A.32 The evolution of probability

This note looks at conservation of probability, and the resulting definitions of the reflection and transmission coefficients in scattering. It also explains the concept of the “probability current” that you may occasionally run into.

For the unsteady Schrödinger equation to provide a physically
correct description of nonrelativistic quantum mechanics, particles
should not be able to disappear into thin air. In particular, during
the evolution of the wave function of a single particle, the total
probability of finding the particle if you look everywhere should stay
one at all times:

Fortunately, the Schrödinger equation

does indeed conserve this total probability, so all is well.

To verify this, note first that

To get an expression for that, take the Schrödinger equation above times

times

Now it can be verified by differentiating out that the right hand side can be rewritten as a derivative:

For reasons that will become evident below,

If (A.232) is integrated over all

Therefore, the total probability of finding the particle does not change with time. If a proper initial condition is provided to the Schrödinger equation in which the total probability of finding the particle is one, then it stays one for all time.

It gets a little more interesting to see what happens to the
probability of finding the particle in some given finite region

and it can change with time. A wave packet might enter or leave the region. In particular, integration of (A.232) gives

This can be understood as follows:

The probability current can be generalized to more dimensions using
vector calculus:

where

Returning to the one-dimensional case, it is often desirable to relate
conservation of probability to the energy eigenfunctions of the
Hamiltonian,

because the energy eigenfunctions are generic, not specific to one particular example wave function

To do so, first an important quantity called the “Wronskian” must be introduced. Consider any two eigenfunctions

If you multiply the first equation above by

The constant

The quantity

As an application, consider the example potential of figure
A.11 in addendum {A.27} that bounces a
particle coming in from the far left back to where it came from. In
the left region, the potential

At the far right, the potential grows without bound and the eigenfunction becomes zero rapidly. To make use of the Wronskian, take the first solution

If that is zero, the magnitude of

This can be understood as follows: if a wave packet is created from
eigenfunctions with approximately the same energy, then the terms

Next consider a generic scattering potential like the one in figure
7.22. To the far left, the eigenfunction is again of the
form

while at the far right it is now of the form

The Wronskian can be found the same way as before:

The fraction of the incoming wave packet that ends up being reflected
back towards the far left is called the “reflection coefficient”

The reflection coefficient gives the probability that the particle can be found to the left of the scattering region at large times.

Similarly, the fraction of the incoming wave packet that passes
through the potential barrier towards the far right is called the
“transmission coefficient”

Alternatively, because of the Wronskian expression above, the
transmission coefficient can be explicitly computed from the
coefficient of the eigenfunction in the far right region as

If the potential energy is the same at the far right and far left, the two classical momenta are the same,