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Journal of Russian Laser Research

Wave function of the harmonic oscillator in classical statistical mechanics

Vladimir N. Chernega, Vladimir I. Man’ko

Published Nov 1, 2007• 12 citations
Publisher
DOI
10.1007/s10946-007-0041-4

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The von Neumann trace form of quantum statistical mechanics is transformed to an integral over classical phase space. Formally exact expressions for the resultant position-momentum commutation function are given. A loop expansion for wave function symmetrization is also given. The method is tested for quantum harmonic oscillators. For both the boson and fermion cases, the grand potential and the average energy obtained by numerical quadrature over classical phase space are shown to agree with the known analytic results. A mean field approximation is given which is suitable for condensed matter, and which allows the quantum statistical mechanics of interacting particles to be obtained in classical phase space.
The time dependent harmonic oscillator is solved explicitly for quantum mechanics by the operator method with an auxiliary condition as the classical solution. Two classical invariant quantities which determine whether or not the system is bound are derived by the classical equation of motion. We obtain the invariant operator from one classical invariant quantity. Its eigenfunction is related to the solution of Schrodinger equation of the system and its eigenvalue is related to another classical quantity. The wave function is evaluated exactly by the eigenfunction of the invariant operator but it is not the eigenfunction of the Hamiltonian of the system. The uncertainty which calculates with the wave function is not a minimum one. We will confirm that the function which holds minimum uncertainty is a eigenfunction of the Hamiltonian.
The seventh chapter is concerned with the formulation of quantum statistical mechanics from first principles. It is shown that interactions with the reservoir or environment cause the subsystem wave function to collapse into a mixture of pure entropy states, which obey the axioms of classical probability theory. A dual representation in position and momentum states allows a permutation loop expansion for particle symmetrisation, and a second expansion accounting for the non-commutativity of the position and momentum operators. The leading term gives classical statistical mechanics, and higher order terms gives successive quantum corrections that become increasingly important for low temperatures, high densities, and small masses. From the permutation loop expansion, explicit results for the quantum ideal gas and the quantum harmonic oscillator are derived. The role of entropy in the classical manifestation of the universe given the underlying quantum mechanical wave equation is discussed.