### Energy States of Electrons in a Plasma I

For uniform field, cylindrical symmetry apply cylindrical coordinates , , . Then

From the symmetry of the problem, we can guess (and verify) that . These variables will be constants of the motion and we therefore choose

Let (dummy variable, not the coordinate) and . Then

In the limit ,

while in the other limit ,

Try a solution of the form . Then

A well behaved function

Plugging this in, we have

We can turn this into the hydrogen equation for

and hence

Transforming the equation we get

Compare this to the equation we had for hydrogen

with . The equations are the same if WE set our . Recall that our . This gives us the energy eigenvalues

As in Hydrogen, the eigenfunctions are

We can localize electrons in classical orbits for large E and . This is the classical limit.

Max when

Now let's put in some numbers: Let . Then

This can be compared to the purely classical calculation for an electron with angular momentum which gives . This simple calculation neglects to count the angular momentum stored in the field.

Jim Branson 2013-04-22