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Unfortunately, analytic studies sometimes are unrevealing. This is particularly true if the hypotheses were not well founded at the outset. It is an axiom of field epidemiology that if you cannot generate good hypotheses (for example, by talking to some casepatients or local staff and examining the descriptive epidemio... | {
"Header 1": "**Steps of an Outbreak Investigation**",
"Header 2": "*Step 9: Reconsider, refine, and re-evaluate hypotheses*",
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While epidemiology can implicate vehicles and guide appropriate public health action, laboratory evidence can confirm the findings. The laboratory was essential in both the outbreak of salmonellosis linked to marijuana and in the Legionellosis outbreak traced to the grocery store mist machine. You may recall that the i... | {
"Header 1": "**Steps of an Outbreak Investigation**",
"Header 2": "*Step 9: Reconsider, refine, and re-evaluate hypotheses*",
"Header 3": "*Step 10: Compare and reconcile with laboratory and environmental studies*",
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In most outbreak investigations, the primary goal is control of the outbreak and prevention of additional cases. Indeed, although implementing control and prevention measures is listed as Step 11 in the conceptual sequence, in practice control and prevention activities should be implemented as early as possible. The he... | {
"Header 1": "**Steps of an Outbreak Investigation**",
"Header 2": "*Step 9: Reconsider, refine, and re-evaluate hypotheses*",
"Header 3": "*Step 11: Implement control and prevention measures*",
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Once control and prevention measures have been implemented, they must continue to be monitored. If surveillance has not been ongoing, now is the time to initiate active surveillance. If active surveillance was initiated as part of case finding efforts, it should be continued. The reasons for conducting active surveilla... | {
"Header 1": "**Steps of an Outbreak Investigation**",
"Header 2": "*Step 12: Initiate or maintain surveillance*",
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As noted in Step 1, development of a communications plan and communicating with those who need to know during the investigation is critical. The final task is to summarize the investigation, its findings, and its outcome in a report, and to communicate this report in an effective manner. This communication usually take... | {
"Header 1": "**Steps of an Outbreak Investigation**",
"Header 2": "*Step 13: Communicate findings*",
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Outbreaks occur frequently. Not every outbreak comes to light, but of those that do, public health agencies must decide whether to handle them without leaving the office, or spend the time, energy, and resources to conduct field investigations. The most important reason to investigate is to learn enough about the situa... | {
"Header 1": "**Summary**",
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Nine cases of cancer in a community represents a cluster — a group of cases in a given area over a particular period of time that seems to be unusual, although we do not actually know the size of the community, the background rate of cancer, and the number of cases that might be expected. Nonetheless, either the health... | {
"Header 1": "**Summary**",
"Header 2": "*Exercise 6.1*",
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| Initial Case Definition | | Revised Case Definition |
|------------------------------------------------------|----------------|------------------------------------------------------|
| Patient 1: No, eosinophil count < 2,000 | ... | {
"Header 1": "**Summary**",
"Header 2": "*Exercise 6.3*",
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Cell phones are quite popular. Noting that most if not all of the 17 patients had used cell phones does not indicate that cell phones are the cause of brain cancer. An epidemiologic study that compares the exposure experience of the case-patients with the exposure experience of persons without brain cancer is necessary... | {
"Header 1": "**Summary**",
"Header 2": "*Exercise 6.7*",
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- 1. The appropriate measure of association for a case-control study is the odds ratio.
- 2. The odds ratio is calculated as the cross-product ratio: ad / bc.
Odds ratio = 15 x 23 / 8 x 7 = 6.16 = 6.2
3. With a chi-square of 9.41 and a 95% confidence interval of 1.6–25.1, this study shows a very strong (odds ratio ... | {
"Header 1": "**Summary**",
"Header 2": "*Exercise 6.8*",
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The first step in answering this question is to compare the attack rates (% ill) among those who ate the meal and those who did not eat the meal. Since the % ill is a measure of risk of illness, you could calculate a risk ratio for each meal.
| | | | Risk Ratio |
|------|-------------|-... | {
"Header 1": "**Summary**",
"Header 2": "*Exercise 6.8*",
"Header 3": "*Exercise 6.9*",
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- 1. C, D. Most outbreaks come to the attention of health authorities because an alert clinician or a concerned case-patient (or parent of a case-patient) calls. The other methods listed occasionally detect outbreaks, but less frequently.
- 2. A, B, C, D. Factors influencing a health department's decision to conduct a ... | {
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The key feature that characterizes an analytic (epidemiologic) study is presence of a comparison group. Single case reports and case series do not have comparison groups and are not analytic studies. Cohort studies (compares disease experience among exposed and
unexposed groups) and case-control studies (compares exp... | {
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**case definition** a set of uniformly applied criteria for determining whether a person should be identified as having a particular disease, injury, or other health condition. In epidemiology, particularly for an outbreak investigation, a case definition specifies clinical criteria and details of time, place, and pers... | {
"Header 1": "**C**",
"Header 3": "**case-control study** see **study, case-control**.",
"token_count": 297,
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**cause, sufficient** a factor or collection of factors whose presence is always followed by the occurrence of a particular health problem.
