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American Journal of Physics

Critique and correction of the textbook comparison between classical and quantum harmonic oscillator probability densities

C. Leubner, Margot Alber, N. Schupfer

Published Dec 1, 1988• 13 citations
Publisher

Abstract

This comparison, although well meant, is unsuitable for demonstrating in introductory quantum mechanics the convergence of classical and quantum predictions for large quantum numbers, for it really tries to match ‘‘classical apples’’ with ‘‘quantum pears’’! First, it compares a classical oscillatory state with a quantum stationary state, instead of comparing it to the corresponding quantum nonstationary state. Second, it compares probabilities of different quality, namely, a classical probability of the classical oscillator (which could be circumvented by specifying the initial conditions precisely) with a quantum probability of the quantum oscillator (which cannot be avoided by any means). However, it is shown that both defects can be nicely corrected, illuminating the fact that—if properly formulated— classical and quantum oscillators have more in common than the textbooks are trying to make the beginner believe. The associated quantum position probability density, because it is now truly analogous to the classical situation, agrees much better with the classical curve than the textbook one does, and its analytical form also explains why the textbook quantum curve, derived from an unwarranted comparison, nevertheless resembles the classical curve in some sense.

DOI
10.1119/1.15736

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This comparison, although well meant, is unsuitable for demonstrating in introductory quantum mechanics the convergence of classical and quantum predictions for large quantum numbers, for it really tries to match ‘‘classical apples’’ with ‘‘quantum pears’’! First, it compares a classical oscillatory state with a quantum stationary state, instead of comparing it to the corresponding quantum nonstationary state. Second, it compares probabilities of different quality, namely, a classical probability of the classical oscillator (which could be circumvented by specifying the initial conditions precisely) with a quantum probability of the quantum oscillator (which cannot be avoided by any means). However, it is shown that both defects can be nicely corrected, illuminating the fact that—if properly formulated— classical and quantum oscillators have more in common than the textbooks are trying to make the beginner believe. The associated quantum position probability density, because it is now truly analogous to the classical situation, agrees much better with the classical curve than the textbook one does, and its analytical form also explains why the textbook quantum curve, derived from an unwarranted comparison, nevertheless resembles the classical curve in some sense.
Quantum mechanics is often compared with classical mechanics because both may deal with a single particle subject to a potential. Classical mechanics is said to be deterministic describing x(t) and its derivatives i.e showing how a particle moves and accelerates while bound state quantum mechanics produces a real wavefunction W(x) such that W(x)W(x) yields a spatial density profile. The comparison is carried further with the correspondence principle which states that at high energy levels quantum mechanics should mimic classical mechanics. Classical mechanics does not simply describe x(t) and its derivatives i.e. deterministic motion. It also includes two body scattering, force and torque balances for a stationary object etc. In the case of the classical statistical mechanics of an equilibrium ideal gas, the Maxwell-Boltzmann (MB) distribution in the absence of a potential is C1 exp(- pp/(2mT) ). If a potential V(x) is added, one may find P(x) two different ways using Newtonian ideas. The first is to consider the gas as a collective entity like a tiny macroscopic classical object at x. Then one may argue that equilibrium means the system is stationary and apply Newtonian statics i.e. a balance of pressures and -dV/dx to obtain exp(-V(x)/T) as is often done in textbooks. Alternatively, one may consider a single particle and apply Newtonian mechanics differently, namely use a conservation of energy equation: p1p1/2m + V(x1) = p2p2/2m + V(x2). Again one obtains exp(-V(x)/T). The spatial part of probability may be based on Newtonian mechanics indicating that it is not stochastic. The stochastic function is C1 exp(-pp/2mT). Interestingly, however, the two Newtonian approaches are different with respect to a single particle. A collective nonmoving set of particles acts as a stationary object, yet within it each particle seems to be “bouncing” off of an imaginary wall creating pressure. Thus one has an impulse free particle picture yet obtains the correct result exp(-V(x)/T). Alternatively one may think of a single particle accelerating which it does if it is not colliding with other particles. This is a very different picture of the single particle than of one bouncing back and forth hitting an imaginary wall. In fact, there is usually a wall holding the gas together and pressure and bouncing particles are the real picture there, but within the gas one would normally think of a single particle accelerating. Nevertheless there are two scenarios which yield the same result. The idea of pressure balance is more in keeping with an overall statistical picture of the gas. In other words, one may actually ignore the idea of following a single particle as it accelerates. (That is not to say that single particle acceleration does not exist - it does.) Quantum mechanics we argue is a statistical theory. In the case of a single bound particle one has a momentum distribution W(x)=wavefunction = Sum over p a(p)exp(ipx). One may apply conservation of average energy at each x to obtain the time-independent