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Journal of Physics A Mathematical and General

Quantisation of velocity-dependent forces αx2and αx4

J. Geicke

Published Sep 7, 1989• 1 citation
Publisher

Abstract

The eigenvalue spectrum E l (V,n,α) of normalisable states of the equation (αℏ/im) n ϕ (n-1) (x)-ℏ 2 /2m) ϕ"(x)+(V(x)-E)ϕ(x) = 0, recently proposed as a quantisation of classical forces -αx n - δV/δx, is discussed for even n. For n = 4 and a rectangular potential well an analytic argument shows that bound states for 0 < α < , → 0, can exist only if E l (V,4,α) → E l (0) (V) as α → 0 where E l (0) (V) are the eigenvalues of the corresponding conservative Schrodinger equation (α = 0). Numerical results indicate E l (V,4,α) - E l (0) (V) = c l (V) α 2 + O(α 3 ) as α → 0. For n = 2, E l (V,2,α) - E l (0) (V)= 1/2 α 2 ℏ 2 m -3 is exact and independent of V(x), and at a potential step (to the left since α > 0) a continuous band of normalisable states exists instead of the scattering states for α =0.

DOI
10.1088/0305-4470/22/17/016

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Physical Review A•Feb 1, 1996•2 citations
The eigenvalue spectrum E l (V,n,α) of normalisable states of the equation (αℏ/im) n ϕ (n-1) (x)-ℏ 2 /2m) ϕ"(x)+(V(x)-E)ϕ(x) = 0, recently proposed as a quantisation of classical forces -αx n - δV/δx, is discussed for even n. For n = 4 and a rectangular potential well an analytic argument shows that bound states for 0 < α < , → 0, can exist only if E l (V,4,α) → E l (0) (V) as α → 0 where E l (0) (V) are the eigenvalues of the corresponding conservative Schrodinger equation (α = 0). Numerical results indicate E l (V,4,α) - E l (0) (V) = c l (V) α 2 + O(α 3 ) as α → 0. For n = 2, E l (V,2,α) - E l (0) (V)= 1/2 α 2 ℏ 2 m -3 is exact and independent of V(x), and at a potential step (to the left since α > 0) a continuous band of normalisable states exists instead of the scattering states for α =0.
In order to avoid the known problems of canonical quantisation of the dissipative equation mx+ alpha x n + delta V/ delta x=0, a modified approach is proposed. It is based on the classically conserved quantity E=mx 2 /2+V(x)+ alpha integral x n dx rather than on a Hamilton function. Linear, quadratic and cubic damping (n=1, 2 and 3) are discussed explicitly. For n=1 and n=3 normalisable (one-particle) states cannot be stable. However, for quadratic friction, which is classically a reversible process, the author shows, using the example of the harmonic oscillator, that stable normalisable eigenstates can exist.
Stationary states General properties We shall consider first some general characteristics of the ‘stationary states’. These are eigenfunctions of the Hamiltonian operator: The requirement that ψ satisfies the Schrödinger equation fixes its time dependence: Notice that (4.1.2) can only be a solution if the external potential V is independent of the time: V = V ( x ). The initial function ψ 0 is not arbitrary but must satisfy the time-independent Schrödinger equation (4.1.1). Writing ψ = R e iS /ħ one thus has The following deductions may be made from (4.1.3): ( a ) The probability density is independent of the time: |ψ| 2 = R 2 0 ( X ). ( b ) The quantum potential Q = (- ħ2/2 mR 0 )∇ 2 R 0 is time-independent. Therefore so is the total effective potential: ( c ) The velocity field v = (l/ m )∇ S 0 is independent of time and hence the trajectories, found by solving m = v ( x )| x = x ( t ) , are solutions of Moreover, ψ 0 ( x ) is often a real function for many bound state problems of interest (although not for scattering states) which implies that the velocity is zero. The particle is at rest where one would classically expect it to move since the quantum force (- ∇ Q ) cancels the classical force (- ∇ K ). ( d ) The energy of all particles in the ensemble, -∂ S /∂ t , is a constant of the motion and equal to the energy eigenvalue: with (4.1.4), we have an example of the conservation of energy (cf. §3.9.3). ( e ) The total field energy is conserved: 〈Ĥ〉 = E . […]