**census** the enumeration of an entire population, usually including details on residence, age, sex, occupation, racial/ethnic group, marital status, birth history, and relation... | {
"Header 1": "**C**",
"Header 3": "**cause-specific mortality rate** see **mortality rate, cause-specific**.",
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**comparison group** a group in an analytic study (e.g., a cohort or case-control study) with whom the primary group of interest (exposed group in a cohort study or case-patients in a case-control study) is compared. The comparison group provides an estimate of the background or expected incidence of disease (in a coho... | {
"Header 1": "**C**",
"Header 2": "**community immunity** see **immunity, herd**.",
"Header 3": "**community trial** see **trial, community**.",
"token_count": 289,
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.
**effect** the result of a cause.
**effectiveness** the ability of an intervention or program to produce the intended or expected results in the field.
**efficacy** the ability of an intervention or program to produce the intended or expected results under ideal conditions.
**efficiency** the ability of an in... | {
"Header 1": "**D**",
"Header 2": "**E**",
"token_count": 645,
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**high-risk group** a group of persons whose risk for a particular disease, injury, or other health condition is greater than that of the rest of their community or population.
**HIPAA** the Health Insurance Portability and Accountability Act, enacted in 1996, which addresses the privacy of a person's medical informa... | {
"Header 1": "**D**",
"Header 2": "**herd immunity** see **immunity, herd**.",
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**latency period** the time from exposure to a causal agent to onset of symptoms of a (usually noninfectious) disease (see also **incubation period**).
**life expectancy** a statistical projection of the average number of years a person of a given age is expected to live, if current mortality rates continue to apply.... | {
"Header 1": "**D**",
"Header 2": "**L**",
"token_count": 261,
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.
**map, area (shaded, choropleth)** a visual display of the geographic pattern of a health problem, in which a marker is placed on a map to indicate where each affected person lives, works, or might have been exposed.
**mean (or average)** commonly called the average; it is the most common measure of central tende... | {
"Header 1": "**D**",
"Header 2": "**M**",
"token_count": 206,
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**midrange** the halfway point, or midpoint, in a set of observations. For the majority of data, the midrange is calculated by adding the smallest observation and the largest observation and dividing by two. The midrange is usually calculated as an intermediate step in determining other measures.
**mode** the most fr... | {
"Header 1": "**D**",
"Header 2": "**M**",
"Header 3": "**medical surveillance** see **surveillance, medical**.",
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**outbreak** the occurrence of more cases of disease, injury, or other health condition than expected in a given area or among a specific group of persons during a specific period. Usually, the cases are presumed to have a common cause or to be related to one another in some way. Sometimes distinguished from an epidemi... | {
"Header 1": "**D**",
"Header 2": "**observational study** see **study, observational**.",
"Header 3": "**ordinal scale** see **scale, ordinal**.",
"token_count": 267,
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**range** in statistics, the difference between the largest and smallest values in a distribution; in common use, the span of values from smallest to largest.
**rate** an expression of the relative frequency with which an event occurs among a defined population per unit of time, calculated as the number of new cases ... | {
"Header 1": "**Q**",
"Header 2": "**race/ethnic-specific mortality rate** see **mortality rate, race/ethnic-specific**.",
"Header 3": "**random sample** see **sample, random**.",
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.
**specificity** the ability or a test, case definition, or surveillance system to exclude persons without the health condition of interest; the proportion of persons without a health condition that are correctly identified as such by a screening test, case definition, or surveillance system.
**spectrum of illness... | {
"Header 1": "**S**",
"Header 3": "**source case** see **case, source**.",
"token_count": 1092,
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.
**table** an arrangement of data in rows and columns. In epidemiology, the data are usually summaries of the frequency of occurrence of an event or characteristic occurring among different groups.
**table shell** a table that is completely drawn and labeled but contains no data.
**table, two-by-two** a two-vari... | {
"Header 1": "**S**",
"Header 2": "**T**",
"token_count": 539,
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.
**validity** the degree to which a measurement, questionnaire, test, or study or any other datacollection tool measures what it is intended to measure.
