How Does a Magnetic Field Affect Eigenfrequencies in a Harmonic Oscillator?

[qspeechc]

Homework Statement

Hello everyone. I'm trying to find the eiegenfrequencies in a problem.

We have a particle of mass m, charge e, that is subject to a linear restoring force
$$\textbf{F} = -k\textbf{r}$$
and the particle is in a magnetic field [tex]\textbf{B} = B\textbf{k}[/itex]
(in the z-direction)

The Attempt at a Solution

I am a bit confused ecause I only get one eigen-frequency: sqrt(k/m), which is as though the magnetic field is not there.

In finding the freq.s you determine the kinetic and potential energy. Th magnetic field has no addittion to the potential energy? This is what I did. I took:
U = 0.5k(x^2 + y^2 + z^2)
T = 0.5mv^2
where v = <dx/dt, dy/dt, dz/dt>
which gave me the above answer, which I suspect is wrong, partly because the question goes on to say:
"Write your answer in terms of sqrt(k/m) and the cyclotron freq. Hint: use the variable u=x+iy"

I would be grateful for any help.

 

[Irid]

Well, electromagnetic Lagrangian is a little different, because there is a velocity-dependent force. I think it is

$$L = T - q\phi + q \mathbf{v\cdot A}$$

where A is the vector potential of magnetic field.

 

[qspeechc]

Hhmm. I'll have to look it up. We never did the Lagrangian when considering velocity-dependent forces. I guess I'll have to look in another book. Thanks for the help Irid.

 

[qspeechc]

I'm sorry, this still isn't working. Do I assume each component x, y, z undergoes SHM, ie are of the form cos(wt), where w is the eigenfrequency, and then solve for w?