Animation 1 | Animation 2 | Animation 3 | Animation 4

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One of the most important concepts of quantum mechanics is that of a probability distribution in the form of a probability density. Understanding classical probability distributions can help us understand quantum-mechanical probability distributions.  Restart.

The relative probability distribution, P_R(x), for a classical system can be thought of as the amount of time that a particle spends in a small region of space, |dx|, relative to some same-sized region of reference. What is the time spent in a region |dx|?  Since |v| = |dx|/dt, we have that dt = |dx|/|v|. The magnitude of the velocity, |v|, is related to the particle's total energy, E, via its kinetic energy (T = 1/2 mv² = p²/2m) as:

|v| = [2(E − V(x))/m]^1/2,

since the kinetic energy of the particle is its total energy minus its potential energy. Therefore the time spent in a region |dx| is

dt = |dx| / [2(E − V(x))/m]^1/2 .

The relative probability distribution, therefore, is a ratio of the time spent in the region of interest divided by the time spent in the region of reference as long as both regions are the same size and thus:

P_R(x) = dt/dt₀ = |dx| / [2(E − V(x))/m]^1/2 / |dx| / [2(E − V₀)/m]^1/2
= [(E − V₀)]^1/2 / [(E − V(x))]^1/2.

where V₀ is the potential energy in the reference region and V(x) is the potential energy in the region of interest.

Now consider Animation 1 which depicts a 1-kg particle moving to the right with an initial velocity of 1 m/s.  Ghost images are also shown to depict the particle's position at equal time intervals. In the graph above the animation, the relative probability distribution is shaded (position is given in meters and time is given in seconds) so that it is easier to see. What do you notice about it?  The relative probability distribution is constant throughout the particle's motion. This occurs because the magnitude of the particle's velocity never changes due to its constant potential energy (recall that F_x = − dV/dx, and hence when dV/dx = 0, F_x = 0 and a_x = 0). Therefore, for any same-sized region of space, the time spent is the same as anywhere else.

What about Animation 2?  In this animation there is a change in the potential energy at x = 0 from zero to 3/8 J.  Here the particle slows down from a constant 1 m/s for x < 0 m to a constant 0.5 m/s for x > 0. What happens to P_R(x)?  There is a distinct change in P_R(x) at x = 0 m.  If we pick a point with x < 0 as our reference point, we note that any point with x > 0 has a larger relative probability. Why does this occur?  The particle is traveling more slowly at points with x > 0 relative to points with x < 0, therefore it takes the particle more time to traverse the same distance interval, dx.  Hence the relative probability is greater for x > 0 than for x < 0.

For bound systems, a particle is confined to move in a finite region of space (between the two classical turning points), and the relative probability distribution can be normalized to yield a normalized probability distribution, P_N(x).  This is done by integrating the relative probability distribution over the finite region and then dividing P_R(x) by this result. What this does is change the relative probability distribution into a normalized probability distribution, P_N(x).  Because of the normalization, the normalized classical probability distribution now has the unit of one over length. It is P_N(x) dx that is interpreted as the probability of finding the particle between x and x + dx.  Consider Animation 3 in which a particle is confined to move in a one-dimensional box with infinitely hard walls at x = −5 m and x = 5 m. The relative probability distribution is uniform from −5 m to 5 m and can be set equal to 1. What is the normalized classical probability distribution?   Since P_R(x) is a constant, the integral of P_R(x) from x_i  = −5 m to x_f  = 5 m is just 10 m. Therefore P_N(x) = 0.1 m⁻¹.

In Animation 4 the particle is again confined to move in a one-dimensional box with infinitely hard walls at x = −5 m and x = 5 m.  But this time there is a change in the potential energy function at x = 0 from V = 0 J to V = 3/8 J. The relative probability distribution is uniform from −5 m to 0 m and is 1, while from 0 m to 5 m the relative probability distribution is uniform and is 2. What is the classical probability distribution now? We must now be careful when integrating because P_R(x) changes at x = 0 m. We get that the integral of P_R(x) from x_i = −5 to x_f = 5 is 15 m. Therefore for x between −5 m and 0 m, P_N(x) = 0.066 m⁻¹ and for x between 0 m and 5 m, P_N(x) = 0.133 m⁻¹.