**variable** any characteristic or attribute that can be measured and can have different values.
**variable** (or **data**), **discrete** a variable that is lim... | {
"Header 1": "**V**",
"token_count": 364,
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David J. Griffiths
Reed College
#### **Fundamental Equations**
Schrödinger equation:
$$i\hbar \frac{\partial \Psi}{\partial t} = H\Psi$$
Time independent Schrödinger equation:
$$H\psi = E\psi, \qquad \Psi = \psi e^{-iEt/\hbar}$$
Standard Hamiltonian:
$$H = -\frac{\hbar^2}{2m}\nabla^2 + V$$
Time depend... | {
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"Header 2": "Introduction to Quantum Mechanics",
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Fine structure constant: $\alpha = e^2/4\pi\epsilon_0\hbar c$ = 1/137.036
Bohr radius: $a = 4\pi\epsilon_0 \hbar^2/m_e e^2 = \hbar/\alpha m_e c = 5.29177 \times 10^{-11} \text{ m}$
Bohr energies: $E_n = E_1/n^2 (n = 1, 2, 3, ...)$
Ground state energy: $-E_1 = m_e e^4/2(4\pi\epsilon_0)^2 \hbar^2 = \alpha^2 m_... | {
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- 1.1 The Schrodinger Equation, 1
- 1.2 The Statistical Interpretation, 2
- 1.3 Probability, 5
- 1.4 Normalization, 11
- 1.5 Momentum, 14
- 1.6 The Uncertainty Principle, 17
#### CHAPTER 2
#### THE TIME-INDEPENDENT SCHRODINGER EQUATION, 20
- 2.1 Stationary States, 20
- 2.2 The Infinite Square Well, 24
- 2.3 The H... | {
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7.1 Theory, 256
7.2 The Ground State of Helium, 261
7.3 The Hydrogen Molecule Ion, 266
Further Problems for Chapter 7, 271
#### **CHAPTER 8**
#### THE WKB APPROXIMATION, 274
8.1 The "Classical" Region, 275
8.2 Tunneling, 280
8.3 The Connection Formulas, 284
Further Problems for Chapter 8, 293
#### *... | {
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Unlike Newton's mechanics, or Maxwell's electrodynamics, or Einstein's relativity, quantum theory was not created—or even definitively packaged—by one individual, and it retains to this day some of the scars of its exhilirating but traumatic youth. There is no general consensus as to what its fundamental principles are... | {
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"Header 2": "CHAPTER 11 SCATTERING, 352",
"Header 3": "**PREFACE**",
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#### 1.1 THE SCHRÖDINGER EQUATION
Imagine a particle of mass m, constrained to move along the x-axis, subject to some specified force F(x,t) (Figure 1.1). The program of classical mechanics is to determine the position of the particle at any given time: x(t). Once we know that, we can figure out the velocity (v = dx/... | {
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Indeed, it would be tough to prove that the particle was really found at C in the first instance if this could not be confirmed by immediate repetition of the measurement. How does the orthodox interpretation account for the fact that the second measurement is bound to give the value C? Evidently the first measurement ... | {
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Trouble is, of course, that you get zero, since, by the nature of the average, $\Delta j$ is as often negative as positive:
$$\langle \Delta j \rangle = \sum (j - \langle j \rangle) P(j) = \sum_{j} j P(j) - \langle j \rangle \sum_{j} P(j) = \langle j \rangle - \langle j \rangle = 0.$$
(Note that $\langle j \rang... | {
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What is the probability that the needle will cross a line? [Hint: Refer to Problem 1.4.]
#### \*Problem 1.6 Consider the Gaussian distribution
$$\rho(x) = Ae^{-\lambda(x-a)^2},$$
where A, a, and $\lambda$ are constants. (Look up any integrals you need.)
- (a) Use Equation 1.16 to determine A.
- **(b)** Find ... | {
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(a) Show that
$$\frac{dP_{ab}}{dt} = J(a,t) - J(b,t)$$
where
$$J(x,t) \equiv \frac{i\hbar}{2m} \left( \Psi \frac{\partial \Psi^*}{\partial x} - \Psi^* \frac{\partial \Psi}{\partial x} \right).$$
What are the units of J(x,t)? [J is called the **probability current**, because it tells you the rate at which prob... | {
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Kinetic energy, for example, is
$$T = \frac{1}{2}mv^2 = \frac{p^2}{2m},$$
and angular momentum is
$$\mathbf{L} = \mathbf{r} \times m\mathbf{v} = \mathbf{r} \times \mathbf{p}$$
(the latter, of course, does not occur for motion in one dimension). To calculate the expectation value of such a quantity, we simply re... | {
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#### 2.1 STATIONARY STATES
In Chapter 1 we talked a lot about the wave function and how you use it to calculate various quantities of interest. The time has come to stop procrastinating and confront what is, logically, the prior question: How do you $get \Psi(x, t)$ in the first place—how do you go about solving th... | {
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As we're about to discover, the time-independent Schrödinger equation (Equation 2.4) yields an infinite collection of solutions $(\psi_1(x), \psi_2(x), \psi_3(x), \ldots)$ , each with its associated value of the separation constant $(E_1, E_2, E_3, \ldots)$ ; thus there is a different wave function for each **allowed... | {
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Well, how *do* we fix the constant A? Answer: We *normalize* $\psi$ :
$$\int_0^a |A|^2 \sin^2(kx) \, dx = |A|^2 \frac{a}{2} = 1, \quad \text{so} \quad |A|^2 = \frac{2}{a}.$$
This only determines the *magnitude* of A, but it is simplest to pick the positive real root: $A = \sqrt{2/a}$ (the phase of A carries no p... | {
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The completeness of the $\psi$ 's (confirmed in this case by Dirichlet's theorem) guarantees that I can always express $\Psi(x, 0)$ in this way, and their orthonormality licenses the use of Fourier's trick to determine the actual coefficients:
$$c_n = \sqrt{\frac{2}{a}} \int_0^a \sin\left(\frac{n\pi}{a}x\right) ... | {
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Formally, if we expand V(x) in a **Taylor series** about the minimum:
$$V(x) = V(x_0) + V'(x_0)(x - x_0) + \frac{1}{2}V''(x_0)(x - x_0)^2 + \cdots,$$
subtract $V(x_0)$ [you can add a constant to V(x) with impunity, since that doesn't change the force], recognize that $V'(x_0) = 0$ (since $x_0$ is a minimum), ... | {
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\text{ QED}$$
[Notice that whereas the ordering of $a_+$ and $a_-$ does matter, the ordering of $a_\pm$ and any constants (such as $\hbar$ , $\omega$ , and E) does not.] By the same token, $a_-\psi$ is a solution with energy $(E - \hbar \omega)$ :
$$(a_{-}a_{+} - \frac{1}{2}\hbar\omega)(a_{-}\psi) = a_{-... | {
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(You'll have to normalize $\psi_0$ "by hand".) Answer:
$$A_n = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} \frac{(-i)^n}{\sqrt{n!(\hbar\omega)^n}}.$$
[2.54]
\*Problem 2.13 Using the methods and results of this section,
- (a) Normalize $\psi_1$ (Equation 2.51) by direct integration. Check your answer against t... | {
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Thus Equation 2.65 determines $h(\xi)$ in terms of two arbitrary constants $(a_0 \text{ and } a_1)$ —which is just what we would expect, for a second-order differential equation.
