On matrices whose eigenvalues are in arithmetic progression

1951 ◽  
Vol 47 (3) ◽  
pp. 585-590 ◽  
Author(s):  
P. T. Landsberg
Keyword(s):  
Quantum Mechanics ◽  
Harmonic Oscillator ◽  
Arithmetic Progression ◽  
Matrix Elements ◽  
One Dimensional ◽  
The Matrix ◽  
The One ◽  
Matrix Problems ◽  
Image Position

The following matrix problems are well known in quantum mechanics:(a) The one-dimensional harmonic oscillator. Givendetermine the eigenvalues hjj of H, and the matrix elements of X, P if H is diagonal. It is found (Wigner (4)) that

One-Dimensional Harmonic Oscillator in Quantum Phase Space

2011 ◽  
Vol 110-116 ◽  
pp. 3750-3754
Author(s):  
Jun Lu ◽  
Xue Mei Wang ◽  
Ping Wu
Keyword(s):  
Phase Space ◽  
Harmonic Oscillator ◽  
Quantum Phase ◽  
Space Representation ◽  
Wave Mechanics ◽  
One Dimensional ◽  
Matrix Mechanics ◽  
The Matrix ◽  
Quantum Wave ◽  
The One

Within the framework of the quantum phase space representation established by Torres-Vega and Frederick, we solve the rigorous solutions of the stationary Schrödinger equations for the one-dimensional harmonic oscillator by means of the quantum wave-mechanics method. The result shows that the wave mechanics and the matrix mechanics are equivalent in phase space, just as in position or momentum space.

Quantum mechanics on (anti)-de Sitter background II: Ramsauer–Townsend effect and WKB method

2018 ◽  
Vol 33 (26) ◽  
pp. 1850150 ◽  
Author(s):  
Won Sang Chung ◽  
Hassan Hassanabadi
Keyword(s):  
Quantum Mechanics ◽  
Energy Level ◽  
Harmonic Oscillator ◽  
Wkb Method ◽  
De Sitter ◽  
Quantum Harmonic Oscillator ◽  
One Dimensional ◽  
Sitter Background ◽  
The One

Based on the one-dimensional quantum mechanics on (anti)-de Sitter background [W. S. Chung and H. Hassanabadi, Mod. Phys. Lett. A 32, 26 (2107)], we discuss the Ramsauer–Townsend effect. We also formulate the WKB method for the quantum mechanics on (anti)-de Sitter background to discuss the energy level of the quantum harmonic oscillator and quantum bouncer.

One-dimensional quantum mechanics with Dunkl derivative

2019 ◽  
Vol 34 (24) ◽  
pp. 1950190
Author(s):  
Won Sang Chung ◽  
Hassan Hassanabadi
Keyword(s):  
Quantum Mechanics ◽  
Harmonic Oscillator ◽  
Scalar Product ◽  
Momentum Operator ◽  
Inner Product ◽  
One Dimensional ◽  
Coordinate Representation ◽  
The One ◽  
Infinite Wall

In this paper, we consider the quantum mechanics with Dunkl derivative. We use the Dunkl derivative to obtain the coordinate representation of the momentum operator and Hamiltonian. We introduce the scalar product to find that the momentum is Hermitian under this inner product. We study the one-dimensional box problem (the spin-less particle with mass m confined to the one-dimensional infinite wall). Finally, we discuss the harmonic oscillator problem.

scholarly journals The Generalized OTOC from Supersymmetric Quantum Mechanics—Study of Random Fluctuations from Eigenstate Representation of Correlation Functions

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 44
Author(s):  
Kaushik Y. Bhagat ◽  
Baibhab Bose ◽  
Sayantan Choudhury ◽  
Satyaki Chowdhury ◽  
Rathindra N. Das ◽  
...  
Keyword(s):  
Quantum Mechanics ◽  
Harmonic Oscillator ◽  
Quantum Statistical Mechanics ◽  
Supersymmetric Quantum Mechanics ◽  
The Other ◽  
Potential Well ◽  
One Dimensional ◽  
Quantum Randomness ◽  
The One ◽  
The Impact

