Chinese Physics B

Normal coordinate in harmonic crystal obtained by virtue of the classical correspondence of the invariant eigen-operator

Meng Xiang-Guo, Fan Hong-Yi, Ji‐Suo Wang

Published Jul 1, 2010• 1 citation

Abstract

Noticing that the equation with double-Poisson bracket, where On is normal coordinate, Hc is classical Hamiltonian, is the classical correspondence of the invariant eigen-operator equation (2004 Phys. Lett. A. 321 75), we can find normal coordinates in harmonic crystal by virtue of the invariant eigen-operator method.

DOI: 10.1088/1674-1056/19/7/070303

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Chinese Physics B

Normal coordinate in harmonic crystal obtained by virtue of the classical correspondence of the invariant eigen-operator

Meng Xiang-Guo, Fan Hong-Yi, Ji‐Suo Wang

Published Jul 1, 2010• 1 citation

Publisher

Abstract

DOI: 10.1088/1674-1056/19/7/070303

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Chinese Physics BSep 1, 20102 citations

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Relating the free particle with the harmonic oscillator

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Revista Mexicana de FísicaMay 1, 20251 citation

We describe an

p

-mechanical (see funct-an/9405002 and quant-ph/9610016) brackets which generate quantum (commutator) and classic (Poisson) brackets in corresponding representations of the Heisenberg group. We \emph{do not} use any kind of semiclassic approximation or limiting procedures for

\hbar \to 0

. Harmonic oscillator considered within the approach. Keywords: Classic and quantum mechanics, Hamilton and Heisenberg equations, Poisson brackets, commutator, Heisenberg group.

We consider two-dimensional harmonic oscillator in the complex Bargmann-Fock-Segal representation with

T^*{\mathbb R}^{2}={\mathbb C}^2

as classical phase space. We show that the eigenfunctions

ψ_n

of the quantum Hamiltonian correspond to complex radial coordinates in the reduced phase space

{\mathbb C}^2/{\mathbb Z}_n\subset{\mathbb C}^2

. They describe

{\mathbb Z}_n

-invariant motion of particle along a circle

S^1

in lens space

S^3/{\mathbb Z}_n\subset{\mathbb C}^2/{\mathbb Z}_n

, where

{\mathbb Z}_n

is the cyclic group of rotation by an angle

2π/n

on the circle

S^1

n=1,2,...\,

. Thus the general solution of the Schrödinger equation carries information about an infinite number of admissible classical states

ψ_n

that can be mapped to other states after lifting into the quantum bundle. We show that in the Kepler/hydrogen atom problem there is a similar correspondence between classical and quantum states.