$_ ；此即是里斯表示定理的陳述，同時也啟發了量子力學之數學描述時所用的狄拉克符號。
類似雙重代數對偶，對連續線性算子亦有連續單射 $\psi :V\rightarrow V''$ ，此映射實為等距同構，即 $\left\Vert \psi (x)\right\Vert =\left\Vert x\right\Vert $ 對一切 _$V$_ 中 $x$ 皆真。使 $\psi $ 為雙射之空間稱自反空間。
連續對偶賦 _$V$_ 以一新拓撲，稱之為弱拓撲。
若 _V_ 之對偶可分，則 _$V$_ 亦可分。反之則不然：考慮空間 $I_{1}$ ，則其對偶 $I\infty $ 不可分。
## 引用
* Bourbaki, Nicolas. Elements of mathematics, Algebra I. Springer-Verlag. 1989. ISBN 3-540-64243-9.
* Paul Halmos. Finite dimensional vector spaces. Springer. 1974. ISBN 0387900934.
* Walter Rudin. Functional analysis. McGraw-Hill Science. 1991. ISBN 978-0070542365. | [
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光的衍射 | **繞射**（英語：Diffraction），又稱**衍射**，是指波遇到障礙物時偏離原來直線傳播的物理現象。:559-560
在古典物理學中，波在穿過狹縫、小孔或圓盤之類的障礙物後會發生不同程度的彎散傳播。假設將一個障礙物置放在光源和觀察屏之間，則會有光亮區域與陰暗區域出現於觀察屏，而且這些區域的邊界並不銳利，是一種明暗相間的複雜圖樣。這現象稱為繞射，當波在其傳播路徑上遇到障礙物時，都有可能發生這種現象。:519 除此之外，當光波穿過折射率不均勻的介質時，或當聲波穿過聲阻抗不均勻的介質時，也會發生類似的效應。在一定條件下，不僅水波、光波能夠產生肉眼可見的繞射現象，其他類型的電磁波（例如 X 射線和無線電波等）也能夠發生繞射。由於原子尺度的實際物體具有類似波的性質，它們也會表現出繞射現象，可以透過量子力學進行研究其性質。
在適當情況下，任何波都具有繞射的固有性質。然而，不同情況中波發生繞射的程度有所不同。如果障礙物具有多個密集分布的孔隙，就會造成較為複雜的繞射強度分布圖樣。這是因為波的不同部分以不同的路徑傳播到觀察者的位置，發生波疊加而形成的現象。
繞射的形式論還可以用來描述有限波（量度為有限尺寸的波）在自由空間的傳播情況。例如，雷射束的發散性質、雷達天線的波束形狀以及超音波傳感器的視野範圍都可以利用繞射方程式來加以分析。
## 繞射與干涉的關係
美國物理學家、諾貝爾物理學獎得主理察・費曼指出：:30-1
他還提到，如果只有少數的波源（例如兩個的時候），我們稱這現象為「干涉」，例如我們稱楊氏雙縫實驗實驗中雙縫所產生的兩束光源產生了干涉現象。而當大量波源存在時，對應的過程被稱作是「繞射」。在實際情況中，繞射和干涉往往是同時出現的。有文獻這樣總結：干涉是有限多個波束「相加」的結果，而繞射則是無限多個波束「積分」的結果。
## 研究歷史
光的繞射效應最早是由弗朗西斯科・格里馬第（Francesco Grimaldi）發現並加以描述，他也是「繞射」一詞的創始人。這個詞源於拉丁語詞彙 _diffringere_，意為「成為碎片」，即波原來的傳播方向被「打碎」、彎散至不同的方向。格里馬第觀察到的現象直到 1665 年才被發表，這時他已經去世。他提出
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] |
光的衍射 | > 「光不僅會沿直線傳播、折射和反射，還能夠以第四種方式傳播，即透過繞射的形式傳播。」（"Propositio I. Lumen propagatur seu diffunditur non solum directe, refracte, ac reflexe, sed etiam alio quodam quarto modo, diffracte."）:149:95
>
>
