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?

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