reperiendi

An analogy

Posted in Math, Quantum by Mike Stay on 2010 July 24
 Stat mech Quant mech column vector distribution $\displaystyle |p\rangle = \sum_j p_j|j\rangle$ amplitude distribution (wave function) $\displaystyle |\psi\rangle = \sum_j \psi_j |j\rangle$ row vector population $\displaystyle \langle p| = \sum_j p^*_j \langle j|$, where $p_j^*$ are the coefficients that, when normalized, maximize the inner product with $|p\rangle$ $\displaystyle \langle \psi| = \sum_j \psi_j^* \langle j|$, where $\psi_j^*$ is the complex conjugate of $\psi_j$. normalization $\displaystyle \sum_j p_j = 1$ $\displaystyle \sum_j |\psi_j|^2 = 1$ transitions stochastic unitary harmonic oscillator many HOs at temperature $T$ one QHO evolving for time $t$ uniform distribution over $n$ states $\displaystyle |U\rangle = \frac{1}{n}\sum_j |j\rangle$ $\displaystyle |U\rangle = \frac{1}{\sqrt{n}}\sum_j |j\rangle$ special distribution Gibbs distribution $\displaystyle p(T) = \sum_j e^{-E_j / k_B T} |j\rangle$ Free evolution $\displaystyle \psi(t) = H|U\rangle = \sum_j e^{-E_j \; it/\hbar} |j\rangle$ partition function = inner product with $\langle U|$ $\displaystyle Z(T) = \langle U|p(T)\rangle = \frac{1}{n}\sum_j e^{-E_j / k_B T}$ $\displaystyle Z(t) = \langle U|\psi(t)\rangle = \frac{1}{\sqrt{n}}\sum_j e^{-E_j \; it/\hbar}$ $\displaystyle \langle Q\rangle$ $\displaystyle = \frac{1}{Z(T)}\langle U| \; |Qp(T)\rangle$ $\displaystyle = \frac{1}{Z(t)}\langle U| \; |Q\psi(t)\rangle$ path integrals $E/T$ = maximum entropy? $Et$ = least action?