Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

European Journal of Physics

The biharmonic oscillator and asymmetric linear potentials: from classical trajectories to momentum-space probability densities in the extreme quantum limit

L J Ruckle, Mario Belloni, Richard W. Robinett

Published Sep 3, 2012• 1 citation
Publisher

Abstract

The biharmonic oscillator and the asymmetric linear well are two confining power-law-type potentials for which complete bound-state solutions are possible in both classical and quantum mechanics. We examine these problems in detail, beginning with studies of their trajectories in position and momentum space, evaluation of the classical probability densities for both x and p, and calculation of the corresponding quantum-mechanical solutions which give |?n(x)|2 and |?n(p)|2 for comparison to their classical counterparts in the?classically allowed regions. We then focus on the behavior of ?n(p) for large momenta, motivated by recent studies of the very direct connections between the continuity behavior of V(x) and the large-p limit of the momentum-space wavefunction.

DOI
10.1088/0143-0807/33/6/1505

Related papers

Quantum and classical probability distributions for position and momentum

Richard W. Robinett

The classical and quantum probability distributions for both position and momentum are compared for several model systems admitting bound states including the harmonic oscillator, the infinite well, and the linear confining potential (V(x)=F‖x‖). Examples corresponding to unbound systems, including the uniformly accelerating particle and the motion of a particle moving away from a point of unstable equilibrium, i.e., the ‘‘unstable oscillator’’ defined by V(x)=−kx2/2, are also considered. The quantum and classical distribution of kinetic and potential energy for the harmonic oscillator is briefly discussed.

American Journal of Physics•Sep 1, 1995•78 citations

Quantum solution for the biharmonic oscillator

W. Edward Gettys, H. W. Graben

The quantum-mechanical solution for the biharmonic oscillator is presented. This one-dimensional asymmetric oscillator is described by the potential energy function V(x) = mω12x2/2, x ≳ 0, and V(x) = mω22x2/2, x < 0. One of the more interesting features of the solution is that the energy levels are very nearly equally spaced. The biharmonic oscillator provides a relatively simple model for systems which lack the symmetry of the harmonic oscillator.

American Journal of Physics•Jul 1, 1975•13 citations

Comparing classical and quantum probability distributions for an asymmetric infinite well

M. A. Doncheski, Richard W. Robinett

We compare the classical and quantum mechanical position-space probability densities for a particle in an asymmetric infinite well. In an idealized system with a discontinuous step in the middle of the well, the classical and quantum probability distributions agree fairly well, even for relatively small quantum numbers, except for anomalous cases which arise due to the unphysical nature of the potential. We are able to derive upper and lower bounds on the differences between the quantum and classical results. We also qualitatively discuss the momentum-space probability densities for this system using intuitive ideas about the amount of time a classical particle spends in various parts of the well. This system provides an excellent example of a non-trivial, but tractable, quantum mechanical bound state problem where the correlations between the amplitude and curvature of quantum mechanical wavefunctions can be easily compared to classical intuition about particle motion, with quantitative success, but also warning of possible surprises in non-physical limiting cases.

arXiv:quant-ph/0307014•European Journal of Physics•Apr 5, 2000•27 citations

Visualizing classical and quantum probability densities for momentum using variations on familiar one-dimensional potentials

Richard W. Robinett

After briefly reviewing the definitions of classical probability densities for position, P CL ( x ), and for momentum, P CL ( p ), we present several examples of classical mechanical potential systems, mostly variations on such familiar cases as the infinite well and the uniformly accelerated particle for which the classical distributions can be easily derived and visualized. We focus especially on a simple potential which interpolates between the symmetric linear potential, V ( x ) = F | x |, and the infinite well, which can illustrate, in a mathematically straightforward way, how the divergent δ-function classical probability density for momentum for the infinite well can be seen to arise. Such examples can help students understand the quantum mechanical momentum-space wavefunction (and its corresponding probability density) in much the same way that other semiclassical techniques, such as the WKB approximation, can be used to visualize position-space wavefunctions.

arXiv:quant-ph/0307056•European Journal of Physics•Jan 30, 2002•11 citations

On Semiclassical Scattering

John Adair Barker, C. H. J. Johnson

The behavior of the quantum-mechanical phase shifts for scattering by a potential well is examined in the classical limit. The phase shifts tend to zero in the classical limit for all cases in which the corresponding classical trajectory does not pass through the well. This disproves by counterexample the result of Curtiss and Powers which indicated that there should be a contribution to the phase shifts even in the classical limit from classically inaccessible but energetically possible regions between two inner classical turning points. The error in the latter result appears to arise from the neglect of bound states of positive energy which exist in the classical limit (and only in that limit).

