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5: Time-dependent Perturbation Theory
https://chem.libretexts.org/Courses/New_York_University/G25.2666%3A_Quantum_Chemistry_and_Dynamics/5%3A_Time-dependent_Perturbation_Theory
Contributors and Attributions Mark Tuckerman (New York University)
6: The Grand Canonical Ensemble
https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Advanced_Statistical_Mechanics_(Tuckerman)/06%3A_The_Grand_Canonical_Ensemble
In the grand canonical ensemble, the control variables are the chemical potential $\mu$ , the volume $V$ and the temperature $T$ . The total particle number $N$ is therefore allowed to fluctuate...In the grand canonical ensemble, the control variables are the chemical potential $\mu$ , the volume $V$ and the temperature $T$ . The total particle number $N$ is therefore allowed to fluctuate. It is therefore related to the canonical ensemble by a Legendre transformation with respect to the particle number $N$ . Its utility lies in the fact that it closely represents the conditions under which experiments are often performed and gives direct access to the equation of state.
5.1: Basic Thermodynamics
https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Advanced_Statistical_Mechanics_(Tuckerman)/05%3A_The_Isothermal-Isobaric_Ensemble/5.01%3A_Basic_Thermodynamics
The isothermal-isobaric ensemble is generated by transforming the volume $V$ in favor of the pressure $P$ so that the natural variables are $N$ , $P$ , and $T$ (which are conditions under whic...The isothermal-isobaric ensemble is generated by transforming the volume $V$ in favor of the pressure $P$ so that the natural variables are $N$ , $P$ , and $T$ (which are conditions under which many experiments are performed, e.g., `standard temperature and pressure'. $dG = \left(\frac {\partial G}{\partial P}\right)_{N,T} dP+ \left(\frac {\partial G}{\partial T}\right)_{N,P} dT+ \left(\frac {\partial G}{\partial N}\right)_{P,T} dN \nonumber$
4.3: Total spin
https://chem.libretexts.org/Courses/New_York_University/G25.2666%3A_Quantum_Chemistry_and_Dynamics/4%3A_Molecular_Quantum_Mechanics/4.3%3A_Total_spin
If the Hamiltonian is independent of spin, then it is clear that the total spin of an $N$ -particle system It can be shown (see problem set # 2) from the Dirac equation that when relativistic correct...If the Hamiltonian is independent of spin, then it is clear that the total spin of an $N$ -particle system It can be shown (see problem set # 2) from the Dirac equation that when relativistic corrections are accounted for, a term in the Hamiltonian appears that is explicitly spin dependent and takes the form We expect that they can be composed of tensor products of the basis vectors of the corresponding individual angular momenta but will not be equal to them.
11.1.4: The Continuous Limit
https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Advanced_Statistical_Mechanics_(Tuckerman)/11%3A_Introduction_to_path_integrals_in_quantum_mechanics_and_quantum_statistical_mechanics/11.01%3A_Discretized_and_Continuous_Path_Integrals/11.1.04%3A_The_Continuous_Limit
The integral is also referred to as a path integral because it implies an integration over all paths that a particle might take between $\tau = 0$ and $\tau = \beta \hbar$ such that \(x (0) = x (...The integral is also referred to as a path integral because it implies an integration over all paths that a particle might take between $\tau = 0$ and $\tau = \beta \hbar$ such that $x (0) = x (\beta \hbar$ , where the paths are parameterized by the variable $\tau$ (which is not time!).
