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arXiv:1512.05083 · European Journal of Physics

Quantum and classical probability distributions for arbitrary Hamiltonians

Claude Semay, Ludovic Ducobu

Published May 16, 2016• 12 citations
arXivHTMLPDF

Abstract

In the limit of large quantum excitations, the classical and quantum probability distributions for a Schrödinger equation can be compared by using the corresponding WKBJ solutions whose rapid oscillations are averaged. This result is extended for one-dimensional Hamiltonians with a non-usual kinetic part. The validity of the approach is tested with a Hamiltonian containing a relativistic kinetic energy operator.

DOI
10.1088/0143-0807/37/4/045403

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In the limit of large quantum excitations, the classical and quantum probability distributions for a Schrödinger equation can be compared by using the corresponding WKBJ solutions whose rapid oscillations are averaged. This result is extended for one-dimensional Hamiltonians with a non-usual kinetic part. The validity of the approach is tested with a Hamiltonian containing a relativistic kinetic energy operator.
The one-dimensional Wigner quasiprobability function is considered rigorously in the classical limit ($\ensuremath{\hbar}\ensuremath{\rightarrow}0$). With the implicit $\ensuremath{\hbar}$-dependence of the wave function $\ensuremath{\psi}$ ignored in the integral representation, this limit differs from ${|\ensuremath{\psi}(q)|}^{2}\ensuremath{\delta}(p)$ by terms $\frac{o(\ensuremath{\hbar})}{\ensuremath{\hbar}}$. With the implicit $\ensuremath{\hbar}$ dependence of $\ensuremath{\psi}$ accounted for (as it should be), rigorous asymptotic methods are applied, with the results that: (1) in the "classically allowed" region, the limit is an asymptotic exponential function of qqq, multiplied by an energy-conserving $\ensuremath{\delta}$ function; (2) in the "nonclassically allowed" region, this limit is zero. The stationary-phase approach leads nowhere in this regard. A WKBJ-type solution for the radial Schr\"odinger equation is developed, with rigorous error bounds in the different situations. The Wigner function is computed for an ensemble of one-dimensional harmonic oscillators, and an associated phase-space "density matrix" is constructed therefrom. This is compared with the Slater sum for the ensemble---in general, and in the particular cases of high temperature and of $\ensuremath{\hbar}\ensuremath{\rightarrow}0$; where $\ensuremath{\hbar}\ensuremath{\rightarrow}0$, the asymptotic behavior of the Wigner function is explicitly manifest.
In Part III we argued that a Schrodinger equation of the form: a d/dx d/dx W(x) + bxx W(x) = E W(x) may be scaled x→y such that yy = xx sqrt(ba) and E→ E sqrt(a/b) so that E is proportional to sqrt(b/a). We applied this idea to the quantum oscillator and noted that classically w, the angular frequency, for the same equation is sqrt(b/a). We further argued that this implies the momentum operator in quantum mechanics should scale as 1/xx as d/dx d/dx does. In this note we use a slightly different argument to analyze this same problem. Instead of treating the classical oscillator and time-independent Schrodinger equation as separate, we link them and show that scaling must occur with E being proportional to w. Furthermore we argue that if quantum mechanically one wishes to use probabilities f(p,x) to calculate average kinetic energy at x i.e. KE(at x) = {Sum over p pp/2m f(p,x) } / {Sum over p f(p,x)}, then f(p,x) = f(px) and pp should scale as 1/xx. Thus comparison of the classical oscillator equation directly with a statistical average type of quantum equation which uses V(x), E and KE(at x) yields information about E, f(p,x) and p i.e. introduces the idea of wavelength. Furthermore the discrete quantum energy values E=hbar w (n+.5) are dependent on w not through any boundary value considerations.
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.
The Schrodinger equation for a single spinless particle is formally obtained via a classical phonon model, namely the Frenkel-Kontorova model. Starting from a one-dimensional lattice of coupled harmonic oscillators, we show that the continuous limit of the corresponding Newtonian equation of motion yields the Klein-Gordon equation for a real-valued field. By introducing a complex-valued change of variables mixing the real-valued displacement and velocity fields, and by separating fast and slow time scales, the Klein-Gordon equation is written as the Schrodinger equation within the non-relativistic limit. This complex change of variable also allows to rewrite classical global observables of the phonon field, such as the total energy or momentum, as the corresponding quantum observables. Additionally, we show that when a friction force is incorporated into the classical model, the corresponding Klein-Gordon equation can be rewritten as a Schrodinger equation with a non-Hermitian Hamiltonian. While the global approach is limited here to the non-relativistic regime and does not address the measurement problem, quantization or relativistic effects, it nonetheless illustrates how quantum algebra and complex-valued wave functions can be exactly reproduced using classical dynamics. The relativistic regime for a spinless particle and the link between commutators and Poisson brackets is addressed in the second part of this series.