The Journal of Chemical Physics•Apr 1, 1965•6 citations

On the classical limit of quantum mechanics I

Bruce K. Driver, Pun Wai Tong

This paper is devoted to the study of the classical limit of quantum mechanics. In more detail we will elaborate on a method introduced by Hepp in 1974 for studying the asymptotic behavior of quantum expectations in the limit as Plank's constant ($\hbar)$ tends to zero. Our goal is to allow for unbounded observables which are (non-commutative) polynomial functions of the position and momentum operators. This is in contrast to Hepp's original paper where the observables were, roughly speaking, required to be bounded functions of the position and momentum operators. As expected the leading order contributions of the quantum expectations come from evaluating the observables along the classical trajectories while the next order contributions are computed by evolving the $\hbar=1$ observables by a linear canonical transformations which is determined by the second order pieces of the quantum mechanical Hamiltonian.

arXiv:1511.02447•Nov 8, 2015•1 citation
The biharmonic oscillator and the asymmetric linear well are two confining power-law-type potentials for which complete bound-state solutions are possible in both classical and quantum mechanics. We examine these problems in detail, beginning with studies of their trajectories in position and momentum space, evaluation of the classical probability densities for both x and p, and calculation of the corresponding quantum-mechanical solutions which give |?n(x)|2 and |?n(p)|2 for comparison to their classical counterparts in the?classically allowed regions. We then focus on the behavior of ?n(p) for large momenta, motivated by recent studies of the very direct connections between the continuity behavior of V(x) and the large-p limit of the momentum-space wavefunction.
The classical and quantum probability distributions for both position and momentum are compared for several model systems admitting bound states including the harmonic oscillator, the infinite well, and the linear confining potential (V(x)=F‖x‖). Examples corresponding to unbound systems, including the uniformly accelerating particle and the motion of a particle moving away from a point of unstable equilibrium, i.e., the ‘‘unstable oscillator’’ defined by V(x)=−kx2/2, are also considered. The quantum and classical distribution of kinetic and potential energy for the harmonic oscillator is briefly discussed.
We compare the classical and quantum mechanical position-space probability densities for a particle in an asymmetric infinite well. In an idealized system with a discontinuous step in the middle of the well, the classical and quantum probability distributions agree fairly well, even for relatively small quantum numbers, except for anomalous cases which arise due to the unphysical nature of the potential. We are able to derive upper and lower bounds on the differences between the quantum and classical results. We also qualitatively discuss the momentum-space probability densities for this system using intuitive ideas about the amount of time a classical particle spends in various parts of the well. This system provides an excellent example of a non-trivial, but tractable, quantum mechanical bound state problem where the correlations between the amplitude and curvature of quantum mechanical wavefunctions can be easily compared to classical intuition about particle motion, with quantitative success, but also warning of possible surprises in non-physical limiting cases.
After briefly reviewing the definitions of classical probability densities for position, P CL ( x ), and for momentum, P CL ( p ), we present several examples of classical mechanical potential systems, mostly variations on such familiar cases as the infinite well and the uniformly accelerated particle for which the classical distributions can be easily derived and visualized. We focus especially on a simple potential which interpolates between the symmetric linear potential, V ( x ) = F | x |, and the infinite well, which can illustrate, in a mathematically straightforward way, how the divergent δ-function classical probability density for momentum for the infinite well can be seen to arise. Such examples can help students understand the quantum mechanical momentum-space wavefunction (and its corresponding probability density) in much the same way that other semiclassical techniques, such as the WKB approximation, can be used to visualize position-space wavefunctions.
The behavior of the quantum-mechanical phase shifts for scattering by a potential well is examined in the classical limit. The phase shifts tend to zero in the classical limit for all cases in which the corresponding classical trajectory does not pass through the well. This disproves by counterexample the result of Curtiss and Powers which indicated that there should be a contribution to the phase shifts even in the classical limit from classically inaccessible but energetically possible regions between two inner classical turning points. The error in the latter result appears to arise from the neglect of bound states of positive energy which exist in the classical limit (and only in that limit).
This paper is devoted to the study of the classical limit of quantum mechanics. In more detail we will elaborate on a method introduced by Hepp in 1974 for studying the asymptotic behavior of quantum expectations in the limit as Plank's constant (ℏ)\hbar)ℏ) tends to zero. Our goal is to allow for unbounded observables which are (non-commutative) polynomial functions of the position and momentum operators. This is in contrast to Hepp's original paper where the observables were, roughly speaking, required to be bounded functions of the position and momentum operators. As expected the leading order contributions of the quantum expectations come from evaluating the observables along the classical trajectories while the next order contributions are computed by evolving the ℏ=1\hbar=1ℏ=1 observables by a linear canonical transformations which is determined by the second order pieces of the quantum mechanical Hamiltonian.