10.4: A simple example - the quantum harmonic oscillator
https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Advanced_Statistical_Mechanics_(Tuckerman)/10%3A_Fundamentals_of_quantum_statistical_mechanics/10.04%3A_A_simple_example_-_the_quantum_harmonic_oscillator
$S = k\ln Q(\beta) + {E \over T} =-k\ln \left(1-e^{-\beta \hbar \omega} \right ) + {\hbar \omega \over T}{e^{-\beta \hbar \omega}\over 1-e^{-\beta \hbar \omega}} \nonumber$ \[ Q(\beta) = {1 \over ... $S = k\ln Q(\beta) + {E \over T} =-k\ln \left(1-e^{-\beta \hbar \omega} \right ) + {\hbar \omega \over T}{e^{-\beta \hbar \omega}\over 1-e^{-\beta \hbar \omega}} \nonumber$ $Q(\beta) = {1 \over h} \int dp dx e^{-\beta \left({p^2 \over 2m} + {1 \over 2} m\omega ^2 x^2 \right )} = {1 \over h} \left ( {2 \pi m \over \beta} \right )^{1/2} = {2\pi \over \beta \omega h} ={1 \over \beta \hbar \omega} \nonumber$
12.7: Relation to Spectra
https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Advanced_Statistical_Mechanics_(Tuckerman)/12%3A_Time-dependent_Processes_-_Classical_case/12.07%3A_Relation_to_Spectra
\[ \begin{align*} \Phi_{AB}(-t) &= {i \over \hbar} \langle \left[e^{-iH_0t/\hbar}Ae^{iH_0t/\hbar},B\right]\rangle _0 \\[4pt] &= { {i \over \hbar} \langle \left(e^{-iH_0t/\hbar}Ae^{iH_0t/\hbar}B -Be^{-...\[ \begin{align*} \Phi_{AB}(-t) &= {i \over \hbar} \langle \left[e^{-iH_0t/\hbar}Ae^{iH_0t/\hbar},B\right]\rangle _0 \\[4pt] &= { {i \over \hbar} \langle \left(e^{-iH_0t/\hbar}Ae^{iH_0t/\hbar}B -Be^{-iH_0t/\hbar}Ae^{iH_0t/\hbar}\right)\rangle _0 } \\[4pt] &= {i \over \hbar} \langle \left(Ae^{iH_0t/\hbar}Be^{-iH_0t/\hbar} -e^{iH_0t/\hbar}Be^{-iH_0t/\hbar}A\right)\rangle _0 \\[4pt] &= {i \over \hbar} \langle \left(AB(t)-B(t)A\right)\rangle _0 \\[4pt] &= -{i \over \hbar} \langle \left[B(t),A\right…
13.1: Calculation of spectra from perturbation theory
https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Advanced_Statistical_Mechanics_(Tuckerman)/13%3A_Time-dependent_Processes_-_Quantum_Case/13.01%3A_Calculation_of_spectra_from_perturbation_theory
3: The Microcanonical Ensemble
https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Advanced_Statistical_Mechanics_(Tuckerman)/03%3A_The_Microcanonical_Ensemble
The microcanonical ensemble is built upon the so called postulate of equal a priori probabilities:
3: Systems of Identical Particles
https://chem.libretexts.org/Courses/New_York_University/G25.2666%3A_Quantum_Chemistry_and_Dynamics/3%3A_Systems_of_Identical_Particles
Contributors and Attributions Mark Tuckerman (New York University)
13.1.2: The Transition Rate
https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Advanced_Statistical_Mechanics_(Tuckerman)/13%3A_Time-dependent_Processes_-_Quantum_Case/13.01%3A_Calculation_of_spectra_from_perturbation_theory/13.1.02%3A_The_Transition_Rate
\[\begin{align*} C_>(\omega) &= {1 \over 2\pi\hbar}\int_{-\infty}^{\infty}\;dt\sum_{i,f}w_i \vert\langle i\vert B\vert f\rangle \vert^2 e^{-i(E_f-E_i-\hbar\omega )t/\hbar} \\[4pt] &= {1 \over 2\pi\hba...\[\begin{align*} C_>(\omega) &= {1 \over 2\pi\hbar}\int_{-\infty}^{\infty}\;dt\sum_{i,f}w_i \vert\langle i\vert B\vert f\rangle \vert^2 e^{-i(E_f-E_i-\hbar\omega )t/\hbar} \\[4pt] &= {1 \over 2\pi\hbar}\int_{-\infty}^{\infty}\;dt\;e^{i\omega t}\sum_{i,f}w_i \vert\langle i\vert B\vert f\rangle \vert^2 e^{-i(E_f-E_i)t/\hbar} \\[4pt] &= {1 \over 2\pi\hbar}\int_{-\infty}^{\infty}\;dt\;e^{i\omega t}\sum _{i, f} w_i \langle i \vert B\vert f\rangle \langle f\vert B\vert i\rangle e^{-iE_f t/\hbar}e^{iE…

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