The concept of the out-of-time-ordered correlation (OTOC) function is treated as a very strong theoretical probe of quantum randomness, using which one can study both chaotic and non-chaotic phenomena in the context of quantum statistical mechanics. In this paper, we define a general class of OTOC, which can perfectly capture quantum randomness phenomena in a better way. Further, we demonstrate an equivalent formalism of computation using a general time-independent Hamiltonian having well-defined eigenstate representation for integrable Supersymmetric quantum systems. We found that one needs to consider two new correlators apart from the usual one to have a complete quantum description. To visualize the impact of the given formalism, we consider the two well-known models, viz. Harmonic Oscillator and one-dimensional potential well within the framework of Supersymmetry. For the Harmonic Oscillator case, we obtain similar periodic time dependence but dissimilar parameter dependences compared to the results obtained from both micro-canonical and canonical ensembles in quantum mechanics without Supersymmetry. On the other hand, for the One-Dimensional Potential Well problem, we found significantly different time scales and the other parameter dependence compared to the results obtained from non-Supersymmetric quantum mechanics. Finally, to establish the consistency of the prescribed formalism in the classical limit, we demonstrate the phase space averaged version of the classical version of OTOCs from a model-independent Hamiltonian, along with the previously mentioned well-cited models.

scholarly journals Quantum mechanics of electrons in crystal lattices

1931 ◽  
Vol 130 (814) ◽  
pp. 499-513 ◽  
Keyword(s):  
Three Dimensional ◽  
Matrix Elements ◽  
Periodic Field ◽  
Crystal Lattices ◽  
One Dimensional ◽  
Physical Problems ◽  
Radiative Transitions ◽  
First Approximation ◽  
The Matrix ◽  
The One

Introduction .–Through the work of Bloch our understanding of the behaviour of electrons in crystal lattices has been much advanced. The principal idea of Bloch’s theory is the assumption that the interaction of a given electron with the other particles of the lattice may be replaced in first approximation by a periodic field of potential. With this model an interpretation of the specific heat, the electrical and thermal conductivity, the magnetic susceptibility, the Hall effect, and the optical properties of metals could be obtained. The advantages and limitations inherent in the assumption of Bloch will be much the same as those encountered when replacing the interaction of the electrons in an atom by a suitable central shielding of the unclear field, as in the work of Thomas and Hartree. In the papers quoted a number of general results were given regarding the behaviour of electrons in any periodic field of potential. To obtain a clearer idea of the details of this behaviour with a view to the application in special problems, however, it appeared worth while to investigate the mechanics of electrons in periodic fields of potential somewhat similar to those met with in practice and of such nature that the energy values W and eigenfunctions Ψ of the wave-equation can actually be computed. It is the purpose of this article to discuss a case where the integration is possible. In Section 1 the energy values and in Section 2 the wave-functions in their dependence on the binding introduced by the potential field are discussed for the one dimensional problem. In Section 3 the matrix elements of the linear momentum, which furnish the electric current associated with the various stationary states, are well as the probability of radiative transitions between these states, are evaluated. In Section 4 the results are extended to the three dimensional case and those features considered which one may expect to find in the case of more general periodic fields of potential. Section 5 deals with some applications to physical problems.

scholarly journals Matrix Elements for the one-dimensional Harmonic Oscillator

2000 ◽  
Vol 0044 ◽  
pp. 61-65
Author(s):  
José L. López-Bonilla ◽  
G. Ovando
Keyword(s):  
Harmonic Oscillator ◽  
Matrix Elements ◽  
One Dimensional ◽  
The One

Orthogonalitätsrelationen der mehrzeitigen erzeugenden Funktionale des Harmonischen Oszillators

1971 ◽  
Vol 26 (2) ◽  
pp. 220-223 ◽  
Author(s):  
R Weber
Keyword(s):  
Quantum Mechanics ◽  
Harmonic Oscillator ◽  
Scattering Theory ◽  
Functional Integration ◽  
One Dimensional ◽  
Eigen Values ◽  
The One

AbstractWe treat the one-dimensional harmonic oscillator completely in the field theoretic calculus of many time generating functionals. Without the results of common quantum mechanics we compute eigen values and functionals of the energy preparing all information of the harmonic oscillator. As an example of functional integration and for applications in scattering theory we prove orthonormality relations of these functionals.

Efficient computation of one-electron two-centre exchange integrals in ion–atom collisions

1976 ◽  
Vol 54 (9) ◽  
pp. 944-949 ◽  
Author(s):  
Alfred Msezane
Keyword(s):  
Principal Quantum Number ◽  
Computing Time ◽  
Translational Motion ◽  
Matrix Elements ◽  
Efficient Computation ◽  
Close Coupling ◽  
One Dimensional ◽  
State Approximation ◽  
The Matrix ◽  
Exchange Matrix

A scheme is presented for the reduction to one-dimensional integrals of any one-electron two-centre exchange matrix elements which incorporate the momentum associated with the translational motion of the electron. These elements are of the types occurring in close coupling-based treatments of ion–atom collisions. It is shown in a six state approximation, by coupling both eigenstates and pseudostates for the asymmetric He2+–H collision process, that computing time for the evaluation of the matrix elements is determined mainly by the number of different exponents in the matrix elements. The coupling of additional states with the same principal quantum number as the already coupled ones alters computing time insignificantly.

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