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Journal of Mathematical Physics

Mass dependence of Schrödinger wavefunctions

C. N. Leung, Jonathan L. Rosner

Published Jul 1, 1979• 8 citations
Publisher

Abstract

It is shown that for nonrelativistic central potentials of specific forms, the bound state wavefunctions u (r) have the property that G (r) ≡ℱ r0dx u (x)(∂u (x)/∂μ) ⩽0 for all r. Here μ is the reduced mass. Thus, for such potentials, the probability that a particle lies within a spherical shell of radius r is a monotonically increasing function of μ. The forms for which this property has been established are (a) V (r) =Crε, −2<ε<∞; (b) V (ir) =C lnr, and (c) S waves in the potential, V (r) =−V0 (0⩽r<a), V (r) =0 (a⩽r<∞). For all monotonically increasing potentials, G (r) ≳0 for nodeless states. It is shown explicitly that G (r) cannot be nonnegative for all bound states in an arbitrary monotonically increasing potential. The question remains open regarding the widest class of potentials for which G (r) is nonnegative for all bound states.

DOI
10.1063/1.524226

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It is shown that for nonrelativistic central potentials of specific forms, the bound state wavefunctions u (r) have the property that G (r) ≡ℱ r0dx u (x)(∂u (x)/∂μ) ⩽0 for all r. Here μ is the reduced mass. Thus, for such potentials, the probability that a particle lies within a spherical shell of radius r is a monotonically increasing function of μ. The forms for which this property has been established are (a) V (r) =Crε, −2<ε<∞; (b) V (ir) =C lnr, and (c) S waves in the potential, V (r) =−V0 (0⩽r<a), V (r) =0 (a⩽r<∞). For all monotonically increasing potentials, G (r) ≳0 for nodeless states. It is shown explicitly that G (r) cannot be nonnegative for all bound states in an arbitrary monotonically increasing potential. The question remains open regarding the widest class of potentials for which G (r) is nonnegative for all bound states.
The asymptotic properties of a Schrödinger wave function which represents the bound ground state of a system of three interacting particles are examined. It is assumed that the interaction can be described by a static potential which is the sum of three two-body potentials and one three-body potential, where the potentials have the property that if any one of the particles is separated from the other two by a distance which tends to infinity, then the part of the potential energy which depends on the position of that particle tends to zero. The problem is treated nonrelativistically. Decreasing exponential bounds on the ground-state three-body wave function are established in configuration space. It is shown that these bounds depend only on the masses of the three particles, on the ground-state energy of the three-body system, and on the lower bounds on the spectra of the Hamiltonians for the three two-body systems arising when one of the three particles is removed and the remaining two interact through the corresponding two-body potential. It is furthermore shown that the three-body wave function in momentum space admits of analytic continuation into a tube-region in complex momentum space, which tube-region is the product of real momentum space and a convex bounded region in imaginary momentum space. The tube-region is explicitly determined. The implications of the results for the theory of the vertex function in quantum field theory are discussed. The relevancy of the results obtained in this paper to the variational computation of energy levels and wave functions of three-body systems is also briefly discussed. In the course of the derivation some generalizations are pointed out; in particular, the final results remain valid for a class of more general interactions, which cannot be described by static potentials. Likewise, the results remain valid for a class of spin-dependent interactions.
There have been various derivations of the Schrodinger equation in the literature (e.g. (1)) based on hydrodynamics (continuity and conservation of momentum equations) with the added assumption that a velocity field u(x,t) = d/dx S(x,t) where S(x,t) is a new field. Furthermore a mathematical transformation Wv= sqrt(d(x,t)) exp(i S(x,t)) is used leading to the new function Wv(x). In (2) we compared the bound state Schrodinger scenario to the hydrodynamic equations in (1). In (1) d u(x,t)/dt = -dV/dx -dW/dx ((1)). V(x) is the usual potential and W(x) is described as a fluid stress potential. We also compared (1) to a free quantum particle and found that S(x,t) must contain a -Et piece if the hydrodynamical approach is to match quantum mechanics. In (1), ((1)) is shown to be equivalent to: d/dt partial S + ½ u(x,t)u(x,t) + W(x) + V(x) = 0 ((2)) with W being expressed in terms of spatial derivatives of density. d/dt partial S yields -E and u(x,t)=0 for a bound state so ((2)) is of the form of a conservation of energy equation. If a free quantum particle is described spatially by exp(ipx) then one may create a distribution Sum over p a(p)exp(ipx) and calculate average kinetic energy [-1/2m Sum over p a(p)pp/2m exp(ipx)] / [Sum over p a(p)exp(ipx)] and set it equal to W(x) as done in (2). In this note, we wish to examine the idea of a free particle conditional probability exp(-iEt+ipx) and compare it to the Schrodinger bound state which itself may be considered a localized solution if one ignores the internal wavefunction and considers only the center of mass features exp(-i (masses +En) t + ipoverall x) where p(overall) =0. We argue that center of mass features are contained in the Schrodinger bound state solution if one integrates over x relative to cm, the x variable in the Schrodinger equation. The idea of time t only applies to the cm, there is no time in Sum over p a(p)exp(ipx). Furthermore p(overall) =0 only if one integrates W(x) (-id/dx W) over all x (relative to cm). Also the form -i d/dx Wavefunction(free particle p) = p Wavefunction(free particle) where Conditional Prob= exp(-iEt+ipx) carries over to the bound state with: -i d/dx exp(ln(Wv)) ={ -i d/dx ln(Wv)} exp(ln(Wv)) where Wv=Sum over p a(p)exp(ipx). In other words there is an imaginary momentum -id/dx ln(Wv(x)) at each x in the bound solution so u(x,t)=0 holds for the real part. As a result, the hydrodynamical approach to quantum mechanics is not as straightforward as it may appear on the surface. We try to investigate a number of ideas related to the treatment of quantum bound states using hydrodynamics.
In this paper, Schrodinger equation has been solved analytically with position dependent mass by the Rosen-Morse potential. The position dependent mass defined as the function 1/(1 — tanh(η x )). Then corresponding position dependent mass substituted into Schrodinger equation. After that the obtained equation compared with associated Jacobi differential equation. Therefore, the eigenvalue and eigenfunctionhave been calculated, and from there the bound state has been found in terms of quantum numbers and the coefficients of potential. Also, the wave function has been obtained in terms of the associated Jacobi equation by μ = —3 and v = 2.