Schrodinger equation. In such a picture, like with the statistical mechanical gas, we argue that one may imagine impulse hits with the potential instead of considering acceleration. The quantity vrms(x) from KEclassical= .5m vrms(x)vrms(x) = -1/2m d/dx dW/dx / W does display acceleration, but this is a mathematical average. As energy levels become higher the correspondence principle is said to apply, but a classical particle exhibits acceleration. The quantum bound state does not, but like in the case of the statistical mechanical gas it may be that two scenarios lead to the same result. In other words, the quantum bound W(x) still has crests and troughs, but with low resolution one may only see an envelope roughly touching W(x)W(x) which matches a classical C2/v(x) where v(x) is classical velocity. Given that quantum mechanics is a statistical theory it may not be necessary to have it explicitly display acceleration.
An earlier study of quantum-mechanical time has suggested that, from the quantum-theoretic point of view, time is a random variable. The present work explores how such a description of time can be used to bring greater detail into the analysis of the one-dimensional harmonic oscillator. Specifically, the classical equations for harmonic motion are used as a point transformation between phase space and time-energy space, permitting probability distributions in the two spaces to be different representations of the same thing. The familiar energy-quantization hypothesis is supplemented with the hypothesis that quantized energy values are means (averages) of energy distributions characteristic of each energy state. With the probability distribution of time determined from the previously cited study, the desired quantized energy distributions follow from the requirements: (1) that they should be normalized, non-negative, and with means equal to the quantized energy values; (2) that the equivalent representation of distribution in position-momentum space should have marginal densities that are proportional (except near the origin) to the densities in position and momentum, ${|\ensuremath{\phi}(q)|}^{2}$ and ${|\ensuremath{\psi}(p)|}^{2}$, which are postulated in quantum theory. Requirement (2) is a weakening of the Born postulate. In the time-energy space, the probability distributions exclude energies smaller than the mode (energy value with maximum probability). In position-momentum space this exclusion requirement becomes the dynamical restriction: $\frac{{q}^{2}m{\ensuremath{\omega}}^{2}}{2}+\frac{{p}^{2}}{2m}\ensuremath{\ge}\frac{{\ensuremath{\Lambda}}_{n}\ensuremath{\hbar}\ensuremath{\omega}}{2}$, where, for the first three energy states, ${\ensuremath{\Lambda}}_{0}=0$, ${\ensuremath{\Lambda}}_{1}=\frac{3}{2}$, ${\ensuremath{\Lambda}}_{2}=2+\ensuremath{\surd}\frac{3}{2}$. In general terms it is thus shown that the quantized harmonic oscillator can be described as a statistical ensemble of classical oscillators. Some comparisons with work on phase-space formulations of quantum mechanics (especially with the work of Wigner) are made and some aspects of the theory that should submit to experimental verification are pointed out.
The classical and quantum mechanical solutions to three simple physical systems are directly compared by using the classical equations of motion to determine a probability density function. This function is compared to the probability density function for the quantum mechanical stationary state with the same energy. The nonclassical behavior of the quantum mechanical systems is thus made more distinct. This comparison allows students to gain experience using probability functions to describe physical systems and to see how the quantum mechanical solutions reduce to classical behavior in the high-energy/short-wavelength limit.
Both quantum mechanics and classical statistical mechanics predict the same momentum and spatial density forms forms for the solution of an oscillator. In the case of quantum mechanics, the ground state is considered. One may ask whether this is a coincidence, and if not, why two seemingly different theories should lead to the same result for the oscillator case. We argue in this note that it is the form of the conservation equation of the oscillator which leads to x and p having the same probability distribution.Furthermore, given that both quantum mechanics and classical statistical mechanics are of a statistical nature, one may inquire into the actual nature of the statistics being used, the meaning behind averages being used. For example, the classical deterministic solution of an oscillator can be written in statistical terms by removing time dependence and using density = dx/v. This does not mean, however, that the actual physical motion is no longer deterministic. In addition, we try to argue that quantum mechanics may be a more detailed statistical model including dynamics which satisfies the more static statistical result of spatial density for classical statistical mechanics.
Starting with a real abstract algebra which encapsulates the algebraic structures of both classical and quantum mechanics, this paper presents a self–contained real-ization of the latter in phase space. Having both mechanics formulated in the same space opens two new windows into the comparative study of foundations. This stems from the fact that the same physical problem, as defined by a given Hamil-tonian, can be solved in several independent ways. The exact solutions can then be compared. Thus, comparison of the classical and quantum solutions in phase space offers new epistemological insights into Bohr’s correspondence principle, while com-parison of the quantum solutions in the different formalisms of Hilbert space and phase space yields new physical insights. These general ideas are then tested on the harmonic oscillator. The analytic ground work is presented in Part I, the exact solutions will be derived in Part II. 1.