scholarly journals Langer Modification, Quantization Condition and Barrier Penetration in Quantum Mechanics

Universe ◽  
2020 ◽  
Vol 6 (7) ◽  
pp. 90
Author(s):  
Bao-Fei Li ◽  
Tao Zhu ◽  
Anzhong Wang
Keyword(s):  
Quantum Mechanics ◽  
Bound States ◽  
Analytical Approximation ◽  
Quantization Condition ◽  
Wave Functions ◽  
Potential Barriers ◽  
Transmission Coefficients ◽  
Clear Explanation ◽  
Exactly Solvable Potentials ◽  
Exactly Solvable

The WKB approximation plays an essential role in the development of quantum mechanics and various important results have been obtained from it. In this paper, we introduce another method, the so-called uniform asymptotic approximations, which is an analytical approximation method to calculate the wave functions of the Schrödinger-like equations, and it is applicable to various problems, including cases with poles (singularities) and multiple turning points. A distinguished feature of the method is that in each order of the approximations the upper bounds of the errors are given explicitly. By properly choosing the freedom introduced in the method, the errors can be minimized, which significantly improves the accuracy of the calculations. A byproduct of the method is to provide a very clear explanation of the Langer modification encountered in the studies of the hydrogen atom and harmonic oscillator. To further test our method, we calculate (analytically) the wave functions for several exactly solvable potentials of the Schrödinger equation, and then obtain the transmission coefficients of particles over potential barriers, as well as the quantization conditions for bound states. We find that such obtained results agree with the exact ones extremely well. Possible applications of the method to other fields are also discussed.

WKB method for potentials unbounded from below

2016 ◽  
Vol 30 (03) ◽  
pp. 1650003 ◽  
Author(s):  
Aleksandar Demić ◽  
Vitomir Milanović ◽  
Jelena Radovanović ◽  
Milenko Musić
Keyword(s):  
Quantum Mechanics ◽  
Bound States ◽  
Wave Functions ◽  
Wkb Method ◽  
Symmetric Potential ◽  
Overlap Integral ◽  
Entire Spectrum ◽  
One Dimensional ◽  
Model Based ◽  
Exactly Solvable

Bound states degenerated in energy (and differing in parity) may form in one-dimensional quantum mechanics if the potential is unbounded from below. We focus on symmetric potential and present quasi-exactly solvable (QES) model based on WKB method. The application of this method is limited on slow-changing potentials. We consider the overlap integral of WKB wave functions [Formula: see text] and [Formula: see text] which correspond to energies [Formula: see text] and [Formula: see text], and by setting [Formula: see text], we determine the type of spectrum depending on parameter [Formula: see text] which arises from this method. For finite value [Formula: see text], we show that the entire spectrum will consist of degenerated bound states.

scholarly journals HIGHER-ORDER SUSY, EXACTLY SOLVABLE POTENTIALS, AND EXCEPTIONAL ORTHOGONAL POLYNOMIALS

2011 ◽  
Vol 26 (25) ◽  
pp. 1843-1852 ◽  
Author(s):  
C. QUESNE
Keyword(s):  
Quantum Mechanics ◽  
Orthogonal Polynomials ◽  
Higher Order ◽  
Wave Functions ◽  
Supersymmetric Quantum Mechanics ◽  
Solvable Potentials ◽  
Exactly Solvable Potentials ◽  
Exactly Solvable ◽  
Exceptional Orthogonal Polynomials

Exactly solvable rationally-extended radial oscillator potentials, whose wave functions can be expressed in terms of Laguerre-type exceptional orthogonal polynomials, are constructed in the framework of kth-order supersymmetric quantum mechanics, with special emphasis on k = 2. It is shown that for μ = 1, 2, and 3, there exist exactly μ distinct potentials of μth type and associated families of exceptional orthogonal polynomials, where μ denotes the degree of the polynomial gμ arising in the denominator of the potentials.