However, not all the solutions so obtained are normalizable. For at very large j, the recursion formula becomes (approximately)
$$a_{j... | {
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Notice that it oscillates sinusoidally. What is the amplitude of the oscillation? What is its (angular) frequency?
- (d) Use your result in (c) to determine $\langle p \rangle$ . Check that Ehrenfest's theorem holds for this wave function.
- (e) Referring to Figure 2.5, sketch the graph of $|\Psi|$ at t = 0, $\pi/\... | {
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For a free particle the solution has the form of Equation 2.83; the only remaining question is how to determine $\phi(k)$ so as to fit the initial wave function:
$$\Psi(x,0) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} \phi(k) e^{ikx} dk.$$
[2.84]
This is a classic problem in Fourier analysis; the answer is p... | {
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- \*\*Problem 2.20 This problem is designed to guide you through a "proof" of Plancherel's theorem, by starting with the theory of ordinary Fourier series on a *finite* interval, and allowing that interval to expand to infinity.
- (a) Dirichlet's theorem says that "any" function f(x) on the interval [-a, +a] can be e... | {
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Technically, it's not a function at all, since it is not finite at x=0 (mathematicians call it a **generalized function**, or **distribution**).<sup>24</sup> Nevertheless, it is an extremely useful construct in theoretical physics. (For example, in electrodynamics the charge density of a point charge is a delta functio... | {
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So Equation 2.107 says
$$\kappa = \frac{m\alpha}{\hbar^2},\tag{2.108}$$
and the allowed energy (Equation 2.99) is
$$E = -\frac{\hbar^2 \kappa^2}{2m} = -\frac{m\alpha^2}{2\hbar^2}.$$
[2.109]
Finally, we normalize $\psi$ :
$$\int_{-\infty}^{+\infty} |\psi(x)|^2 dx = 2|B|^2 \int_0^{\infty} e^{-2\kappa x} dx = \... | {
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As long as we've got the relevant equations on the table, let's look briefly at the case of a delta-function barrier (Figure 2.11). Formally, all we have to do is change the sign of $\alpha$ . This kills the bound state, of course (see Problem 2.2). On the other hand, the reflection and transmission coefficients, wh... | {
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The advantage of this is that we need only impose the boundary conditions on one side (say, at +a); the other side is then automatic, since $\psi(-x) = \pm \psi(x)$ . I'll work out the even solutions; you get to do the odd ones in Problem 2.28. The cosine is even (and the sine is odd), so I'm looking for solutions of ... | {
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**Problem 2.30** The Dirac delta function can be thought of as the limiting case of a rectangle of area 1, as the height goes to infinity and the width goes to zero. Show that the delta-function well (Equation 2.96) is a "weak" potential (even though it is
infinitely deep), in the sense that $z_0 \to 0$ . Determin... | {
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For example, in the case of the finite square well,
$$S_{21} = \frac{e^{-2ika}}{\cos(2la) - i\frac{\sin(2la)}{2kl}(k^2 + l^2)}$$
(Equation 2.150). Substituting $k \to i\kappa$ , we see that $S_{21}$ blows up whenever
$$\cot(2la) = \frac{l^2 - \kappa^2}{2\kappa l}.$$
Using the trigonometric identity
$$\ta... | {
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In view of your conclusions in (b), does the electron tend to draw the nuclei together, or push them apart? (Of course, there is also the internuclear repulsion to consider, but that's a separate problem.)