A consistent scheme is proposed for quantizing the potential amplitude in the Schrödinger equation in the case of negative energies (lying in the discrete-spectrum domain). The properties of the eigenfunctions φν(r) and eigenvalues ανcorresponding to zero, small, and large absolute values of energy E < 0 are analyzed. Expansion in the set {φν(r)} is used to develop a regular perturbation theory (for E < 0), and a general expression is found for the Green function associated with the time-independent Schrödinger equation. A similar method is used to solve several physical problems: the polarizability of a bound quantum-mechanical system, the two-center problem, and the elastic scattering of slow particles. The proposed approach is advantageous in that it does not require the use of continuum states (for E > 0).
A consistent scheme is proposed for quantizing the potential amplitude in the Schrödinger equation in the case of negative energies (lying in the discrete-spectrum domain). The properties of the eigenfunctions φν(r) and eigenvalues αν corresponding to zero, small, and large absolute values of energ y E<0 are analyzed. Expansion in the set φν(r) is used to develop a regular perturbation theory (for E<0), and a general expression is found for the Green function associated with the time-independent Schrödinger equation. A similar method is used to solve several physical problems: the polarizability of a bound quantum-mechanical system, the two-center problem, and the elastic scattering of slow particles. The proposed approach is advantageous in that it does not require the use of continuum states (for E>0).
A general formulation of the eigenvalue condition is pointed out for a one-dimensional stationary part of a separable Schrodinger equation with finite bounds, i.e. where the wavefunction vanishes at two arbitrary values of the independent variable. The procedure is general and applies equally to other eigenvalue problems of second-order, homogeneous differential equations. Where appropriate a continuous density of eigenvalues can be defined and evaluated without first solving the eigenvalue problem. Illustrative examples are provided, including gravitational and harmonic oscillator potentials.
Using the classical trajectory we quantize the motion of a pointlike particle of mass m and energy 1/2${\mathit{mx}\mathit{\ifmmode \dot{}\else \.{}\fi{}}}_{0}^{2}$ under the influence of the force Fn{\mathit{F}}_{\mathit{n}}Fn​=-\ensuremath{\alpha}\ensuremath{\Vert}x\ifmmode \dot{}\else \.{}\fi{}${\mathrm{\ensuremath{\Vert}}}^{\mathit{n}}$, \ensuremath{\alpha}\ensuremath{\gtrsim}0, n=1,2,.... Such forces are frictional only for x\ifmmode \dot{}\else \.{}\fi{}\ensuremath{\gtrsim}0 and antifrictional for x\ifmmode \dot{}\else \.{}\fi{}0 and can be simulated through the gradient of a one-parameter class of energy-dependent potentials Vn{\mathit{V}}_{\mathit{n}}Vn​(x;x${\mathrm{\ifmmode \dot{}\else \.{}\fi{}}}_{0}$), which increase monotonically in x. This permits one to construct a class of Lagrange and Hamilton functions that do not contain any integrating functions. We then proceed to quantization and study the quantum scattering problem assuming that Fn{\mathit{F}}_{\mathit{n}}Fn​ acts on a finite interval of the x axis only. The stationary problem can be solved analytically for all n, and quantitative examples are shown for n=2. The time-dependent scattering problem is discussed with emphasis on Ehrenfest's theorems and on linear damping n=1. When the quantum-mechanical probability for reflection is small the method provides an approximation to the motion of a wave packet through a dissipative medium, with velocity-power forces Fdn{\mathit{F}}_{\mathit{dn}}Fdn​=-\ensuremath{\alpha}\ensuremath{\Vert}x\ifmmode \dot{}\else \.{}\fi{}${\mathrm{\ensuremath{\Vert}}}^{\mathit{n}\mathrm{\ensuremath{-}}1}$x\ifmmode \dot{}\else \.{}\fi{}. \textcopyright{} 1996 The American Physical Society.