Energy eigenvalues of the three-dimensional quantum harmonic oscillator from
                    SU
                    (3) cubic Casimir operator

In this paper the small relativistic correction for the energy eigenvalues of the two- and three-dimensional anisotropic quantum harmonic oscillator are calculated, using as eigenstates [Formula: see text], for different values of the relativistic parameters βi ≡ ħwi / m0c2 with i = 1, 2 and 3.

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STUDY OF A RELATIVISTIC QUANTUM HARMONIC OSCILLATOR BY THE METHOD OF STATE-DEPENDENT DIAGONALIZATION

International Journal of Modern Physics C ◽

10.1142/s0129183196000545 ◽

1996 ◽

Vol 07 (05) ◽

pp. 645-653

Author(s):

H. C. LEE ◽

K. L. LIU ◽

C. F. LO

Keyword(s):

Harmonic Oscillator ◽

Relativistic Quantum ◽

Quantum Harmonic Oscillator ◽

Relativistic Limit ◽

New Approach ◽

Energy Eigenvalues ◽

State Dependent ◽

Anharmonic Potential ◽

Relativistic Parameter ◽

Special Case

We apply the method of State-dependent Diagonalization to study the eigenstates of the relativistic quantum harmonic oscillator in the low relativistic limit. The relativistic corrections of the energy eigenvalues of the quantum harmonic oscillator are evaluated for different values of the relativistic parameter α ≡ ħω0 / m0c2. Unlike the conventional exact diagonalization, this new method is shown to be very efficient for evaluating the energy eigenvalues and eigenfunctions. We have also found that for non-zero α the eigenfunctions of the system become more localized in space and that the ground state of the SHO (i.e., the α = 0 case) turns into a squeezed state. Furthermore, since our system is a special case of the quantum harmonic oscillator with a velocity-dependent anharmonic potential, this new approach should be very useful for investigating the cases with more complicated velocity-dependent anharmonic potentials.

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Symmetry algebra of the planar anisotropic quantum harmonic oscillator with rational ratio of frequencies

HNPS Proceedings ◽

10.12681/hnps.2881 ◽

2020 ◽

Vol 4 ◽

pp. 153

Author(s):

Dennis Bonatsos ◽

C. Daskaloyannis ◽

P. Kolokotronis ◽

D. Lenis

Keyword(s):

Harmonic Oscillator ◽

Linear Extension ◽

Symmetry Algebra ◽

Quantum Harmonic Oscillator ◽

Dimensional Representation ◽

General Applicability ◽

Finite Dimensional Representation ◽

Energy Eigenvalues ◽

Finite Dimensional ◽

Superintegrable Systems

The symmetry algebra of the two-dimensional quantum harmonic oscillator with rational ratio of frequencies is identified as a non-linear extension of the u(2) algebra. The finite dimensional representation modules of this algebra are studied and the energy eigenvalues are de- termined using algebraic methods of general applicability to quantum superintegrable systems.

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NUMERICAL CALCULATIONS OF THE SMALL RELATIVISTIC CORRECTIONS OF QUANTUM HARMONIC OSCILLATOR ENERGY EIGENVALUES

International Journal of Modern Physics C ◽

10.1142/s0129183195000630 ◽

1996 ◽

Vol 06 (06) ◽

pp. 773-780

Author(s):

GHEORGHE ARDELEAN

Keyword(s):

Harmonic Oscillator ◽

Relativistic Correction ◽

Numerical Calculations ◽

Quantum Harmonic Oscillator ◽

Energy Eigenvalues ◽

Relativistic Corrections ◽

Relativistic Parameter

The relativistic correction of the energy eigenvalues of quantum harmonic oscillator (QHO) are calculated using [Formula: see text] as eigenstates, for different values of the relativistic parameter α ≡ ħω/m0c2.

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A New Framework for the Determination of the Eigenvalues and Eigenfunctions of the Quantum Harmonic Oscillator

Journal of Mathematics Research ◽

10.5539/jmr.v13n6p20 ◽

2021 ◽

Vol 13 (6) ◽

pp. 20

Author(s):

Francis T. Oduro ◽

Amos Odoom

Keyword(s):

Harmonic Oscillator ◽

The Other ◽

Erential Equation ◽

Quantum Harmonic Oscillator ◽

Eigenvalues And Eigenfunctions ◽

Energy Eigenvalues ◽

Alternative Approach ◽

Schmidt Orthogonalization ◽

New Framework

This study was designed to obtain the energy eigenvalues and the corresponding Eigenfunctions of the Quantum Harmonic oscillator through an alternative approach. Starting with an appropriate family of solutions to a relevant linear di erential equation, we recover the Schr¨odinger Equation together with its eigenvalues and eigenfunctions of the Quantum Harmonic Oscillator via the use of Gram Schmidt orthogonalization process in the usual Hilbert space. Significantly, it was found that there exists two separate sequences arising from the Gram Schmidt Orthogonalization process; one in respect of the even eigenfunctions and the other in respect of the odd eigenfunctions.

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Quantified Energy of the Three‐Dimensional Quantum Harmonic Oscillator

10.1002/9781119881483.app1 ◽

2021 ◽

pp. 311-322

Keyword(s):

Harmonic Oscillator ◽

Three Dimensional ◽

Quantum Harmonic Oscillator

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Quantization of fractional harmonic oscillator using creation and annihilation operators

Open Physics ◽

10.1515/phys-2021-0035 ◽

2021 ◽

Vol 19 (1) ◽

pp. 395-401

Author(s):

Mohamed Al-Masaeed ◽

Eqab. M. Rabei ◽

Ahmed Al-Jamel ◽

Dumitru Baleanu

Keyword(s):

Harmonic Oscillator ◽

Annihilation Operator ◽

The State ◽

Hermite Functions ◽

Fractional Operators ◽

Quantum Harmonic Oscillator ◽

Eigenvalues And Eigenfunctions ◽

Creation And Annihilation Operators ◽

Energy Eigenvalues ◽

Fractional Harmonic Oscillator

Abstract In this article, the Hamiltonian for the conformable harmonic oscillator used in the previous study [Chung WS, Zare S, Hassanabadi H, Maghsoodi E. The effect of fractional calculus on the formation of quantum-mechanical operators. Math Method Appl Sci. 2020;43(11):6950–67.] is written in terms of fractional operators that we called α \alpha -creation and α \alpha -annihilation operators. It is found that these operators have the following influence on the energy states. For a given order α \alpha , the α \alpha -creation operator promotes the state while the α \alpha -annihilation operator demotes the state. The system is then quantized using these creation and annihilation operators and the energy eigenvalues and eigenfunctions are obtained. The eigenfunctions are expressed in terms of the conformable Hermite functions. The results for the traditional quantum harmonic oscillator are found to be recovered by setting α = 1 \alpha =1 .

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