arXiv:quant-ph/0211026 · Physical Review A

Three-dimensional harmonic oscillator and time evolution in quantum mechanics

Pavel Kundrát, M. V. Lokajíček

Published Jan 13, 2003• 5 citations

Abstract

The problem of defining a time (or phase) operator for a three-dimensional harmonic oscillator has been analyzed. A formula for this operator has been derived. The results have been used to demonstrate a possibility of representing quantum-mechanical time evolution in the framework of an extended Hilbert-space structure. The physical interpretation of the extended structure has been discussed briefly, too.

DOI: 10.1103/physreva.67.012104

Related papers

Wave Mechanics in Quantum Phase Space: Harmonic Oscillator

Jun Lu

In this paper, we solve the rigorous solutions of the stationary Schrödinger equations for the harmonic oscillator within the framework of the quantum phase-space representation established by Torres-Vega and Frederick. We obtain the phase-space eigenfunctions of the three-dimensional isotropic harmonic oscillator in rectangular and spherical coordinate systems, and discuss the relationship between the eigenfunctions of the two coordinate systems. We also extend the "Fourier-like" projection transformations of the phase-space wave function to three-dimensional systems, and obtain the eigenfunctions in position and momentum representation through the transformations of the eigenfunction in the phase-space representation.

Physica ScriptaJan 1, 20044 citations

Thermodynamics of the harmonic oscillator using coherent states

Jürgen Schnack

The ongoing discussion whether thermodynamic properties can be extracted from a (possibly approximate) quantum-mechanical time evolution using time averages is fed with an instructive example. It is shown for the harmonic oscillator how the Hilbert space or an appropriately defined phase space must be populated in terms of coherent states in order to obtain the quantum result, respectively the classical one.

arXiv:cond-mat/9808287Europhysics Letters (EPL)Mar 15, 199926 citations

Two Wave Interpretation of the 3-Dimensional Harmonic Oscillator

Shun Liu

The three-dimensional harmonic oscillator is studied by means of two wave interpretation quantum theory.The time evolution equation and some constants of the motion are obtained that describe the single three dimensional harmonic oscillator and the classical limits.

Journal of Guangdong University of TechnologyJan 1, 20010 citations

Displacement and Squeeze Operators of a Three-Dimensional Harmonic Oscillator and Their Associated Quantum States

Mehdi Miri, Sina Ataollah Khorasani

We generalized the squeeze and displacement operators of the one-dimensional harmonic oscillator to the three-dimensional case and based on these operators we construct the corresponding coherent and squeezed states. We have also calculated the Wigner function for the three-dimensional harmonic oscillator and from the analysis of time evolution of this function, the quantum Liouville equation is also presented. Further properties of the quantum states including Mandel's Q and quadrature squeezing parameters are discussed as well.

arXiv:0912.3335International Journal of Optics and PhotonicsDec 17, 20090 citations

THE QUANTUM OSCILLATOR IN PHASE SPACE Part I

Emile Grgin, Guido V. Sandri

Starting with a real abstract algebra which encapsulates the algebraic structures of both classical and quantum mechanics, this paper presents a self–contained real-ization of the latter in phase space. Having both mechanics formulated in the same space opens two new windows into the comparative study of foundations. This stems from the fact that the same physical problem, as defined by a given Hamil-tonian, can be solved in several independent ways. The exact solutions can then be compared. Thus, comparison of the classical and quantum solutions in phase space offers new epistemological insights into Bohr’s correspondence principle, while com-parison of the quantum solutions in the different formalisms of Hilbert space and phase space yields new physical insights. These general ideas are then tested on the harmonic oscillator. The analytic ground work is presented in Part I, the exact solutions will be derived in Part II. 1.

Fizika BJan 1, 19960 citations

The Imaginary Unit as a Structural Invariant

Dai, Chuanjie

In standard presentations of mathematics and quantum physics the imaginary unit i is introduced axiomatically as a formal symbol with i 2 = −1, and the complex field C is then postulated as the natural home for wave functions and amplitudes. The physical reasons why such a structure is singled out are usually left implicit. In this paper we give a systematic derivation and reinterpretation of i from three complementary viewpoints, organised within the PAC–µ 8 framework. Part I adopts an axiomatic + vibrational (AV) viewpoint: starting from a real Hilbert space describing vibrational modes, together with three simple assumptions about time– translation, isotropy and phase completeness, we show that there is a distinguished linear operator J with J 2 = −1 acting as a quarter–period phase shift. Complex numbers arise by representing J as multiplication by i, so that “i” is nothing but the abstract 90◦ rotation operator on the vibrational plane. Part II adopts an axiomatic + quantum (AQ) viewpoint. We reformulate standard quantum postulates in terms of a real Hilbert space equipped with a complex structure J and a positive spectral generator K, with Ω = K1/2 and H = ℏΩ. Recovering the usual unitary time evolution Ut = exp(−iHt/ℏ) then requires no additional postulate about i: the symbol i is simply the matrix representation of J in a preferred basis. Part III connects i to the Phase–Arithmetic Spectral (PAS) axis of PAC–µ 8 . We discuss how phase geometry, irrational numbers such as √ 2 and π, Euler’s formula e iθ, Dirichlet characters and arithmetic L–functions can all be understood as expressions of the same underlying complex structure. In this sense i is the pivot that relates geometric rotations, vibrational phases and arithmetic spectra. Part IV illustrates the structural role of i on the physical side through four canonical examples: the harmonic oscillator, spin–1/2, two–path interference, and one–qubit gates in quantum information. In each case we show how i implements a 90◦ phase rotation in an appropriate two–dimensional vibrational plane. Part V distils these insights into concrete recommendations for rewriting standard text book and axiomatic treatments of complex numbers and quantum mechanics. We propose replacing the bare postulate “work over C” by a vibrationally motivated assumption of a real Hilbert space equipped with a complex structure J, and we outline how this shift clarifies the role of i in both physics and arithmetic. The overarching message is that i need not be treated as an arbitrary symbol: in the PAC–µ 8 framework it is a derived and geometrically natural object—the abstract quarter– period operator that organises vibrations, quantum phases and arithmetic characters into a single spectral story.

Nov 30, 20250 citations