#### \*\*\*Problem 2.45
(a) Show that
$$\Psi(x,t) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} \exp\left[-\... | {
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#### 3.1 LINEAR ALGEBRA
The purpose of this chapter is to develop the *formalism* of quantum mechanics—terminology, notation, and mathematical background that illuminate the structure of the theory, facilitate practical calculations, and motivate a fundamental extension of the statistical interpretation. I begin with... | {
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#### 3.1.2 Inner Products
In three dimensions we encounter two kinds of vector products: the dot product and the cross product. The latter does not generalize in any natural way to *n*-dimensional vector spaces, but the former *does*—in this context it is usually called the **inner product**. The inner product of t... | {
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For suppose that
$$\hat{T}|e_{1}\rangle = T_{11}|e_{1}\rangle + T_{21}|e_{2}\rangle + \cdots + T_{n1}|e_{n}\rangle,$$
$$\hat{T}|e_{2}\rangle = T_{12}|e_{1}\rangle + T_{22}|e_{2}\rangle + \cdots + T_{n2}|e_{n}\rangle,$$
$$\vdots$$
$$\hat{T}|e_{n}\rangle = T_{1n}|e_{1}\rangle + T_{2n}|e_{2}\rangle + \cdots + T_{n... | {
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[3.45]
To construct the (complex) **conjugate** of a matrix (which we denote, as usual, with an asterisk: $T^*$ ), you take the complex conjugate of every element:
$$\mathbf{T}^* = \begin{pmatrix} T_{11}^* & T_{12}^* & \dots & T_{1n}^* \\ T_{21}^* & T_{22}^* & \dots & T_{2n}^* \\ \vdots & \vdots & & \vdots \\ T_{n... | {
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and Equation 3.62—multiplying both sides by $S^{-1}$ —entails<sup>10</sup> $\mathbf{a}^e = S^{-1}\mathbf{a}^f$ , so
$$\mathbf{a}^{f'} = \mathbf{S}\mathbf{a}^{e'} = \mathbf{S}(\mathbf{T}^e\mathbf{a}^e) = \mathbf{S}\mathbf{T}^e\mathbf{S}^{-1}\mathbf{a}^f.$$
Evidently
$$\mathbf{T}^f = \mathbf{S}\mathbf{T}^e\math... | {
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\***Problem 3.15** Prove that $Tr(T_1T_2) = Tr(T_2T_1)$ . It follows immediately that $Tr(T_1T_2T_3) = Tr(T_2T_3T_1)$ , but is it the case that $Tr(T_1T_2T_3) = Tr(T_2T_1T_3)$ , in general? Prove it, or disprove it. *Hint*: The best disproof is always a counterexample—and the simpler the better!
**Problem 3.16**... | {
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We may as well pick $a_1 = 1$ (since any multiple of an eigenvector is still an eigenvector):
$$\mathbf{a}^{(1)} = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}, \text{ for } \lambda_1 = 0.$$
[3.75]
For the second eigenvector (recycling the same notation for the components) we have
$$\begin{pmatrix} 2 & 0 & -2 \\ -... | {
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#### 3.1.5 Hermitian Transformations
In Equation 3.48 I defined the Hermitian conjugate (or "adjoint") of a *matrix* as its transpose conjugate: $\mathbf{T}^{\dagger} = \tilde{\mathbf{T}}^*$ . Now I want to give you a more fundamental definition for the Hermitian conjugate of a *linear transformation*: It is that ... | {
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**Proof**: Suppose $\hat{T}|\alpha\rangle = \lambda |\alpha\rangle$ and $\hat{T}|\beta\rangle = \mu |\beta\rangle$ , with $\lambda \neq \mu$ . Then
$$\langle \alpha | \hat{T}\beta \rangle = \langle \alpha | \mu \beta \rangle = \mu \langle \alpha | \beta \rangle,$$
and if $\hat{T}$ is Hermitian,
$$\langle \a... | {
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$$P_{0} = 1$$
$$P_{1} = x$$
$$P_{2} = \frac{1}{2}(3x^{2} - 1)$$
$$P_{3} = \frac{1}{2}(5x^{3} - 3x)$$
$$P_{4} = \frac{1}{8}(35x^{4} - 30x^{2} + 3)$$
$$P_{5} = \frac{1}{8}(63x^{5} - 70x^{3} + 15x)$$
this restriction is also *sufficient*—if f and g are both square integrable, then the integral in Equation 3.... | {
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In fact, it's a Hermitian transformation, since (obviously)
$$\int_{-1}^{1} [f(x)]^* [xg(x)] dx = \int_{-1}^{1} [xf(x)]^* [g(x)] dx.$$
But what are its eigenfunctions? Well,
$$x(a_0 + a_1x + a_2x^2 + \cdots) = \lambda(a_0 + a_1x + a_2x^2 + \cdots),$$
for all x, means
$$0 = \lambda a_0,$$
$$a_0 = \lambda a_1... | {
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And even if the sequence does have a limit, the limit function may not be square integrable, so we can't include it in an inner product space. To complete the space, then, we throw in all square-integrable convergent sequences of functions in the space. Notice that completing a space does not involve the introduction o... | {
