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

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.

Quantization of fractional harmonic oscillator using creation and annihilation operators

Open Physics ◽  
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 .

NUMERICAL CALCULATIONS OF THE SMALL RELATIVISTIC CORRECTIONS OF ANISOTROPIC QUANTUM HARMONIC OSCILLATOR ENERGY EIGENVALUES: THE TWO- AND THREE-DIMENSIONAL CASE

1996 ◽  
Vol 07 (04) ◽  
pp. 563-571
Author(s):  
GHEORGHE ARDELEAN ◽  
ION I. COTĂESCU
Keyword(s):  
Harmonic Oscillator ◽  
Three Dimensional ◽  
Relativistic Correction ◽  
Numerical Calculations ◽  
Dimensional Case ◽  
Quantum Harmonic Oscillator ◽  
Energy Eigenvalues ◽  
Relativistic Corrections

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.

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

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.

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

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.

scholarly journals Determination of quasi-primary odors by endpoint detection

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Hanxiao Xu ◽  
Koki Kitai ◽  
Kosuke Minami ◽  
Makito Nakatsu ◽  
Genki Yoshikawa ◽  
...  
Keyword(s):  
Machine Learning ◽  
Arbitrary Number ◽  
The Other ◽  
Endpoint Detection ◽  
Alternative Approach ◽  
The Given

AbstractIt is known that there are no primary odors that can represent any other odors with their combination. Here, we propose an alternative approach: “quasi” primary odors. This approach comprises the following condition and method: (1) within a collected dataset and (2) by the machine learning-based endpoint detection. The quasi-primary odors are selected from the odors included in a collected odor dataset according to the endpoint score. While it is limited within the given dataset, the combination of such quasi-primary odors with certain ratios can reproduce any other odor in the dataset. To visually demonstrate this approach, the three quasi-primary odors having top three high endpoint scores are assigned to the vertices of a chromaticity triangle with red, green, and blue. Then, the other odors in the dataset are projected onto the chromaticity triangle to have their unique colors. The number of quasi-primary odors is not limited to three but can be set to an arbitrary number. With this approach, one can first find “extreme” odors (i.e., quasi-primary odors) in a given odor dataset, and then, reproduce any other odor in the dataset or even synthesize a new arbitrary odor by combining such quasi-primary odors with certain ratios.

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

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.

SU(d)-INVARIANT MULTIDIMENSIONAL q-OSCILLATORS WITH BOSONIC DEGENERACY

2000 ◽  
Vol 15 (19) ◽  
pp. 1237-1242 ◽  
Author(s):  
A. ALGIN ◽  
M. ARIK ◽  
N. M. ATAKISHIYEV
Keyword(s):  
Harmonic Oscillator ◽  
Quantum Group ◽  
Lie Group ◽  
The Other ◽  
Quantum Harmonic Oscillator ◽  
Standard Quantum ◽  
Newton Equation ◽  
Two Parameter

Multidimensional two-parameter (q1, q2)-oscillators are of two kinds: one is invariant under the (ordinary) Lie group SU (d), whereas the other is invariant under the quantum group SU q(d) where q = q1/q2. It is shown that the q1 = q2 limit of both of these two-parameter oscillators coincide and give the q-deformed Newton oscillator which can be derived from the standard quantum harmonic oscillator Newton equation. The bosonic degeneracies of the excited levels of these oscillators are different for q1 ≠ q2, but coincide in the q1 = q2 limit.

Determination of the lattice translation across {110} APBs in GaAs

1988 ◽  
Vol 46 ◽  
pp. 600-601
Author(s):  
D.R. Rasmussen ◽  
N.-H. Cho ◽  
C.B. Carter
Keyword(s):  
Grain Boundaries ◽  
Lower Energy ◽  
Antiphase Boundary ◽  
The Other ◽  
Boundary Structure ◽  
Interface Plane ◽  
Inversion Symmetry ◽  
High Angle ◽  
Equilibrium Distance

Domains in GaAs can exist which are related to one another by the inversion symmetry, i.e., the sites of gallium and arsenic in one domain are interchanged in the other domain. The boundary between these two different domains is known as an antiphase boundary [1], In the terminology used to describe grain boundaries, the grains on either side of this boundary can be regarded as being Σ=1-related. For the {110} interface plane, in particular, there are equal numbers of GaGa and As-As anti-site bonds across the interface. The equilibrium distance between two atoms of the same kind crossing the boundary is expected to be different from the length of normal GaAs bonds in the bulk. Therefore, the relative position of each grain on either side of an APB may be translated such that the boundary can have a lower energy situation. This translation does not affect the perfect Σ=1 coincidence site relationship. Such a lattice translation is expected for all high-angle grain boundaries as a way of relaxation of the boundary structure.

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