The left-to-right motion of a free quantum Gaussian wave packet can be accompanied by the right-to-left flow of the probability density, the effect recently studied by Villanueva. Using the Wigner representation of the wave packet, we analyze the effect in phase space and demonstrate that its physical origin is rooted in classical mechanics.

^{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,

*x*and

*t*are the position and time variables, respectively. Here,

*t*, with

*x*

_{0}and

*p*

_{0}being the initial mean position and momentum, respectively;

*m*is the particle mass. The complex-valued function

*α*is defined as

_{t}*α*

_{0}, must satisfy $Im\u2009 \alpha 0 >0$ for the wave packet to be normalizable. Finally, the complex-valued function

*γ*encapsulates both the normalization constant and global phase; the imaginary part of

_{t}*γ*is related to that of

_{t}*α*via

_{t}*q*be some fixed point on the

*x*-axis, and consider the scenario in which

*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,

*α*

_{0}is such that

Here, we point out that, if considered in phase space, the above effect has a simple intuitive explanation. In fact, the effect is rooted in classical mechanics: the same negative flow of probability takes place in an ensemble of free classical particles with an appropriate Gaussian distribution of positions and momenta. We also present a phase-space-based derivation of condition (7).

^{2,3}

^{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

*x*>

*q*reads

*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

*x*>

*q*at time

*t*equals $ N \Pi cl ( t ) $, where

*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

^{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 ) | \u2264 1 / \pi \u210f $ 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 $ \Pi ( t ) $ for a Gaussian quantum state, described by an initial Wigner function $ W ( x , p , 0 ) $, is identical to the behavior of $ \Pi cl ( t ) $ for an ensemble of free classical particles, initially distributed in accordance with the phase-space density $ W cl (x,\u2009p,\u20090)=W(x,\u2009p,\u20090)$. In particular, this means that the negative flow of probability addressed by Villanueva

^{1}is essentially a classical-mechanical effect.

*L*. We achieve this by noticing that, according to Eq. (3), the quantity $ Im ( 1 / \alpha t ) = \u2212 ( Im \alpha t ) / | \alpha t | 2 $ does not depend on time, i.e., $ Im ( 1 / \alpha t ) = Im ( 1 / \alpha 0 ) $. Thus,

*L*can be defined as

*x*>

*q*) at time

*τ*(or

*t*) is given by

*τ*, the wave packet is parametrized by three dimensionless real numbers: $ \xi \tau $,

*η*

_{0}, and $ \u03f5 \tau $. Figure 1 shows a phase-space curve of a constant value of $ \Omega ( \xi , \eta , \tau ) $. The curve is an ellipse centered at $ ( \xi \tau , \eta 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

*π*to 0 as time

*τ*increases from $ \u2212 \u221e $ to $ + \u221e $. This is shown in Fig. 2. According to Eq. (21), $ \u03f5 \tau = 0 $ when time $ \tau = \u2212 \u03f5 0 $. At this instant, the Wigner function representing the wave packet reads

This is a minimal uncertainty state. The phase-space contour lines representing minimal
uncertainty states are circles, and so the angle $ \theta \tau $ is not defined at $ \tau = \u2212 \u03f5 0 $. In Fig. 2, the value of *ϵ*_{0} is chosen to be negative.

*τ*and $ \tau + \Delta \tau $, i.e.,

*ξ*-axis by $ \eta 0 \Delta \tau $,

*τ*and $ \tau + \Delta \tau $ is given by

In summary, we have shown that the effect of the negative probability flow, studied by
Villanueva,^{1} has an intuitive explanation
when considered in phase space. The effect is classical-mechanical in nature and occurs not
only for a free quantum particle with a Gaussian wave function but also for an ensemble of
free classical particles with a Gaussian distribution of positions and momenta.

### APPENDIX A: DERIVATION OF EQ. (23)

*C*> 0 is a constant. In polar coordinates,

*r*and, therefore, satisfies the equation

*r*.

### APPENDIX B: DERIVATION OF EQS. (30), (31), (33), and (34)

### APPENDIX C: GEOMETRICAL MEANING OF EQ. (35)

Figure 5 shows the set of angle pairs $ ( \theta \xaf \tau , \varphi \tau ) $ fulfilling condition (C1). The condition becomes especially simple in the limit of small $ \theta \xaf \tau $: if $ \theta \xaf \tau \u226a 1 $, then negative probability flow at $ \xi = \delta $ occurs for $ \varphi \tau < \theta \xaf \tau $.

## REFERENCES

The equivalence between Eqs. (8) and (11) is a particular case of the
following fact: If the system potential is of the form $V(x)=ax2+bx+c$, where *a*, *b*, and *c* are some constants, then the quantum evolution equation for the
Wigner function is identical to the classical Liouville equation for the phase-space
probability density. The free particle case addressed in the present paper corresponds
to $a=b=0$.