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Taking the inner product with $|f_{\mu}\rangle$ , and exploiting the "orthonormality" of the basis (Equation 3.108), we obtain the "components" $a_{\lambda}$ :
$$\langle f_{\mu}|f\rangle = \int_{-\infty}^{\infty} a_{\lambda} \langle f_{\mu}|f_{\lambda} \rangle d\lambda = \int_{-\infty}^{\infty} a_{\lambda} \delta... | {
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Classical dynamical quantities (such as position, velocity, momentum and kinetic energy) can be expressed as functions of the "canonical" variables x and p (and, in rare cases, t): Q(x, p, t). To each such classical observable we associate a quantum-mechanical *operator*, $\hat{Q}$ , obtained from Q by the substitutio... | {
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(or rather, they can always be *chosen* so); the completeness relation takes the form of an *integral*:
$$|\Psi\rangle = \int_{-\infty}^{\infty} c_k |e_k\rangle \, dk; \qquad [3.125]$$
the "components" are given by "Fourier's trick":
$$c_k = \langle e_k | \Psi \rangle, \tag{3.126}$$
and the probability of get... | {
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\end{split}$$
Similarly,
$$\langle g|f\rangle = \langle \hat{B}\hat{A}\rangle - \langle A\rangle\langle B\rangle,$$
SO
$$\langle f|g\rangle - \langle g|f\rangle = \langle \hat{A}\hat{B}\rangle - \langle \hat{B}\hat{A}\rangle = \langle [\hat{A},\hat{B}]\rangle,$$
where
$$[\hat{A}, \hat{B}] \equiv \hat{A}\hat... | {
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Equation 3.146 is often paired with the **energy-time uncertainty principle**,
$$\Delta t \ \Delta E \ge \frac{\hbar}{2}.\tag{3.147}$$
Indeed, in the context of special relativity the energy-time form might be thought of as a consequence of the position-momentum version, because x and t (or rather, ct) go together ... | {
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Therefore,
$$\Delta E \, \Delta t = \frac{p \Delta p}{m} \frac{m \Delta x}{p} = \Delta x \, \Delta p \ge \frac{\hbar}{2}.$$
**Example 3.** The $\Delta$ particle lasts about $10^{-23}$ seconds before spontaneously disintegrating. If you make a histogram of all measurements of its mass, you get a kind of bell-sha... | {
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In general,
$$\langle Q(x, p, t) \rangle = \begin{cases} \int \Psi^* \hat{Q}\left(x, \frac{\hbar}{i} \frac{\partial}{\partial x}, t\right) \Psi \, dx, & \text{in position space;} \\ \int \Phi^* \hat{Q}\left(-\frac{\hbar}{i} \frac{\partial}{\partial p}, p, t\right) \Phi \, dp, & \text{in momentum space.} \end{cases}$$... | {
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The most general state is a normalized linear combination:
$$|\Psi\rangle = a|1\rangle + b|2\rangle = \begin{pmatrix} a \\ b \end{pmatrix}, \text{ with } |a|^2 + |b|^2 = 1.$$
Suppose the Hamiltonian matrix is
$$\mathbf{H} = \begin{pmatrix} h & g \\ g & h \end{pmatrix},$$
where g and h are real constants. The ... | {
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The generalization to three dimensions is straightforward. Schrödinger's equation says
$$i\hbar \frac{\partial \Psi}{\partial t} = H\Psi; \tag{4.1}$$
the Hamiltonian operator H is obtained from the classical energy
$$\frac{1}{2}mv^2 + V = \frac{1}{2m}(p_x^2 + p_y^2 + p_z^2) + V$$
by the standard prescription (a... | {
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Dividing by YR and multiplying by $-2mr^2/\hbar^2$ :
$$\left\{ \frac{1}{R} \frac{d}{dr} \left( r^2 \frac{dR}{dr} \right) - \frac{2mr^2}{\hbar^2} [V(r) - E] \right\}$$
$$+\frac{1}{Y}\left\{\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial Y}{\partial\theta}\right) + \frac{1}{\sin^2... | {
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We encountered the latter (Equation 3.91) as orthogonal polynomials on the interval (-1, +1); for our present purposes it is more convenient to define them by the **Rodrigues formula**:
$$P_l(x) \equiv \frac{1}{2^l l!} \left(\frac{d}{dx}\right)^l (x^2 - 1)^l.$$
[4.28]
For example,
$$P_0(x) = 1$$
, $P_1(x) = \fra... | {
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As we shall prove later on, they are automatically orthogonal, so
$$\int_{0}^{2\pi} \int_{0}^{\pi} [Y_{l}^{m}(\theta,\phi)]^{*} [Y_{l'}^{m'}(\theta,\phi)] \sin\theta \, d\theta \, d\phi = \delta_{ll'} \delta_{mm'}. \tag{4.33}$$
In Table 4.2 I have listed the first few spherical harmonics.