英國科學家艾薩克・牛頓對這些現象進行了研究，他認為光線發生了彎曲，並認為光是由粒子構成。在 19 世紀以前，由於牛頓在學界的權威，光微粒說在很長一段時間占有主流位置。這樣的情況直到 19 世紀幾項理論和實驗結果的發表，才得以改變。1803 年，托馬斯・楊進行了一項非常著名的實驗，這項實驗展示了兩條緊密相鄰的狹縫造成的干涉現象，後人稱之為「雙縫實驗」。在這個實驗中，一束光照射到具有緊挨的兩條狹縫的遮光擋板上，當光穿過狹縫並照射到擋板後面的觀察屏上，可以產生明暗相間的條紋。他把這歸因於光束透過兩條狹縫後繞射產生的干涉現象，並進一步推測光一定具有波動的性質。奧古斯丁・菲涅耳則對繞射做了更多權威的計算研究，他的結果分別於 1815 年:卷 1，239-281 和 1818 年:33-475 被發表，他提到
> 「這樣，我就展示了人們能夠透過何種方式來構想光以球面波連續不斷地傳播出去……」（ "J'ai donc montré de quelle façon l'on peut concevoir que la lumière s'étend successivement par des ondes sphériques, ..."）:章 1,p18
>
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法國科學院曾經舉辦了一個關於繞射問題的有獎辯論會，菲涅耳贏得了這次辯論。作為反對光波動說的其中一位，西莫恩・德尼・帕松提出，如果菲涅耳聲稱的結論是正確的，那麼當光射向一個球的時候，將會在球後面陰影區域的中心找到亮斑。結果，評審委員會安排了上述實驗，並發現了位於陰影區域中心的亮斑（它後來被稱作帕松光斑）。這個發現極大地支持了菲涅耳的理論。:940 他的研究為克里斯蒂安・惠更斯發展的光的波動理論提供了很大的支持。他與楊的理論共同反駁了牛頓關於光是粒子的理論。 | [
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光的衍射 | 在對繞射現象的探索過程中，人們也不斷積累了對於繞射光柵的認識。17 世紀，蘇格蘭數學家、天文學家詹姆斯・格雷戈里在鳥的羽毛縫間觀察到了陽光的繞射現象。他是第一個發現繞射光柵原理的科學家。在 1673 年 5 月 13 日他寫給約翰・科林斯（John Colins）的一封信中提到了此發現。:卷 2：251-255；1786 年，美國天文學家戴維・里滕豪斯用螺絲和細線第一次人工製成了繞射光柵，細線的密度達到每英寸 100 線，他用這個裝置成功地看到了陽光的繞射。1821 年，約瑟夫・夫琅禾費利用相似的裝置（每厘米 127 線）證明了托馬斯・楊關於繞射的公式（參見段落下方），並對繞射進行了許多重要研究。1867 年，劉易斯・拉塞福（Lewis Morris Rutherfurd）採用水輪機作為動力進行刻線、製作光柵。後來的亨利・奧古斯塔斯・羅蘭 改良了光柵的刻劃技術，並在 1882 年發明了在凹形球面鏡上進行刻劃的凹面光柵。其後的羅伯特・伍德（Robert William Wood）改進了光柵的刻劃形狀，從而提高了光柵的繞射效率。近代的阿爾伯特・邁克生提出利用干涉伺服系統控制光柵的刻划過程，於 1948 年實現了這一想法。20 世紀下半葉，由於雷射、光刻膠等新技術的出現，光柵製造技術取得很大的進步，製造成本顯著降低，製造週期也得以縮短。
## 日常生活中的實例
繞射效應在日常生活中並不罕見。許多有關光的繞射實例都可以用肉眼觀察到。例如，在 CD 或 DVD 光碟的表面，均勻地緊密排列著一系列的光軌，這些光軌相當於繞射光柵的作用。如果以一定的角度觀察它們，會看到光在盤面表現出類似彩虹的彩色圖樣。將上述現象的基本原理加以利用，很多產生有意思繞射圖樣的繞射光柵，都可以被製備出來。繞射也是信用卡等所採用的全像攝影的技術基礎之一。地球的大氣層是由微小粒子組成的，因此它也能夠使空間光源（例如太陽或者月亮）的光在大氣層發生繞射，從而形成光環。此外，當雷射照射到粗糙的光學界面上時，也能夠發生繞射現象，產生散斑。上述所有例子都是光具有波動性的結果。
繞射是一切波的固有屬性。即使是宏觀的海浪，在防波堤或其他障礙物附近也能夠發生繞射。此外，聲波在障礙物邊緣發生繞射，也是人站在障礙物（例如牆壁、樹木）後面仍然能夠聽到聲音的原因之一。:102 不過，繞射也為照相機、望遠鏡和顯微鏡等光學儀器的的解析度設定了一定的限制。:190-192
