Comment on “The negative flow of probability” [Am. J. Phys. 88, 325–333 (2020)]

In a recent paper,¹ Villanueva has explored a seemingly paradoxical effect associated with the motion of a free quantum particle. The particle state is represented by a Gaussian wave packet, where x and t are the position and time variables, respectively. Here, is the wave packet center at time t, with x₀ and p₀ being the initial mean position and momentum, respectively; m is the particle mass. The complex-valued function α_t is defined as and controls the wave packet spread, along with position-momentum correlations. Its initial value, α₀, must satisfy $Im α 0 >0$ for the wave packet to be normalizable. Finally, the complex-valued function γ_t encapsulates both the normalization constant and global phase; the imaginary part of γ_t is related to that of α_t via The wave function $ψ ( x , t )$ satisfies the free particle Schrödinger equation

The effect addressed by Villanueva can be summarized as follows. Let q be some fixed point on the x-axis, and consider the scenario in which In other words, the initial wave packet is localized (almost entirely) on the left of q and is moving to the right. Naively, one might think that, as the wave packet approaches the point q from the left (that is, as long as $x t < q$⁠), the probability of finding the particle in the region x > q, namely, grows monotonically with time, i.e., one might expect that $d / Π ( t ) d t > 0$ for $0 < t < t cl ≡ m ( q − x 0 ) / p 0$⁠. However, as Villanueva demonstrates, this naive intuition can be false: if the value of the parameter α₀ is such that at an instant $t < t cl$⁠, then the probability $Π ( t )$ appears to be decreasing,

Let us regard the motion of a free particle, described in the Schrödinger picture by the wave function $ψ ( x , t )$⁠, as the time evolution of the Wigner phase-space (quasiprobability) density^2,3 The reader is referred to review articles^2,3 for a discussion of many fascinating properties of the Wigner density. Here, we only state the following two properties, particularly relevant to the present discussion. First, the time evolution of $W ( x , p , t )$ in free space is governed by This equation is the phase-space representation to the free-particle Schrödinger equation (5). Second, in terms of the Wigner function, the probability of finding the particle in the region x > q reads This is the phase-space representation of Eq. (6).

Substituting $ψ ( x , t )$⁠, given by Eq. (1), into Eq. (8), performing the integration, and using identity (4), we obtain where are, respectively, the particle position and momentum measured relative to their mean values. The Wigner function (10) is positive at all times, $W ( x , p , t ) > 0$⁠.

Now comes an important (and well known) argument. Imagine an ensemble of N free noninteracting classical particles of mass m. We are interested in the limit of large N. Let $N W cl ( x , p , t )$ be the number density of the particles at the phase-space point (x, p) at time t. So, $W cl ( x , p , t )$ is the probability density for a given particle to be found at (x, p) at time t. Since the momentum of a free particle is conserved, the time evolution of $W cl$ is given by The number of particles in the region x > q at time t equals $N Π cl ( t )$⁠, where represents the probability for a given particle to satisfy x > q at time t. Observe that Eqs. (8) and (9) for the Wigner function W of a quantum particle are identical to Eqs. (11) and (12) for the classical phase-space probability density $W cl$⁠, respectively.⁴ It then immediately follows that provided that $W ( x , p , 0 ) = W cl ( x , p , 0 )$⁠. In general,^2,3 the set of all possible Wigner functions $W ( x , p , 0 )$ is not the same as the set of all possible classical probability densities $W cl ( x , p , 0 )$⁠. For example, $W ( x , p , 0 )$ can have negative values, whereas $W cl ( x , p , 0 )$ is nonnegative by construction; on the other hand, the value of $W cl ( x , p , 0 )$ can, in principle, be arbitrarily large, whereas $| W ( x , p , 0 ) | ≤ 1 / π ℏ$ for all x and p. However, the Wigner function W representing a Gaussian wave packet is everywhere positive, Eq. (10), and, therefore, can be regarded as a valid classical probability density $W cl$⁠. This guaranties that the behavior of $Π ( t )$ for a Gaussian quantum state, described by an initial Wigner function $W ( x , p , 0 )$⁠, is identical to the behavior of $Π cl ( t )$ for an ensemble of free classical particles, initially distributed in accordance with the phase-space density $W cl (x, p, 0)=W(x, p, 0)$⁠. In particular, this means that the negative flow of probability addressed by Villanueva¹ is essentially a classical-mechanical effect.