\*Problem 4.3 Use Equati... | {
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Notice that for small x (where $\sin x \approx x - x^3/3! + x^5/5! - \cdots$ and $\cos x \approx 1 - x^2/2 + x^4/4! - \cdots$ ),
$$j_0(x) \approx 1; \quad n_0(x) \approx \frac{1}{x}; \quad j_1(x) \approx \frac{x}{3}; \quad n_1(x) \approx -\frac{1}{x^2};$$
etc. The point is that the Bessel functions are finite at... | {
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As $\rho \to \infty$ , the constant term in the brackets dominates, so (approximately)
$$\frac{d^2u}{d\rho^2}=u.$$
The general solution is
$$u(\rho) = Ae^{-\rho} + Be^{\rho}, \tag{4.57}$$
but $e^{\rho}$ blows up (as $\rho \to \infty$ ), so B = 0. Evidently,
$$u(\rho) \sim Ae^{-\rho} \tag{4.58}$$
for lar... | {
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Defining
$$n \equiv j_{\text{max}} + l + 1 \tag{4.67}$$
(the so-called principal quantum number), we have
$$\rho_0 = 2n.$$
[4.68]
But $\rho_0$ determines E (Equations 4.54 and 4.55):
$$E = -\frac{\hbar^2 \kappa^2}{2m} = -\frac{me^4}{8\pi^2 \epsilon_0^2 \hbar^2 \rho_0^2},$$
[4.69]
so the allowed energies... | {
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<sup>&</sup>lt;sup>15</sup>If you want to see how the normalization factor is calculated, study (for example), L. Schiff, *Quantum Mechanics*, 2nd ed. (New York: McGraw-Hill, 1968), page 93.
**Table 4.4:** The first few Laguerre polynomials, $L_q(x)$ .
$$L_0 = 1$$
$$L_1 = -x + 1$$
$$L_2 = x^2 - 4x + 2$$
$$... | {
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Check that you get the same answer this way.
- (d) Use $b \approx 10^{-15}$ m and $a \approx 0.5 \times 10^{-10}$ m to get a numerical estimate for P. Roughly speaking, this represents the "fraction of its time that the electron spends inside the nucleus".
#### Problem 4.15
(a) Use the recursion formula (Equati... | {
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#### 4.3.1 Eigenvalues
$L_x$ and $L_y$ do not commute; in fact [providing a test function, f(x, y, z), for them to act upon]:
$$[L_{x}, L_{y}]f = \left(\frac{\hbar}{i}\right)^{2} \left\{ \left(y\frac{\partial}{\partial z} - z\frac{\partial}{\partial y}\right) \left(z\frac{\partial}{\partial x} - x\frac{\partia... | {
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According to the generalized uncertainty principle (Equation 3.139),
$$\sigma_{L_x}^2 \sigma_{L_y}^2 \ge \left(\frac{1}{2i} \langle i\hbar L_z \rangle\right)^2 = \frac{\hbar^2}{4} \langle L_z \rangle^2,$$
or
$$\sigma_{L_x}\sigma_{L_y} \ge \frac{\hbar}{2}|\langle L_z\rangle|.$$
[4.100]
It would therefore be futi... | {
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ag{4.114}$$
Let $\hbar \bar{l}$ be the eigenvalue of $L_z$ at this bottom rung:
$$L_z f_b = \hbar \bar{l} f_b; \quad L^2 f_b = \lambda f_b.$$
[4.115]
Using Equation 4.112, we have
$$L^{2} f_{b} = (L_{+} L_{-} + L_{z}^{2} - \hbar L_{z}) f_{b} = (0 + \hbar^{2} \bar{l}^{2} - \hbar^{2} \bar{l}) f_{b} = \hbar^{2... | {
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meanwhile, $\mathbf{r} = r\hat{r}$ , so
$$\mathbf{L} = \frac{\hbar}{i} \left[ r(\hat{r} \times \hat{r}) \frac{\partial}{\partial r} + (\hat{r} \times \hat{\theta}) \frac{\partial}{\partial \theta} + (\hat{r} \times \hat{\phi}) \frac{1}{\sin \theta} \frac{\partial}{\partial \phi} \right].$$
But $(\hat{r} \times ... | {
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"Header 2": "4.1 SCHRÖDINGER EQUATION IN SPHERICAL COORDINATES",
"token_count": 1997,
"source_pdf": "datasets/websources/Physics_v1/Physics/Griffiths - Introduction to quantum mechanics.pdf"
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[4.133]
But there is a curious twist to this story, for the *algebraic* theory of angular momentum permits l (and hence also m) to take on *half*-integer values (Equation 4.119), whereas the *analytic* method yielded eigenfunctions only for *integer* values (Equation 4.29). You might reasonably guess that the half-in... | {
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But s is fixed, for any given particle, and this makes the theory of spin comparatively simple.<sup>24</sup>
Problem 4.26 If the electron is a classical solid sphere, with radius
$$r_c = \frac{e^2}{4\pi\epsilon_0 mc^2},$$
[4.138]
(the so-called **classical electron radius**, obtained by assuming that the electron... | {
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The eigenspinors are obtained in the usual way:
$$\frac{\hbar}{2} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} \alpha \\ \beta \end{pmatrix} = \pm \frac{\hbar}{2} \begin{pmatrix} \alpha \\ \beta \end{pmatrix} \Rightarrow \begin{pmatrix} \beta \\ \alpha \end{pmatrix} = \pm \begin{pmatrix} \alpha \\ \be... | {
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\*Problem 4.28 An electron is in the spin state
$$\chi = A \begin{pmatrix} 3i \\ 4 \end{pmatrix}.$$
- (a) Determine the normalization constant A.
