###

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