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光的衍射 | 光波（或其他波）傳播的路徑不同，可能造成繞射現象的發生。可以用惠更斯－菲涅耳原理和波的疊加原理對現象進行描述。這個理論認為，可以把波前的每一點考慮為次波（球面波）的點波源，這些次波就是後續時刻的波面。這個原理最早由惠更斯於 17 世紀提出，不過他並未慮及波的時空週期性（他認為光是一種非週期性的、無規則的脈衝）。1818 年左右，菲涅耳在巴黎科學院關於解釋繞射現象的有獎競賽中，吸收了惠更斯「次波」的思想，並加入了他對於干涉現象的理解，使上述理論得以發展和完善。後人將這個理論稱為「惠更斯－菲涅耳原理」。根據這一理論，任意後續位置的波位移等於這些次波求和。求和並非簡單的代數和，而必須慮及這些波各自的相對相位以及振幅。因此，它們疊加之後的振幅範圍介於 0（相互完全抵消）和所有次波振幅的代數總和之間。我們可以透過光學實驗，觀察到光波的繞射圖樣。光的繞射圖樣通常具有一系列明暗條紋（分別對應光波振幅的最大值和最小值）。:189
人們為了分析波的繞射現象，構造了許多數學模型，其中包括從波動方程式推導出的菲涅耳－克希荷夫繞射公式、夫朗和斐繞射模型以及菲涅耳繞射模型。:198-200 設 $a$ 為圓孔半徑或狹縫寬度， $\lambda $ 為入射波的波長， $L$ 為觀察屏距離圓孔、狹縫等繞射物體的距離，如果它們滿足
: $F={\frac {a^{2}}{L\lambda }}\geq 1$
我們就稱其為菲涅耳繞射，它是繞射的近場近似；
如果它們滿足
: $F={\frac {a^{2}}{L\lambda }}\ll 1$
我們就稱其為夫朗和斐繞射，它是繞射的遠場近似。
大多數情況，獲得繞射方程式的嚴格解析解較為困難，:559 可以透過有限元分析和邊界元分析方法來求得數值解。實際的繞射過程通常很複雜，不過，如果能夠將實際情況簡化到二維平面上，則對於繞射的數學描述將變得相對簡單。例如，水波就可以近似地看做是分布在二維平面上的機械波。而對於光波，如果它遇到的繞射物體在某一個方向的尺度遠大於光的波長，從而造成這個方向的繞射現象不顯著，那麼，在分析計算時可以將其忽略，這樣做並不會嚴重影響分析結果。例如，狹縫問題就可以簡化到二維的情況，這是因為其沿著縫隙方向的長度和入射光波長相差甚遠，因此我們只需考慮它寬度和厚度這兩個方向。然而，當我們考慮入射光穿過圓孔時，則必須完整地考慮其三維方向光的傳播細節。:563-570
## 光的繞射
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] |
光的衍射 | 假設有一個不透明擋板，用小刀在上面刻一條狹長、筆直、透光的縫，然後在擋板的後面放置一個觀察屏。照射單色平行光（collimated light）在這個擋板上。按照幾何光學，觀察屏上只會有一條與狹縫輪廓相同的亮條紋。然而，精細的觀察可以發現在這條亮條紋的兩側，對稱地分布著一些亮條紋。發生這樣現象是因為光在狹縫處發生了繞射。
假設狹縫寬度大於光波的波長，那麼當這束光穿過狹縫後，會向擋板後的區域傳播，並在那裡發生干涉現象。實際上，狹縫的縫寬之間均勻分布著大量點光源，繞射圖樣是這些點光源的共同作用結果。為了簡化對於該過程的分析，限定入射光具有單一的波長、都是單色光（頻率相同），並且在波源位置具有相同的初始相位。在狹縫後面的區域中任意位置的光是上述所有點光源的「次光波」在那位置的疊加結果。:189 因為次光波從狹縫的每個點光源到給定點所經過的路徑不同，所以它們的光程不同，因此它們在給定點的相位將會不同。對於縫間任意兩個點光源，假若分別來自它們的次光波在觀察屏給定點的相對相位為 $2\pi $ ，則這兩個次光波會干涉相長；假若相對相位為 $\pi $ ，則這兩個次光波會干涉相消。:104 從這概念，可以找到繞射光強的極大值或極小值。在繞射圖樣中，它們分別表現為明暗條紋。
透過下面的推導，:941-942 可以找到繞射光波的第一個極小值在觀察屏上的位置。將寬度為 $d$ 的狹縫均分為上下兩段，每段長度分別為 $d/2$ 。考慮來自上段頂部的一束光與來自下段頂部（即狹縫中點）的一束光（波長為 $\lambda $ ），這兩個點光源的距離為狹縫長度的一半。當兩束光傳播到觀察屏上距離中央極大值最近的位置（此處到狹縫中點連線與狹縫垂直平分線的夾角為 $\theta $ ），兩束光的光程差等於半個波長，即
