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Journal of the Physical Society of Japan

The Free Fall of Quantum Particles

Miki Wadati

Published Aug 15, 1999• 16 citations
Publisher

Abstract

Quantum mechanics of the free fall is studied. The general solution of the Schrödinger equation is obtained by way of the momentum representation. The initial value problem is explicitly solved. Further, its relations to the macroscopic wavefunctions and the hydrodynamic equations are discussed.
DOI
10.1143/jpsj.68.2543

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The momentum representation is seldom used in quantum mechanics courses. Some students are thence surprised by the change in viewpoint when, in doing advanced work, they have to use the momentum rather than the coordinate representation. In this work, we give an introduction to quantum mechanics in momentum space, where the Schrödinger equation becomes an integral equation. To this end we discuss standard problems, namely, the free particle, the quantum motion under a constant potential, a particle interacting with a potential step, and the motion of a particle under a harmonic potential. What is not so standard is that they are all conceived from momentum space and hence they, with the exception of the free particle, are not equivalent to the coordinate space ones with the same names. All the problems are solved within the momentum representation making no reference to the systems they correspond to in the coordinate representation.
The quantum version of the free fall problem is a topic often skipped in undergraduate quantum mechanics courses, because its discussion usually requires wavepackets built on the Airy functions—a difficult computation. Here, on the contrary, we show that the problem can be nicely simplified both for a single particle and for general many-body systems by making use of a gauge transformation that corresponds to a change of reference frame from the laboratory frame to the one comoving with the falling system. Using this approach, the quantum mechanics problem of a particle in an external gravitational potential reduces to a much simpler one where there is no longer any gravitational potential in the Schrödinger equation. It is instructive to see that the same procedure can be used for many-body systems subjected to an external gravitational potential and a two-body interparticle potential that is a function of the distance between the particles. This topic provides a helpful and pedagogical example of a quantum many-body system whose dynamics can be analytically described in simple terms.
In this chapter, the Schrödinger equation is “derived” for particles that can be described by de Broglie waves. The Schrödinger equation is very different from the corresponding equation of motion in classical mechanics. In order to illustrate the fundamental differences between the two theories, one of the simplest problems of particle dynamics is solved in both Newtonian and quantum mechanics. This simple example also helps to show that quantum mechanics is the fundamental theory and classical mechanics is an approximation, a remarkably good approximation, when considering macroscopic objects. The solution of the Schrödinger equation is presented for a particle inside a box and the quantization condition is derived. The amazing possibility of quantum tunneling when a particle is incident on a barrier of height larger than the energy of the incident particle is also discussed. Finally the three-dimensional Schrödinger equation is solved for the hydrogen atom.
Abstract In this chapter, the main features of quantum theory are presented. The chapter begins with a historical account of the invention of quantum mechanics. The meaning of position and momentum in quantum mechanics is discussed and non-commuting operators are introduced. The Schrödinger equation is presented and solved for a free particle and for a harmonic oscillator potential in one dimension. The meaning of the wavefunction is considered and the probabilistic interpretation is presented. The mathematical machinery and language of quantum mechanics are developed, including Hermitian operators, observables and expectation values. The uncertainty principle is discussed and the uncertainty relations are presented. Scattering and tunnelling by potential wells and barriers is considered. The use of variational principles to estimate ground state energies is explained and illustrated with a simple example.
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.