Physical Review A

General theorem on the Schrödinger equation

X. R. Wang

Published Dec 1, 1992• 17 citations

Abstract

Two discrete energy spectra of a particle moving in two different potential wells are compared. We find a general property: If one potential well ${\mathit{V}}_{1}$(r) is less than or equal to the other potential well ${\mathit{V}}_{2}$(r) everywhere in space, i.e., ${\mathit{V}}_{1}$(r)\ensuremath{\le}${\mathit{V}}_{2}$(r) for all r, then the nth eigenvalue of the particle in the first well, ${\mathit{V}}_{1}$, is always not greater than the nth eigenvalue of the particle in the second well, ${\mathit{V}}_{2}$.

DOI: 10.1103/physreva.46.7295

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Physical Review A

General theorem on the Schrödinger equation

X. R. Wang

Published Dec 1, 1992• 17 citations

Publisher

Abstract

Two discrete energy spectra of a particle moving in two different potential wells are compared. We find a general property: If one potential well

{\mathit{V}}_{1}

(r) is less than or equal to the other potential well

{\mathit{V}}_{2}

(r) everywhere in space, i.e.,

{\mathit{V}}_{1}

(r)\ensuremath{\le}

{\mathit{V}}_{2}

(r) for all r, then the nth eigenvalue of the particle in the first well,

{\mathit{V}}_{1}

, is always not greater than the nth eigenvalue of the particle in the second well,

{\mathit{V}}_{2}

DOI: 10.1103/physreva.46.7295

Related papers

One Dimensional Schrodinger Equation With Two Moving Boundaries

Ercan Yilmaz

arXiv:math-ph/0302006Feb 4, 20032 citations

A peculiar relation between mass and energy with respect to the Schrödinger equation

Dieter Flamm, Franz F. Schöberl, Hikotaro Uematsu

Considering energy eigenvalues E( ${m}_{1}$ , ${m}_{2}$ ) with respect to two particles whose masses are ${m}_{1}$ and ${m}_{2}$ , respectively, one would expect that if M>m then E(M, ${m}_{2}$ )>E(m, ${m}_{2}$ ), but actually the opposite can happen if one solves the Schr\"odinger equation. In other words, the derivative \ensuremath{\partial}E/\ensuremath{\partial} ${m}_{1}$ can be negative. In this paper a general condition of this problem is given.Considering energy eigenvalues E(${m}_{1}$,${m}_{2}$) with respect to two particles whose masses are ${m}_{1}$ and ${m}_{2}$, respectively, one would expect that if M>m then E(M,${m}_{2}$)>E(m,${m}_{2}$), but actually the opposite can happen if one solves the Schr\"odinger equation. In other words, the derivative \ensuremath{\partial}E/\ensuremath{\partial}${m}_{1}$ can be negative. In this paper a general condition of this problem is given.

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We consider the application of the supersymmetric quantum-mechanical formalism to the Schr\"odinger equation describing a particle characterized by a position-dependent effective mass $m(x).$ We show that any one-dimensional quantum system with effective mass has a supersymmetric partner system characterized by the same position dependence of the mass, but with a new potential function. The form of this supersymmetric partner potential ${V}_{2}(x)$ depends on both the form of the original potential ${V}_{1}(x)$ and the form of the mass x dependence. We also generalize the concept of shape invariance to the nonconstant mass scenario. As illustrative examples we provide, for a given form $m(x)$ of the effective mass, shape-invariant potentials exhibiting (a) harmonic-oscillator-like spectra and (b) Morse-like spectra. In both cases the energy eigenvalues and eigenfunctions can be obtained in algebraic fashion.We consider the application of the supersymmetric quantum-mechanical formalism to the Schr\"odinger equation describing a particle characterized by a position-dependent effective mass $m(x).$ We show that any one-dimensional quantum system with effective mass has a supersymmetric partner system characterized by the same position dependence of the mass, but with a new potential function. The form of this supersymmetric partner potential ${V}_{2}(x)$ depends on both the form of the original potential ${V}_{1}(x)$ and the form of the mass x dependence. We also generalize the concept of shape invariance to the nonconstant mass scenario. As illustrative examples we provide, for a given form $m(x)$ of the effective mass, shape-invariant potentials exhibiting (a) harmonic-oscillator-like spectra and (b) Morse-like spectra. In both cases the energy eigenvalues and eigenfunctions can be obtained in algebraic fashion.

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Speculation on the Role of Velocity in Quantum Mechanics

