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[
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"from": "human",
"value": "Suppose I have two free-particles $A,B$ whose positions evolve according to wave functions $\\psi_A(x,t)$ and $\\psi_B(x,t)$. I am interested in talking about the position vector from $A, B$, and ideally I would like to describe it according to a wave function lets call it $\\ps... |
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"from": "human",
"value": "PROBLEM 1.22 Give the hybridization state of each carbon in the following compounds:\n(f) Acrylonitrile (CH2=CHC≡N). What is the hybridization state of carbon in acrylonitrile (CH2=CHC≡N)?"
},
{
"from": "gpt",
"value": "Let's analyze the hybridization state of each c... |
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"from": "human",
"value": "$V(p) $ is the Coulomb potential. I think this theory should be fine because Hamilton's equations are somewhat symmetric in $x$ and $p$. The momentum space Schrodinger equation will be:\n$$i\\frac{d\\psi }{dt}=V(p) \\psi +\\frac{d^2}{dp^2}\\psi$$\nwhich is pretty much the same a... |
[
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"from": "human",
"value": "Let's consider free Maxwell theory:\n$$\nS_{Maxwell} = \\int d^dx \\; -\\frac{1}{4e^2}F_{\\mu\\nu}F^{\\mu\\nu}\n$$\nIn such theory one can define monopole operator using path integral via correlation functions with other operators:\n$$\n\\langle\\mathcal{M}(x) \\mathcal{O}_1(x_1... |
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{
"from": "human",
"value": "The electronic wave function of a molecule is an antisymmetric product of 1-electron molecular spin-orbitals. Using the electronic wave function of \\( \\text{H}_2 \\) molecule derive the expression for the ground state electronic energy of the molecule in terms of 1- and 2-elec... |
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"from": "human",
"value": "In trying to better understand the chiral-anomaly WZW-term interaction of the heavy vector mesons, I have trouble finding much relevant literature, but maybe I am looking in the wrong place. \nOne useful article I found is this one:\n\nS. Ozaki, H. Nagahiro, A. Hosaka, \n\"Magne... |
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{
"from": "human",
"value": "Alkylating butanone with methyl iodide leads to two different products since there are a-protons on either side of the carbonyl group. One of these products is obtained specifically by using a β-keto ester to make the target alkylation site more acidic. What complication arises ... |
[
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"from": "human",
"value": "I'm trying to prove that the QED Lagrangian\n$$\\mathscr{L}=\\bar{\\psi}(i\\!\\!\\not{\\!\\partial}-m)\\psi - \\frac{1}{4}F^{\\mu\\nu}F_{\\mu\\nu} - J^\\mu A_\\mu$$\nIs invariant under P and C. The two fields transform as\n$$A^\\mu(x) \\xrightarrow{P} - \\eta^*\\mathscr{P}^\\mu{... |
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"from": "human",
"value": "Consider the probability of finding an electron within a region:\n$$P= \\iiint |\\psi(x,y,z)|^2 dxdydz$$\nI would think that The probability of finding an electron at a single point in space would be defined as:\n$$P(x,y,z)= \\lim_{dx,dy,dz\\rightarrow 0} \\int_{x}^{x+dx}\\int_... |
[
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"from": "human",
"value": "The yellow crystalline antibiotic frustulosin was isolated from a fungus in 1978 and it was suggested the structure was an equilibrium mixture of **A** and **B**. Apart from the difficulty that the NMR spectrum clearly shows one compound and not an equilibrium mixture of two com... |
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"from": "human",
"value": "The formation of allantoine from uric acid is evidence that the latter contains two urea residues, and the formation of alloxan that it contains the arrangement of atoms represented by the symbols, - \nN-C \nCC \nN-C \nThese facts lead to the view that the structure of uric ... |
[
{
"from": "human",
"value": "Initially, the alkyne is converted into the E-alkene via the intermediate linear cation. The activation energy for this step is labelled AG₁. The E-alkene can convert to the Z isomer via an intermediate, with activation energy AG2. Since AG₁ is smaller than AG₂, the E-alkene for... |
[
{
"from": "human",
"value": "I am currently taking thermodynamics and am finding the concept of thermodynamic equilibrium rather difficult to grasp. My textbook gives a definition in terms of the virtual displacement of with respect to virtual displacements of variable $x_1,x_2,\\dots,x_n$ and writes\n$$\\d... |
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"from": "human",
"value": "Detailed features of the active site of RNA polymerase have been probed using modified RNA nucleotides that can react with nearby targets. In one study an analog of U carrying a reactive chemical group was incorporated at one of two positions in a radiolabeled RNA and then 'walk... |
[
{
"from": "human",
"value": "What do you mean by D- and L-descriptors of α-amino acids? Give examples. Are they related in any way with the CIP system of naming enantiomers containing chiral centres?"
