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6.1 Non-normalisable wave functions

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)=Aeikx+Beikx
(6.1)

Let us investigate the two functions. Remembering that p=ix we find that this represents the sum of two states, one with momentum k, 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)2mk=A2mk
(6.2)

This concept only makes sense for states that are not bound, and thus behave totally different from those I discussed previously.