Square‐well representations for potentials in quantum mechanics

1992 ◽  
Vol 33 (10) ◽  
pp. 3472-3476 ◽  
Author(s):  
Richard L. Hall
Keyword(s):  
Quantum Mechanics ◽  
Square Well

Path integral for quantum dynamics with position-dependent mass within the displacement operator approach

2020 ◽  
Vol 35 (30) ◽  
pp. 2050246
Author(s):  
H. Benzair ◽  
M. Merad ◽  
T. Boudjedaa
Keyword(s):  
Quantum Mechanics ◽  
Path Integral ◽  
Quantum Dynamics ◽  
Translation Operator ◽  
Integral Approach ◽  
Operator Approach ◽  
Square Well ◽  
Dependent Mass ◽  
Position Dependent Mass ◽  
Coulomb Potentials

In the context of quantum mechanics reformulated in a modified Hilbert space, we can formulate the Feynman’s path integral approach for the quantum systems with position-dependent mass particle using the formulation of position-dependent infinitesimal translation operator. Which is similar a deformed quantum mechanics based on modified commutation relations. An illustration of calculation is given in the case of the harmonic oscillator, the infinite square well and the inverse square plus Coulomb potentials.

STICKY SPHERES IN QUANTUM MECHANICS

1994 ◽  
Vol 06 (05a) ◽  
pp. 947-975 ◽  
Author(s):  
M. D. PENROSE ◽  
O. PENROSE ◽  
G. STELL
Keyword(s):  
Quantum Mechanics ◽  
Brownian Motion ◽  
Gaseous Phase ◽  
Virial Coefficient ◽  
Hard Spheres ◽  
Second Virial Coefficient ◽  
Dimensional System ◽  
3 Dimensional ◽  
Fermi Statistics ◽  
Square Well

For a 3-dimensional system of hard spheres of diameter D and mass m with an added attractive square-well two-body interaction of width a and depth ε, let BD, a denote the quantum second virial coefficient. Let BD denote the quantum second virial coefficient for hard spheres of diameter D without the added attractive interaction. We show that in the limit a → 0 at constant α: = ℰma2/(2ħ2) with α < π2/8, [Formula: see text] The result is true equally for Boltzmann, Bose and Fermi statistics. The method of proof uses the mathematics of Brownian motion. For α > π2/8, we argue that the gaseous phase disappears in the limit a → 0, so that the second virial coefficient becomes irrelevant.

Do attractive scattering potentials concentrate particles at the origin in one, two and three dimensions? I. Potentials finite at the origin

1983 ◽  
Vol 388 (1795) ◽  
pp. 401-418 ◽  
Keyword(s):  
Quantum Mechanics ◽  
Classical Mechanics ◽  
Three Dimensions ◽  
Simple Extension ◽  
Final State ◽  
Central Potential ◽  
Scattering State ◽  
Square Well ◽  
Semiclassical Regime ◽  
High Wavenumbers

Paradoxically, in beta decay, for instance, the final-state Coulomb forces pulling the electron inwards accelerate the emission. Quantum mechanics (q. m. ) makes the rate proportional to α ≡ ρ 0 / ρ ∞ ; ρ 0, ∞ (and v 0, ∞ ) are the particle densities (and speeds) at r = 0 and far upstream in the scattering state which describes the electron. Hence, as regards the effects of finalstate interactions, one must base one’s physical intuition on this ratio α . It is shown that according to (non-relativistic) classical mechanics, if the origin is accessible, then any central potential U(r) where v 0 < ∞ (i. e. where U (0) > -∞) gives in 1, 2 and 3 dimensions, α 1 = v ∞ / v 0 , α 2 = 1, α 3 = v 0 / v ∞ ; the remaining course of U(r) is irrelevant to α . The same results hold also in q. m. in the semiclassical regime, i. e. in the W. K. B. approximation which for such potentials becomes valid at high wavenumbers; in 2D it needs rather careful formulation, and in 3D one must avoid the Langer modification. (The W. K. B. results apply even if d U / d r diverges at r = 0, provided U (0) remains finite; these cases are covered by a simple extension of the argument. ) The square-well and exponential potentials are discussed as examples. Potentials which diverge at the origin are treated in the following paper.

Time-dependent versus time-independent probabilities of transmission and reflection of a quantum particle incident upon a step potential and a square potential well

2016 ◽  
Vol 94 (1) ◽  
pp. 9-14 ◽  
Author(s):  
Mark R.A. Shegelski ◽  
Kevin Malmgren ◽  
Logan Salayka-Ladouceur
Keyword(s):  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Quantum Particle ◽  
Time Dependent ◽  
Potential Well ◽  
Step Potential ◽  
Exact Analytical Expression ◽  
Square Well ◽  
Transmission And Reflection

We investigate the transmission and reflection of a quantum particle incident upon a step potential decrease and a square well. The probabilities of transmission and reflection using the time-independent Schrödinger equation and also the time-dependent Schrödinger equation are in excellent agreement. We explain why the probabilities agree so well. In doing so, we make use of an exact analytical expression for the square well for time-dependent transmission and reflection, which reveals additional interesting and unexpected results. One such result is that transmission of a wave packet can occur with the probability of transmission depending weakly on the initial spread of the packet. The explanations and the additional results will be of interest to instructors of and students in upper year undergraduate quantum mechanics courses.

scholarly journals Leaky Modes of Waveguides as a Classical Optics Analogy of Quantum Resonances

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Sara Cruz y Cruz ◽  
Oscar Rosas-Ortiz
Keyword(s):  
Quantum Mechanics ◽  
Short Range ◽  
Bound States ◽  
Waveguide Structure ◽  
One Dimensional ◽  
Leaky Modes ◽  
Scattering States ◽  
Quantum Resonances ◽  
Zone Of Influence ◽  
Square Well

A classical optics waveguide structure is proposed to simulate resonances of short range one-dimensional potentials in quantum mechanics. The analogy is based on the well-known resemblance between the guided and radiation modes of a waveguide with the bound and scattering states of a quantum well. As resonances are scattering states that spend some time in the zone of influence of the scatterer, we associate them with the leaky modes of a waveguide, the latter characterized by suffering attenuation in the direction of propagation but increasing exponentially in the transverse directions. The resemblance is complete because resonances (leaky modes) can be interpreted as bound states (guided modes) with definite lifetime (longitudinal shift). As an immediate application we calculate the leaky modes (resonances) associated with a dielectric homogeneous slab (square well potential) and show that these modes are attenuated as they propagate.

Quantum chaos in a double square well: An approach based on Bohm’s view of quantum mechanics

1998 ◽  
Vol 58 (5) ◽  
pp. 6851-6854 ◽  
Author(s):  
O. F. de Alcantara Bonfim ◽  
J. Florencio ◽  
F. C. Sá Barreto
Keyword(s):  
Quantum Mechanics ◽  
Quantum Chaos ◽  
Square Well
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