: $(d/2)\,\sin \theta =\lambda /2$ | [
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光的衍射 | 時（在等式兩邊同時乘以 2，可以得到 $d\,\sin \theta =\lambda $ ），二者將發生干涉相消。現在考慮上段中點和下段中點發出的兩束光，如果它們在相應位置的光程差也等於半個波長，則也能發生相似的干涉相消現象。注意，在上面的討論中，我們已經假定狹縫與觀察屏的間距遠大於狹縫的寬度 $L\gg d$ ，這樣就可以近似認為狹縫間諸點光源以相同的角度 $\theta $ 平行地傳播到第一極小值位置。可以想像，狹縫其他位置任意兩個點光源，只要滿足 $d\,\sin \theta _{min,1}=\lambda $ ，那麼都會在上述位置形成干涉相消，形成第一級暗紋。
回想先前的假設為狹縫寬度大於光波波長。注意到狹縫寬度越小，同時保持波長不變，則 $\sin \theta _{min,1}=\lambda /d$ 越大， $\theta _{min,1}$ 也越大，因此觀察屏展示的第一級暗紋離開中央越遠，直到當狹縫寬度等於光波波長時， $\theta _{min,1}=\pi /2$ ，在觀察屏表面再也找不到第一級暗紋，整個觀察屏都被明紋覆蓋了。所以，只有當狹縫寬度大於光波波長時，才能夠展示出繞射的干涉圖樣。
上面考慮了第一極小值的情況。可以仿照上面的方法，將狹縫均分為 4 段、6 段、8 段…… $2n$ 段，則 $n$ 級繞射極小值位置的繞射角滿足下面的方程式
: $d\,\sin \theta _{min,n}=n\lambda $
這裡 $n$ 是非零整數，表示第 $n$ 級暗紋（極小值）。
此外，輻照度分布可以由夫朗和斐繞射方程式給出：
: $I(\theta )=I_{0}\,\operatorname {sinc} ^{2}(d\pi \sin \theta /\lambda )$
這裡
* $I(\theta )$ 為給定角度位置處的輻照度
* $I_{0}$ 初始輻照度
* 當 $x\neq 0$ 時， $\operatorname {sinc} (x)=\sin(\pi x)/(\pi x)$ ，在原點處 $\operatorname {sinc} (0)=1$
#### 菲涅耳半波帶法
菲涅耳半波帶法是單縫繞射中一種簡易的分析方法。
將平行入射光分成 $k$ 等分，其中
: $d\,\sin \alpha =2k(\lambda /2)$ ，（入射光半波長的偶數倍），對應暗紋中心。 | [
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光的衍射 | : $d\,\sin \alpha =(2k+1)(\lambda /2)$ ，（入射光半波長的奇數倍），對應明紋中心。
### 雙縫繞射
當我們討論雙縫干涉時，為了簡化問題，常常假設縫的寬度遠小於入射光的波長。這樣，在觀察屏上就可以看到輻照度近似相等的干涉條紋。事實上，在真實的實驗並不總能滿足上述假設。呈現在觀察屏上的亮條紋是中央最亮，兩側亮度逐漸衰減。因此，實際產生的圖樣是干涉、繞射效應的總和。簡單地說，實際雙縫實驗的條紋，具有理想雙縫干涉中條紋的位置，但是輻照度在觀察屏上的分布類似單縫繞射中央強、兩側弱的情況。:950
考慮到繞射效應，實際的雙縫干涉圖樣的輻照度可以用以下公式計算
: $I(\theta )=I_{m}(\cos ^{2}\beta )\left({\frac {\sin \alpha }{\alpha }}\right)^{2}$
其中， $\beta ={\frac {\pi d}{\lambda }}\sin \theta $ 為干涉因子，源於縫間距為 $d$ 的雙縫干涉效應；而 $\alpha ={\frac {\pi a}{\lambda }}\sin \theta $ 為繞射因子，源於縫寬為 $a$ 的單縫繞射效應。:950