- **(b)** Find the expectation values of $S_x$ , $S_y$ , and $S_z$ .
- (c) Find the "uncertainties" $\sigma_{S_x}$ , $\sigma_{S_y}$ , and $\sigma_{S_z}$ .
- (d) C... | {
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Thus
$$\chi(t) = \begin{pmatrix} \cos(\alpha/2)e^{i\gamma B_0 t/2} \\ \sin(\alpha/2)e^{-i\gamma B_0 t/2} \end{pmatrix}.$$
[4.163]
To get a feel for what is happening here, let's calculate the expectation value of the spin $\langle S \rangle$ as a function of time:
$$\langle S_{x} \rangle = \chi(t)^{\dagger} S_{... | {
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"Header 2": "4.1 SCHRÖDINGER EQUATION IN SPHERICAL COORDINATES",
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"source_pdf": "datasets/websources/Physics_v1/Physics/Griffiths - Introduction to quantum mechanics.pdf"
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for in this case $S_z = \hbar/2$ and $p_z = F_z T$ .)
The Stern-Gerlach experiment has played an important role in the philosophy of quantum mechanics, where it serves both as the prototype for the preparation of a quantum state and as an illuminating model for a certain kind of quantum measurement. We casually as... | {
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"Header 2": "4.1 SCHRÖDINGER EQUATION IN SPHERICAL COORDINATES",
"token_count": 1855,
"source_pdf": "datasets/websources/Physics_v1/Physics/Griffiths - Introduction to quantum mechanics.pdf"
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\quad [4.179]$$
Using Equations 4.142 and 4.145, we have
$$\mathbf{S}^{(1)} \cdot \mathbf{S}^{(2)}(\uparrow\downarrow) = (S_x^{(1)} \uparrow)(S_x^{(2)} \downarrow) + (S_y^{(1)} \uparrow)(S_y^{(2)} \downarrow) + (S_z^{(1)} \uparrow)(S_z^{(2)} \downarrow)$$
$$= \left(\frac{\hbar}{2} \downarrow\right) \left(\frac{\h... | {
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"Header 2": "4.1 SCHRÖDINGER EQUATION IN SPHERICAL COORDINATES",
"token_count": 1975,
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These tables also work the other way around:
$$|s_1 m_1\rangle |s_2 m_2\rangle = \sum_s C_{m_1 m_2 m}^{s_1 s_2 s} |s m\rangle.$$
[4.186]
For example, the shaded row in the $3/2 \times 1$ table tells us that
$$|\frac{3}{2}\frac{1}{2}\rangle|10\rangle = \sqrt{\frac{3}{5}}|\frac{5}{2}\frac{1}{2}\rangle + \sqrt{\... | {
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"token_count": 2035,
"source_pdf": "datasets/websources/Physics_v1/Physics/Griffiths - Introduction to quantum mechanics.pdf"
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Express your answer as a function of r, $\theta$ , $\phi$ , and a (the Bohr radius) only—no other variables $(\rho, z, \text{etc.})$ , or functions (Y, v, etc.), or constants $(A, a_0, \text{etc.})$ , or derivatives allowed $(\pi \text{ is okay, and } e, \text{ and } 2, \text{ etc.})$ .
- **(b)** Check that this w... | {
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"token_count": 1930,
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40 Let a be some convenient constant with the dimensions of length (the Bohr radius, say, if we're talking about hydrogen), and define the operators
$$q_{1} \equiv \frac{1}{\sqrt{2}} \left[ x + (a^{2}/\hbar) p_{y} \right]; \quad p_{1} \equiv \frac{1}{\sqrt{2}} \left[ p_{x} - (\hbar/a^{2}) y \right];$$
$$q_{2} \equi... | {
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"Header 2": "4.1 SCHRÖDINGER EQUATION IN SPHERICAL COORDINATES",
"token_count": 1989,
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#### 5.1 TWO-PARTICLE SYSTEMS
For a *single* particle, the wave function $\Psi(\mathbf{r}, t)$ is a function of the spatial coordinates $\mathbf{r}$ and the time t (we'll ignore spin for the moment). The wave function for a two-particle system is a function of the coordinates of particle one $(\mathbf{r}_1)$ , t... | {
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"Header 2": "IDENTICAL PARTICLES",
"token_count": 2009,
"source_pdf": "datasets/websources/Physics_v1/Physics/Griffiths - Introduction to quantum mechanics.pdf"
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Quantum mechanics neatly accommodates the existence of particles that are *indistinguishable in principle*: We simply construct a wave function that is *noncommittal* as to which particle is in which state. There are actually *two* ways to do it:
$$\psi_{\pm}(\mathbf{r}_1, \mathbf{r}_2) = A[\psi_a(\mathbf{r}_1)\psi... | {
"Header 1": "INTRODUCTION TO QUANTUM MECHANICS",
"Header 2": "IDENTICAL PARTICLES",
"token_count": 2003,
"source_pdf": "datasets/websources/Physics_v1/Physics/Griffiths - Introduction to quantum mechanics.pdf"
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