We now present a phase-space interpretation of the effect. For the discussion below, it is important to introduce dimensionless versions of the position, momentum, and time variables. This requires defining a natural length scale L. We achieve this by noticing that, according to Eq. (3), the quantity $Im ( 1 / α t ) = − ( Im α t ) / | α t | 2$ does not depend on time, i.e., $Im ( 1 / α t ) = Im ( 1 / α 0 )$⁠. Thus, L can be defined as Subsequently, we introduce dimensionless positions dimensionless momenta dimensionless time and dimensionless (quasi)probability density A straightforward calculation yields the following dimensionless version of the Wigner function (10): where and

The time dependence of the dimensionless Wigner function is specified by cf. Eq. (2), and cf. Eq. (3), where $ϵ 0 = Re α 0 / Im α 0$⁠. It is easy to check that the dimensionless Wigner function satisfies the free particle evolution equation cf. Eqs. (8) and (11), and that the probability of finding the particle in the region $ξ > δ$ (or x > q) at time τ (or t) is given by cf. Eqs. (9) and (12).

Equations (17), (19), (20), and (21) provide the complete description of the motion of a free Gaussian wave packet. At time τ, the wave packet is parametrized by three dimensionless real numbers: $ξ τ$⁠, η₀, and $ϵ τ$⁠. Figure 1 shows a phase-space curve of a constant value of $Ω ( ξ , η , τ )$⁠. The curve is an ellipse centered at $( ξ τ , η 0 )$⁠. It is easy to show (see  Appendix A) that the angle between the major axis of the ellipse and the ξ-axis is given by

The angle $θ τ$ decreases monotonically from π to 0 as time τ increases from $− ∞$ to $+ ∞$⁠. This is shown in Fig. 2. According to Eq. (21), $ϵ τ = 0$ when time $τ = − ϵ 0$⁠. At this instant, the Wigner function representing the wave packet reads

The time evolution of the Wigner function during the time interval between τ and $τ + Δ τ$⁠, i.e., can be viewed as a result of two consecutive transformations, both illustrated in Fig. 3. The first transformation is a rigid shift of the Wigner distribution along the ξ-axis by $η 0 Δ τ$⁠,

The second transformation is a simple shear leaving the distribution center unchanged, The change in the Wigner density $Δ Ω$ during the time interval between τ and $τ + Δ τ$ is given by where is the change due to the shift transformation and is the change due to the shear transformation. Consequently, the corresponding change in the probability of finding the particle in the region $ξ > δ$ is where and It is straightforward to show (see  Appendix B) that, in the limit of small $Δ τ$⁠, and

In the case of the Wigner function Ω given by Eq. (17), the probability change due to the shift transformation is always positive This follows directly from the fact that $Ω ( ξ , η , τ ) > 0$⁠. However, the probability change due to the shear transformation, $( Δ Π ) shear$⁠, can be negative for some values of the wave packet parameters. Figure 3 provides an example of the situation in which $( Δ Π ) shear < 0$⁠; it is clear that $ϵ τ < 0$ (or, equivalently, $θ τ > 3 π / 4$⁠) is a necessary condition for $( Δ Π ) shear$ to be negative. The negative flow of the net probability, $Δ Π < 0$⁠, occurs if and only if This condition is the phase-space equivalent of Eq. (7).

In order to explicitly recover Eq. (7) from Eq. (32), we substitute Eq. (17) into Eqs. (30) and (31) and evaluate the corresponding momentum integrals. This yields (see the  Appendix B) and Substituting these expressions into Eq. (32), we obtain Using the rescaling transformations (14), (15), (16), and (18), it can be easily verified that Eq. (35) is the dimensionless version of the negative probability flow condition (7). The geometrical meaning of this condition is discussed in  Appendix C.