上述公式表明，實際的雙縫干涉是干涉和繞射的共同效應。如果考慮問題時把縫寬忽略，把入射波考慮成來自少數幾個具有相同相位的波源，那麼就稱看到的現象為「干涉」；如果把入射波考慮成來自同相位的波陣面（縫寬方向的大量點波源），那麼就稱看到的現象為「繞射」。這樣的說法只是為了分析問題方便，事實上二者常常是同時發生的。:950
### 繞射光柵
繞射光柵是狹縫按照一定規律分布的光學裝置，它能夠調整入射光的相位、振幅等屬性，使透過它的光發生繞射、干涉，以達到所需的實驗目的。光穿過繞射光柵後形成的圖樣形狀與光柵的結構和數量都有關係。
所有繞射光柵的 $m$ 級極大繞射角 $\theta _{m}$ 滿足下列光柵方程式:229-230
: $d\left(\sin {\theta _{m}}+\sin {\theta _{i}}\right)=m\lambda .$
這裡
* $\theta _{i}$ i 為光波入射到光柵的角度，如果是垂直入射到平面光柵，則 $\sin {\theta _{i}}=0$
* $d$ 為光柵刻線的間距，也成為光柵常數
* $m$ 為非零整數 | [
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光的衍射 | 繞射光柵後面給定位置的光波，是繞射光柵諸狹縫繞射光的疊加。用於分離白光中不同頻率成分光的分光計，就利用了繞射光柵的原理。:955
下面位於中間的這幅圖顯示了具有相同縫間距的雙縫光柵和五縫光柵的繞射圖樣。可以看出，繞射光加強點的位置是相同的，但是光斑的寬度有所不同。
* * *
#### 繞射光柵強度分布
繞射光柵的強度分布是繞射因子和干涉因子的乘積：
$$
P=D(\theta )\cdot I(\theta )
$$
其中 D 是 繞射因子
$$
D={\frac {\sin(\pi \cdot d\cdot \sin(\theta )/\lambda )^{2}\cdot \lambda ^{2}}{(\pi ^{2}\cdot d^{2}\cdot \sin(\theta )^{2})}}
$$
I 是干涉因子：
$$
I={\frac {\sin(\pi \cdot a\cdot \sin(\theta )\cdot N/\lambda )^{2}}{(N^{2}\cdot \sin(\pi \cdot a\cdot \sin(\theta )/\lambda )^{2})}}
$$
### 圓孔繞射
當平面光波入射穿過圓孔時，會形成圓對稱的繞射圖樣，其中心部分最亮，這表示能量主要集中在其零級繞射斑處。上述圖樣的中心亮斑常被稱為艾里斑（Airy disk）。:214:189-190 根據遠場近似，:189
: $I(\theta )=I_{0}\left({\frac {2J_{1}(ka\sin \theta )}{ka\sin \theta }}\right)^{2}$
式中， $a$ 為環狀孔隙的半徑， $k$ 為波向量量的大小，等於 $2\pi /\lambda $ ， $J_{1}$ 為一階貝索函數。孔隙越小，則給定位置的光斑越大，並且繞射光束偏離原來的傳播方向越嚴重。沿著徑向，艾里斑的繞射圖樣表現為一系列明暗相間的同心圓環，不過徑向距離越大，各個明亮圓環的亮度越低。:189 艾里斑對光學儀器的成像品質有一定的影響。
### 一般孔隙的情況
: 在這篇文章內，向量與純量分別用粗體與斜體顯示。例如，位置向量通常用 $\mathbf {r} \,\!$ 表示；而其大小則用 $r\,\!$ 來表示。檢驗變數或場變數的標記的後面沒有單撇號「 $'\,\!$ 」；源變數的標記的後面有單撇號「 $'\,\!$ 」。 | [
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光的衍射 | 從馬克士威方程式，可以推導出在自由空間裏，電場 $\mathbf {E} $ 、磁場 $\mathbf {B} $ 的波動方程式：:246
: $\nabla ^{2}\mathbf {E} -\ {\frac {1}{c^{2}}}\ {\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}=-{\frac {1}{\epsilon _{0}}}\left(-\nabla \rho -\ {\frac {1}{c^{2}}}\ {\frac {\partial ^{2}\mathbf {J} }{\partial t^{2}}}\right)$