Francesco R. Ruggeri

Velocity plays a central role in classical mechanics, yet quantum mechanics deals with the variables E,t,p and x. P nonrelativistically is mo v, but a given p value corresponds to various m1v1=m2v2=m3v3 and so does not determine v uniquely. (A p, E pair or a p,mo pair is needed for this.). Here we consider a quantum particle in a square well potential. If detection of a particle occurs through impulse= Integral F dt, then m1v1=p and m2v2=p yield the same impulse although the velocities may be completely different. If one seeks a “time-independent” scenario then one would expect the two to yield the same spatial probability distribution even though E is different for each. In such a case E1 and E2 distinguish between v1 and v2. For a particle in an infinite well v1 and v2 represent different frequencies and so this “frequency” information role is taken on by energy. What happens if one considers two different average momenta levels for the same single particle? 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It is not restricted to L and so its probability density should mimic classical mechanics i.e. be a constant everywhere. To allow for this one considers a probability distribution of crests and troughs together with a second spatially shifted distribution indicating the direction and nature motion , but with the constraint that the modulus is 1. This suggests cos(px) and sin(px) for the two distributions. This is very different from the classical mechanical scenario of a bound state spatial density (1) which is proportional to dt with fixed dx i.e. Cdx/v(x) where v(x) is velocity. Such a picture only accounts for relative densities in space. Thus for two different constant velocities v1 and v2 which represent different frequencies of traversing a distance L, the spatial density is the same classically. Thus important information is lost and we argue that Cdx/v(x) is incomplete. In quantum mechanics the envelope of P(x)=W*(x)W(x)dx approaches the classical value for high n (energy level), but quantum mechanics contains much more information, namely the idea that different v for the same mo have different frequencies which must be manifested spatially.Velocity plays a central role in classical mechanics, yet quantum mechanics deals with the variables E,t,p and x. P nonrelativistically is mo v, but a given p value corresponds to various m1v1=m2v2=m3v3 and so does not determine v uniquely. (A p, E pair or a p,mo pair is needed for this.). Here we consider a quantum particle in a square well potential. If detection of a particle occurs through impulse= Integral F dt, then m1v1=p and m2v2=p yield the same impulse although the velocities may be completely different. If one seeks a “time-independent” scenario then one would expect the two to yield the same spatial probability distribution even though E is different for each. In such a case E1 and E2 distinguish between v1 and v2. For a particle in an infinite well v1 and v2 represent different frequencies and so this “frequency” information role is taken on by energy. What happens if one considers two different average momenta levels for the same single particle? In such a case mo is fixed and p1 and p2 correspond to different v1 and v2 i.e. to different frequencies. There is no time in a time-independent scenario so how does one represent the relative frequencies? One may do this spatially, we argue. If p1=n1 3.14/L and p2=n2 3.14/L where L is the well size then p1 has n1 crests and p2, n2 crests. Thus in one case relative velocity for the same p is distinguished by E1 and E2, but not by any spatial distribution. In the second for the same mo and different v1 and v2, there is both a different spatial distribution describing the relative frequencies, but also different E’s which also distinguish and are linked to frequency. This leads to the idea of probability crests with certain base widths proportional to 1/p. What happens in the case of a free particle? Such a particle should also be associated with crests linked to 1/p, but the particle is traveling through space. It is not restricted to L and so its probability density should mimic classical mechanics i.e. be a constant everywhere. To allow for this one considers a probability distribution of crests and troughs together with a second spatially shifted distribution indicating the direction and nature motion , but with the constraint that the modulus is 1. This suggests cos(px) and sin(px) for the two distributions. This is very different from the classical mechanical scenario of a bound state spatial density (1) which is proportional to dt with fixed dx i.e. Cdx/v(x) where v(x) is velocity. Such a picture only accounts for relative densities in space. Thus for two different constant velocities v1 and v2 which represent different frequencies of traversing a distance L, the spatial density is the same classically. Thus important information is lost and we argue that Cdx/v(x) is incomplete. In quantum mechanics the envelope of P(x)=W*(x)W(x)dx approaches the classical value for high n (energy level), but quantum mechanics contains much more information, namely the idea that different v for the same mo have different frequencies which must be manifested spatially.

Jun 30, 20220 citations

Two discrete energy spectra of a particle moving in two different potential wells are compared. We find a general property: If one potential well

{\mathit{V}}_{1}

(r) is less than or equal to the other potential well

{\mathit{V}}_{2}

(r) everywhere in space, i.e.,

{\mathit{V}}_{1}

(r)\ensuremath{\le}

{\mathit{V}}_{2}

(r) for all r, then the nth eigenvalue of the particle in the first well,

{\mathit{V}}_{1}

, is always not greater than the nth eigenvalue of the particle in the second well,

{\mathit{V}}_{2}

Considering energy eigenvalues E(

{m}_{1}

{m}_{2}

) with respect to two particles whose masses are

{m}_{1}

and

{m}_{2}

, respectively, one would expect that if M>m then E(M,

{m}_{2}

)>E(m,

{m}_{2}

), but actually the opposite can happen if one solves the Schr\"odinger equation. In other words, the derivative \ensuremath{\partial}E/\ensuremath{\partial}

{m}_{1}

can be negative. In this paper a general condition of this problem is given.

We consider the application of the supersymmetric quantum-mechanical formalism to the Schr\"odinger equation describing a particle characterized by a position-dependent effective mass

m(x).

We show that any one-dimensional quantum system with effective mass has a supersymmetric partner system characterized by the same position dependence of the mass, but with a new potential function. The form of this supersymmetric partner potential

{V}_{2}(x)

depends on both the form of the original potential

{V}_{1}(x)

and the form of the mass x dependence. We also generalize the concept of shape invariance to the nonconstant mass scenario. As illustrative examples we provide, for a given form

m(x)

of the effective mass, shape-invariant potentials exhibiting (a) harmonic-oscillator-like spectra and (b) Morse-like spectra. In both cases the energy eigenvalues and eigenfunctions can be obtained in algebraic fashion.