Title: ETH_Sem4 / Advanced_Kinetics / QM_Basics

CLASSICAL MECHANICS
- Particles described by position and momentum
- Dynamics according to Newton’s law
- Complete mathematical description of macroscopic objects
- First great success: planetary motion

QUANTUM MECHANICS
- Particles described by wavefunctions
- Dynamics according to Schrödinger’s equation
- Complete mathematical description of microscopic objects
- First great success: hydrogen atom

A very short introduction to Quantum Mechanics
Bras, kets and scalar products:
\(\quad \braket{\psi_a| \psi_b}=c\) a complex number
\(\quad \braket{\psi_b | \psi_a} =c^\ast\) its complex conjugate
Operators, eigenstates and eigenvalues:
\(\quad \hat F \ket{\phi_n}=f_n \ket{\phi_n} \) discrete
\(\quad \hat G \ket{\xi}=g(\xi)\ket{\xi} \) continuous
Eigenstates are orthonormal:
\(\quad \braket{\psi_n | \psi_{n'} } = \delta_{nn'} \) Kronecker delta
\(\quad \braket{\xi | \xi'}=\delta(\xi-\xi')\) Dirac delta function

Q: Completeness / resolution of the identity
\[\hat 1 = [...] \quad \text{discrete}\]\[\hat 1 = [...] d\xi \quad \text{continuous}\]
A: \sum_n \ket{\phi_n}\bra{\phi_n}; \int \ket{\xi}\bra{\xi}

Expectation Value:
\[\langle \hat G \rangle_\psi=\bra{\psi}\hat G \ket{\psi}  = \bra{\psi} \hat G \hat 1 \ket{\psi} = \int \bra{\psi}\hat G \ket{\xi} \braket{\xi | \psi} d \xi = \int \psi(\xi)^* g(\xi) \psi(\xi) d \xi\]

Q: Matrix representation operators:
\[\bra{\psi}\hat A \ket{\psi} = [...]\]where \[A_{nm}=[...]\]
A: \sum_{n, m}c^*_n A_{nm}c_m; \bra{\psi_n}\hat A \ket{\psi_m}

Q: Trace of a matrix:
\[\tr[A] = [...]\]for the continuous case:\[\tr[\hat A]=[...]\]
A: \sum_n A_{nn}; \int \bra{\xi}\hat A \ket{\xi} d \xi

Q: Density operator:\[\hat \rho =[...]\]
A: \sum_k P_k \ket{\psi_k}\bra{\psi_k}

Q: Density operator:\[\hat \rho = [...]\]Ensemble average:\[\langle \hat A \rangle^*_\rho=[...]\]
A: \sum_k P_k \ket{\psi_k}\bra{\psi_k}; \tr[\hat \rho \hat A]=\sum_k P_k \bra{\psi_k}\hat A \ket{\psi_k}

Example partition function harmonic oscillator:
Hamiltonian:\[H=\frac{p^2}{2m}+\frac{1}{2}m\omega^2x^2\]Classical: \(Q_{cl}= \frac{1}{2\pi\hbar}\int_{\R^2} dp dx \ e^{-\beta H(x,p)} = \frac{1}{\beta \hbar \omega} \)
Quantum: \(Q = \tr[e^{-\beta \hat H} ] = \sum_n \exp\bigg(-\beta(n+\frac{1}{2})\hbar\omega\bigg)=\frac{e^{-\beta\hbar\omega/2} }{1-e^{-\beta\hbar\omega} } = \frac{1}{2 \sinh(\beta\hbar\omega/2)} \)
They are equal for
Back Extra: \(\beta \to 0 \) meaning \(T \to \infty\)

Q: Exponential of a matrix: \(\exp(A) = [...]\)
A: \sum_{k=0}^\infty \frac{1}{ k!} A^k

Q: We can evaluate \(Q=\tr[e^{-\beta \hat H}]\) in [...] to give
\[\displaylines{Q = \sum_n \bra{\psi_n} e^{-\beta \hat H}\ket{\psi_n} \\ = [...]\\ = [...] } \]
A: the basis of eigenstates of \(\hat H\); \sum_n \bra{\psi_n} e^{-\beta E_n} \ket{\psi_n}; \sum_n e^{-\beta E_n}

Q: The classical partition function can be defined as\[Q_{cl}=[...]\]where f is [...]
A: \frac{1}{(2\pi\hbar)^f} \int \int e^{-\beta H(x,p)} dx dp; the number of degrees of freedom

Q: Example classical partition function of the free particle in one degree of freedom
\(Q_{cl}/L=[...]\)
using that\[\int e^{-\beta \frac{p^2}{2m} } dp = [...]\]
A: \bigg(\frac{m}{2\pi\beta\hbar^2} \bigg)^{1/2}; \sqrt{\frac{2\pi m}{\beta} }

Q: Time dependent Schrödinger equation (TDSE)\[[...] = [...]\]
A: i \hbar \dede{}{t} \ket{\Psi(t)}; \hat H \ket{\Psi(t)}

Q: Time-evolution operator from TDSE\[[...] = [...]\]The special case of time-independent \(\hat H\) leads to \[\hat U(t,t_0) = [...]\]
A: i \hbar \dede{\hat U(t, t_0)}{t}; \hat H \hat U(t, t_0); \exp\bigg(-\frac{i\hat H(t-t_0)}{\hbar} \bigg)

Q: Time evolution of the expectation value\[\abk{A(t)} = [...]\]
A: \bra{\Psi(0)} e^{i \hat H t/\hbar} \hat A e^{-i \hat H t /\hbar} \ket{\Psi(0)}

Differences between mixed states and superpositions of states
A superposition is a linear combination of states: \(\Psi(x) = c_1 \psi_1(x) + c_2 \psi_2(x)\). Applying the TDSE shows the probability density oscillates as\[\abs{\Psi(x,t)}^2 = \abs{c_1}^2\abs{\psi_1}^2 + \abs{c_2}^2 \abs{\psi_2}^2 + 2Re[c_1^*c_2 \psi_1(x)^* \psi_2(x)] \cos((E_2-E_1)t/\hbar)\]The dynamics of mixed states is simple, as they are independent. We only have time-dependence for the off-diagonal "coherence" terms\[\hat \rho(t) = \abs{c_1}^2 \ket{\psi_1}\bra{\psi_1} + e^{-i(E_1-E_2)t/\hbar} c_1 c_2^* \ket{\psi_1}\bra{\psi_2} + e^{-i(E_2-E_1)t/\hbar} c_2 c_1^* \ket{\psi_2}\bra{\psi_1} + \abs{c_2}^2 \ket{\psi_2}\bra{\psi_2} \]

Q: Correlation functions
In order to gain information about the dynamical processes, we define time-correlation functions.\[\abk{A(0),B(t)} = [...]\]
A: (2\pi\hbar)^{-f} \int \int \rho(\vec p_0, \vec q_0) A(\vec p_0, \vec q_0) B(\vec p_t, \vec q_t) d\vec p_0 d \vec q_0
Back Extra: After a long time the correlation between two measurements decays, i.e. the system decorrelates
and \(\abk{A(0)B(t)} \to \abk A \abk B \text{ as } t \to \infty\). This is a consequence of chaotic dynamics.

Q: [...]: [...]
A: Born-Oppenheimer approximation; Electron and lattice dynamics can be decoupled.