: $\nabla ^{2}\mathbf {B} -\ {\frac {1}{c^{2}}}\ {\frac {\partial ^{2}\mathbf {B} }{\partial t^{2}}}=-\mu _{0}\nabla \times \mathbf {J} $
這裏， $c$ 是光速， $\epsilon _{0}$ 是電常數， $\mu _{0}$ 是磁常數， $\rho $ 是電荷密度， $\mathbf {J} $ 是電流密度。
這些方程式都具有同樣形式的波動方程式：
: $\nabla ^{2}\Psi -\ {\frac {1}{c^{2}}}\ {\frac {\partial ^{2}\Psi }{\partial t^{2}}}=-F(\mathbf {r} ,t)$
這裏， $\Psi (\mathbf {r} ,t)$ 是描述純量波的波函數， $F(\mathbf {r} ,t)$ 是波源分佈。
假設這波源只發射出頻率為 $\omega $ 的單色波，
: $F(\mathbf {r} ,t)=f(\mathbf {r} )e^{-i\omega t}$
則前述含時波動方程式可以寫為不含時波動方程式：:135-152
: $\nabla ^{2}\psi +k^{2}\psi =-f(\mathbf {r} ),$ （1）
這裏， $k=\omega /c$ 是波向量的數值大小， $\psi (\mathbf {r} )$ 是描述波的不含時波函數， $\Psi (\mathbf {r} ,t)=\psi (\mathbf {r} )e^{-i\omega t}$ 。 | [
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光的衍射 | 方程式 **（1）** 乃是非齊次亥姆霍茲方程式。設 ${\hat {\mathbf {r} }}$ 為波的源點，只要能夠找到對應的格林函數 $G(\mathbf {r} ,\mathbf {r} ')$ ，就可以解析這方程式。這對應的格林函數必須滿足方程式:246
: $\nabla ^{2}G(\mathbf {r} ,\mathbf {r} ')+k^{2}G(\mathbf {r} ,\mathbf {r} ')=-\delta (\mathbf {r} -\mathbf {r} ')$
這裏， $\mathbf {r} '$ 為任意位置向量， $\delta (\mathbf {r} -\mathbf {r} ')$ 為三維狄拉克 δ 函數。
假若找到了對應的格林函數，那麼，假若 $\mathbf {r} $ 在積分體積 $\mathbb {V} $ 內，則波函數 $\psi (\mathbf {r} )$ 與格林函數、波源分佈的關係式為
: $\psi (\mathbf {r} )=\int _{\mathbb {V} }f(\mathbf {r} ')G(\mathbf {r} ,\mathbf {r} ')\mathrm {d} ^{3}\mathbf {r} '$
格林函數的形式與邊界條件有關。對於邊界曲面為無窮遠的自由空間，格林函數只與 $\mathbf {R} =\mathbf {r} -\mathbf {r} '$ 有關：
: $\nabla ^{2}G(\mathbf {R} ,\mathbf {O} )+k^{2}G(\mathbf {R} ,\mathbf {O} )=-\delta (\mathbf {R} )$
這裏， $\mathbf {O} $ 是坐標系的原點。
從這方程式，可以觀察到，格林函數具球對稱性質。因此，採用球坐標，格林函數滿足
: ${\frac {1}{R}}{\frac {\mathrm {d} ^{2}}{\mathrm {d} R^{2}}}(RG)+k^{2}G=-\delta (\mathbf {R} )$
透過代換，可以得知，形式為射出的球面波的格林函數能夠滿足以下方程式：:110-116
: $G(\mathbf {R} ,\mathbf {O} )={\frac {e^{ikR}}{4\pi R}}$ | [
