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Quantum-classical correspondence
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Quantum mechanics and classical mechanics are very close relatives.
There are analogous objects for everything of relevance in
classical and quantum statistical mechanics.
Observable f:
classical - real phase space function f(x,p)
quantum - Hermitian linear operator or sesquilinear form f
Lie product f \lp g:
read \lp as 'Lie', and visualize it as a very slanted L;
Macro for LaTeX:
\def\lp{\angle}
classical: f \lp g = {g,f} in terms of the Poisson bracket
quantum: f \lp g = i/hbar [f,g] in terms of the commutator
The Lie product is bilinear in the arguments and satisfies
f \lp g = - g \lp f
f \lp gh = (f \lp g)h + g(f \lp h) (Leibniz)
f \lp (g \lp h) = (f \lp g) \lp h + g \lp (f \lp h) (Jacobi)
Invariant measure:
classical - integral f := integral dxdp f(x,p)
quantum - integral f := trace f
Integrability: integral |f| finite
quantum integrable <==> f trace class
Partial integration formula:
integral f \lp g = 0.
Dynamics: df/dt = X_H f := H \lp f with Hermitian H
canonical transformations = mappings exp(tX_H) with Hermitian H
Liouville's theorem says that
integral f = integral exp(tX_H)f
The infinitesimal form of this is the partial integration formula.
State rho:
classical - real integrable phase space function rho(x,p)>=0
quantum - Hermitian positive semidefinite trace class operator rho
both normalized to integral rho = 1.
expectation of f in state rho:
= integral rho f