It is shown that the imaginary part of ψ*Pψ plays a crucial role in describing the dispersive properties of probability densities in quantum mechanics. The quantity Im(ψ*Pψ) is proportional to the dispersive velocity of the probability density and has the same form as the diffusion current. The dispersive velocity does not contribute to the expectation value of the momentum operator, but contributes to the expectation value of the kinetic energy operator. The interpretation of the dispersive velocity is given for the one-dimensional Gaussian wave packet, the stationary states of a particle in a one-dimensional box, and the stationary states of the hydrogen atom.
It is shown that the imaginary part of ψPψ plays a crucial role in describing the dispersive properties of probability densities in quantum mechanics. The quantity Im(ψPψ) is proportional to the dispersive velocity of the probability density and has the same form as the diffusion current. The dispersive velocity does not contribute to the expectation value of the momentum operator, but contributes to the expectation value of the kinetic energy operator. The interpretation of the dispersive velocity is given for the one-dimensional Gaussian wave packet, the stationary states of a particle in a one-dimensional box, and the stationary states of the hydrogen atom.
Schrödinger's equation can be considered as a diffusion equation with a diffusion coefficient β2=ℏ/2m. In this article, we explore the implications of this view. Rewriting the wave function in a polar form and transforming Schrödinger's equation into two real equations, we show that one of them reduces to the continuity equation, and the other, a nonlinear dynamical equation for the probability density. Considering these two equations as if they were the basic equations of quantum mechanics, we apply them to several one-dimensional quantum systems. We show the dispersive properties in the probability densities of stationary states of a particle in a rigid box and in harmonic potential; quasi-classical Gaussian probability densities of a free particle; and coherent states and squeezed states of the harmonic oscillator. We also present the soliton as a quantum mechanical representation of a free particle. We discuss the meaning of the diffusion coefficient β2 for each quantum system using a density plot.
Position probabilities play a privileged role in the interpretation of quantum mechanics. The quantum wave function (usually designated “Ψ(r)’) cannot itself represent a physical quantity, because it is complex valued. Its square — |Ψ(r)|2 — however, is real; and it is |Ψ(r)|2 which is the fundamental interpreted quantity of the theory. Schrödinger, who originated wave mechanics, interpreted |Ψ(r)|2 as the density of a charged particle; but this suggestion was inadequate. A principal reason is that, for most physical systems, |Ψ(r)|2 spreads out in time, whereas particles themselves do not. Schrödinger's interpretation was replaced by Born who introduced probabilities into quantum mechanics. Born urged that |Ψ(r)|2 represents not the density of the system in space, but rather a probability density. This is almost universally how quantum mechanics is taught today: |Ψ(r)|2 gives the probability that a system in state Ψ(r) is located at r.
Based on the Ehrenfest theorem, the time-dependent expectation value of a momentum operator can be evaluated equivalently in two ways. The integrals appearing in the expressions are taken over two different functions. In one case, the integrand is the quantum mechanical flux density j̲, and in the other, a different quantity j̲̃ appears, which also has the units of a flux density. The quantum flux density j̲ is related to the probability density ρ via the continuity equation, and j̲̃ may as well be used to define a density ρ̃ that fulfills a continuity equation. Employing a model for the coupled dynamics of an electron and a proton, we document the properties of the densities and flux densities. It is shown that although the mean momentum derived from the two quantities is identical, the various functions exhibit a very different coordinate and time-dependence. In particular, it is found that the flux density j̲̃ directly monitors temporal changes in the probability density, and the density ρ̃ carries information about wave packet dispersion occurring in different spatial directions.
In classical statistical mechanics, C exp[ -( mv*v/2 + V(x))/T) represents density i.e. the probability for a particle to have a velocity v and be at x. If applies this probability to Shannon’s entropy formula, one obtains C/T ( mv*v/2 + V(x)) exp[ -( mv*v/2 + V(x))/T) + exp[ -( mv*v/2 + V(x))/T) ln(C). Thus, it seems energy times density + density times ln(C) is the entropy density, a well known result in classical statistical mechanics, although sometimes not explicitly stated. The first term integrated over x is essentially the kinetic energy, while the second integrated over velocity yields the potential energy term. Thus, the first term partial density seems to be a purely momentum dependent kinetic energy piece times probability and the second a purely x dependent potential energy piece. (There is still an integral over velocity for the first term and one over x for the second to obtain the full entropy.) In quantum mechanics, we argue that a(p1)exp(ip1 x) a(p2) exp(ip2 x) serves as the probability where W(x)=Sum over p a(p) exp(ipx) = wavefunction. This leads to an expression for entropy density with two terms, one of which may be immediately integrated over momentum and the other over space. If one wishes to make a link to the classical statistical mechanical case, one may ask what requirement is necessary for the momentum related piece to represent average kinetic energy and the spatial part, average potential energy. The requirement, we find, yields the wavefunction of the ground state harmonic oscillator. For other cases, the partial density is not equivalent to average energy. Thus, we try to establish a link between entropy density and energy density, but require the kinetic energy related term to be strictly a function of momentum and the potential related term strictly one of x. In quantum mechanics, one may form averages at a point x. For example, KEave(x)= [Sum over p p*p/2m a(p) exp(ipx)]/W(x). If one examines a p/-p pair with a(p)=a(-p) (for example), cos(px) terms arise which are positive at some x and negative at others. We argued in previous notes that this represents stochastic flow in the opposite direction from that expected by -dV/dx. In other words, -dV/dx is an average which consists of stochastic force addends which may give momentum hits to the right or left, but which add to yield -dV/dx. As a result, one may have negative contributions to the kinetic energy (and density as well). In this note, we argue one may also have negative contributions to the entropy density due to this phenomenon.
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.