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光的衍射 | 注意到，這格林函數假定點波源位於坐標系的原點 $\mathbf {O} $ 。對於空間中任意點波源位置 $\mathbf {r} '$ ，格林函數為
: $G(\mathbf {r} ,\mathbf {r} ')={\frac {e^{ik|\mathbf {r} -\mathbf {r} '|}}{4\pi |\mathbf {r} -\mathbf {r} '|}}$
假設波源是在 xy - 平面的面積分佈， $f(\mathbf {r} ')=f_{s}({\boldsymbol {\rho }}')\delta (z')$ ，那麼，
: ${\begin{aligned}\psi &=\int _{\mathbb {V} }f(\mathbf {r} ')G(\mathbf {r} ,\mathbf {r} ')\,\mathrm {d} ^{3}\mathbf {r} '=\int _{\mathbb {V} }f_{s}({\boldsymbol {\rho }}')\delta (z')G(\mathbf {r} ,\mathbf {r} ')\,\mathrm {d} ^{3}\mathbf {r} '\\&=\int _{\mathbb {S} }f_{s}({\boldsymbol {\rho }}')G(\mathbf {r} ,{\boldsymbol {\rho }}')\,\mathrm {d} \sigma '=\int _{\mathbb {S} }f_{s}({\boldsymbol {\rho }}'){\frac {e^{ik|\mathbf {r} -{\boldsymbol {\rho }}'|}}{4\pi |\mathbf {r} -{\boldsymbol {\rho }}'|}}\,\mathrm {d} \sigma '\\\end{aligned}}$ | [
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光的衍射 | 這裏， $\mathbf {r} '={\boldsymbol {\rho }}'=\rho '{\hat {\boldsymbol {\rho }}}$ 為在積分曲面 $\mathbb {S} $ 內的波源位置，以直角坐標表示， ${\boldsymbol {\rho }}'=(x',y',0)$ ，又 $\mathbb {S} $ 是積分曲面， $\mathrm {d} \sigma '$ 是微小面積分元素， $\delta (z')$ 是一維狄拉克 δ 函數。
對於一般的面積分佈波源，
: $\psi =\int _{\mathbb {V} }f(\mathbf {r} ')G(\mathbf {r} ,\mathbf {r} ')\,\mathrm {d} ^{3}\mathbf {r} '=\int _{\mathbb {S} }f_{s}(\mathbf {r} '){\frac {e^{ik|\mathbf {r} -\mathbf {r} '|}}{4\pi |\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} \sigma '$
根據惠更斯－菲涅耳原理，波前的每一點都是次波的點波源，這些次波共同形成了稍後時刻的疊加波。:189 假設有波入射於某孔隙，則可以假定這入射波的波前在孔隙內的每一點都是孔隙內的波源，並且推論孔隙內的波源與在同位置的入射波波函數 $\psi _{inc}$ 有關： $f_{s}(\mathbf {r} ')\propto \psi _{inc}(\mathbf {r} ')$ 。雖然這裏並沒有給出確切關係式，必定存在乘法因子 $C$ ，滿足
: $\psi (\mathbf {r} )=C\int _{\mathbb {S} }\psi _{inc}(\mathbf {r} '){\frac {e^{ik|\mathbf {r} -\mathbf {r} '|}}{|\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} \sigma '$ | [
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