scholarly journals SPECTRUM OF THE RELATIVISTIC PARTICLES IN VARIOUS POTENTIALS

2005 ◽  
Vol 20 (12) ◽  
pp. 911-921 ◽  
Author(s):  
RAMAZAN KOÇ ◽  
MEHMET KOCA
Keyword(s):  
Quantum Mechanics ◽  
Hypergeometric Functions ◽  
Two Dimensions ◽  
Dirac Oscillator ◽  
Confluent Hypergeometric Functions ◽  
Solvable Potentials ◽  
Exactly Solvable Potentials ◽  
Exactly Solvable ◽  
Nonrelativistic Quantum Mechanics ◽  
Relativistic Particles

We extend the notion of Dirac oscillator in two dimensions, to construct a set of potentials. These potentials become exactly and quasi-exactly solvable potentials of nonrelativistic quantum mechanics when they are transformed into a Schrödinger-like equation. For the exactly solvable potentials, eigenvalues are calculated and eigenfunctions are given by confluent hypergeometric functions. It is shown that, our formulation also leads to the study of those potentials in the framework of the supersymmetric quantum mechanics.

scholarly journals Precision Measurement of the Position-Space Wave Functions of Gravitationally Bound Ultracold Neutrons

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Y. Kamiya ◽  
G. Ichikawa ◽  
S. Komamiya
Keyword(s):  
Quantum Mechanics ◽  
Particle Physics ◽  
Bound States ◽  
Wave Functions ◽  
Recent Measurement ◽  
Ultracold Neutrons ◽  
Neutron Guide ◽  
Position Space ◽  
Position Sensitive Detectors ◽  
Space Wave

Gravity is the most familiar force at our natural length scale. However, it is still exotic from the view point of particle physics. The first experimental study of quantum effects under gravity was performed using a cold neutron beam in 1975. Following this, an investigation of gravitationally bound quantum states using ultracold neutrons was started in 2002. This quantum bound system is now well understood, and one can use it as a tunable tool to probe gravity. In this paper, we review a recent measurement of position-space wave functions of such gravitationally bound states and discuss issues related to this analysis, such as neutron loss models in a thin neutron guide, the formulation of phase space quantum mechanics, and UCN position sensitive detectors. The quantum modulation of neutron bound states measured in this experiment shows good agreement with the prediction from quantum mechanics.

EXACTLY-SOLVABLE COUPLED-CHANNEL MODELS FROM SUPERSYMMETRIC QUANTUM MECHANICS

2008 ◽  
Vol 22 (23) ◽  
pp. 2277-2286 ◽  
Author(s):  
JEAN-MARC SPARENBERG ◽  
ANDREY M. PUPASOV ◽  
BORIS F. SAMSONOV ◽  
DANIEL BAYE
Keyword(s):  
Inverse Problem ◽  
Quantum Mechanics ◽  
Bound States ◽  
Potential Model ◽  
Analytical Form ◽  
Exactly Solvable Potential ◽  
Exactly Solvable ◽  
Potential Matrix ◽  
Coupled Channel ◽  
Factorization Energy

Starting from a system of N radial Schrödinger equations with a vanishing potential and finite threshold differences between the channels, a coupled N × N exactly-solvable potential model is obtained with the help of a single non-conservative supersymmetric transformation. The obtained potential matrix, which subsumes a result obtained in the literature, has a compact analytical form, as well as its Jost matrix. It depends on N(N + 1)/2 unconstrained parameters and on one upper-bounded parameter, the factorization energy. For N = 2, previous results are reviewed, in particular regarding the number of bound states and resonances of the potential. A schematic inverse problem with one resonance is considered.

scholarly journals Unified approach for exactly solvable potentials in quantum mechanics using shift operators

2000 ◽  
Vol 62 (5) ◽  
Author(s):  
Mo-Lin Ge ◽  
L. C. Kwek ◽  
Yong Liu ◽  
C. H. Oh ◽  
Xiang-Bin Wang
Keyword(s):  
Quantum Mechanics ◽  
Unified Approach ◽  
Shift Operators ◽  
Solvable Potentials ◽  
Exactly Solvable Potentials ◽  
Exactly Solvable
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