I have argued that solutions to the time-independent Schrödinger equation must be normalised, in order to have a the total probability for finding a particle of one. This makes sense if we think about describing a single Hydrogen atom, where only a single electron can be found. But if we use an accelerator to send a beam of electrons at a metal surface, this is no longer a requirement: What we wish to describe is the flux of electrons, the number of electrons coming through a given volume element in a given time.
Let me first consider solutions to the “free” Schrödinger equation, i.e., without potential, as discussed before. They take the form
|
ϕ(x) = A{e}^{ikx} + B{e}^{−ikx}.
| (6.1) |
Let us investigate the two functions. Remembering that p = {ℏ\over i} {∂\over ∂x} we find that this represents the sum of two states, one with momentum ℏk, and the other with momentum − ℏk. The first one describes a beam of particles going to the right, and the other term a beam of particles traveling to the left.
Let me concentrate on the first term, that describes a beam of particles going to the right. We need to define a probability current density. Since current is the number of particles times their velocity, a sensible definition is the probability density times the velocity,
|
|ϕ(x){|}^{2}{ℏk\over
m} = |A{|}^{2}{ℏk\over
m} .
| (6.2) |
This concept only makes sense for states that are not bound, and thus behave totally different from those I discussed previously.