},
{
"from": "gpt",
"value": "### D- and L-Descriptors of α-Amino Acids\n\nThe **D- and L-** sy... |
[
{
"from": "human",
"value": "Answer the Following:\\n\\n(a) (CH3)3N: is a stronger nucleophile than (CF3)3N. Comment on their basicities.\\n\\n(b) The order of reactivity of I- > Br- > Cl- > F- towards CH3Br in water is reversed in DMF solution. Explain.\\n\\n(c) Explain the effect of increase in solvent po... |
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"from": "human",
"value": "(b). H2C=CH-CH2-Br →[Mg] H2C=CH-CH2-MgBr + H2C=CH-CH2-Br →[SN2] Molecular Formula: C10H16"
},
{
"from": "gpt",
"value": "Let's analyze the problem carefully step-by-step.\n\n---\n\n### Given:\n\n- Substrate: **H2C=CH-CH2-Br** (allyl bromide, also called 3-bromopropen... |
[
{
"from": "human",
"value": "Calculate the time required to deposit 7.00 g of silver from a solution of AgNO\\(_3\\) with a current of 4.00 A."
},
{
"from": "gpt",
"value": "**Given:**\n\n- Mass of silver to be deposited, \\( m = 7.00 \\, \\mathrm{g} \\)\n- Current, \\( I = 4.00 \\, \\mathrm{A} ... |
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{
"from": "human",
"value": "I've been studying the interference phenomena observed in the double-slit experiment and came across a concept that piqued my interest. It is well understood that for interference patterns to form, the overlapping waves (or photons) must exhibit a degree of coherence. However, t... |
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{
"from": "human",
"value": "The usual explanation goes as follows. Charge flows until the applied electric field is canceled by the induced field, giving a total field $\\vec{E} = \\vec{0}$. This is reasonable if one gives as an example a rectangular slab of conductor with a constant applied field $\\vec{E... |
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{
"from": "human",
"value": "The industrial production of NH\\(_3\\) by use of a natural gas feedstock can be represented by the following simplified set of reactions:\\n\\n1. \\( \\text{CH}_4 + \\text{H}_2\\text{O} \\rightarrow \\text{CO} + 3\\text{H}_2 \\) \\n2. \\( 2\\text{CH}_4 + \\text{O}_2 \\rightarr... |
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{
"from": "human",
"value": "Find an equation of the normal plane to the curve that is the intersection of 9x^2 + 4y^2 - 36z = 0 and 3x + y + z - z^2 = 1 at the point (2, -3, 2)."
},
{
"from": "gpt",
"value": "Given the curve as the intersection of two surfaces:\n\\[\n\\begin{cases}\n9x^2 + 4y^2... |
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"from": "human",
"value": "The reaction \\(2\\text{HI} \\rightleftharpoons \\text{H}_2 + \\text{I}_2\\), at equilibrium contained 7.8 g, 203.2 g and 1638.4 g of \\(\\text{H}_2\\), \\(\\text{I}_2\\) and \\(\\text{HI}\\), respectively. Calculate \\(K_c\\)."
},
{
"from": "gpt",
"value": "Given th... |
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"from": "human",
"value": "How does one calculate the available energy in the transitions in the pictures below? \n\nIf I have a transition like $\\alpha\\rightarrow\\beta$ surrounded by transitions coupled to a single temperature like on the left diagram, it seems reasonable to me to calculate the availa... |
[
{
"from": "human",
"value": "I am currently dealing with the perturbation theory. For the moment in my lectures we are considering the time independent perturbation theory. Degeneracy might be relevant or not. I have a set of questions, which deal more with an intuitive understanding of this theory:\n\nWhy ... |
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{
"from": "human",
"value": "The following reaction occurs:\\n\\n\\[ \\n2 \\, \\text{Cp}(\\text{CO})_3\\text{Mo-H} + \\text{PhC}(\\text{CH}_3)=\\text{CH}_2 \\rightarrow [\\text{Cp}(\\text{CO})_3\\text{Mo}]_2 + \\text{PhCH}(\\text{CH}_3)_2 \\n\\]\\n\\nOBSERVATIONS\\n\\n1) Second order kinetics are observed, ... |
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