# Classical Harmonic Oscillator in a Magnetic Field

Consider a 3D charged Harmonic Oscillator which is placed in a homogenous magnetic field along the z direction. The equations of motion in plane are then given as: $$m\ddot{x}=kx+a\dot{y},$$ $$m\ddot{y}=ky-a\dot{x},$$ where $$a=qB_z$$. Because I really knew no other way forward, I assumed: $$y=A\exp(i\omega t)+B\exp(-i \omega t)$$ $$x=C\exp(i\omega t)+D\exp(-i\omega t).$$ My first question is this, is this Ansatz even valid or is there a better one? I comforted myself, that both dimensions should have the same $$\omega$$ because mass, charge and magnetic field are the same for both dimensions, is this really valid though?

After some arithmetic I reached:

$$\omega_{1,2,3,4}=\pm\sqrt{\frac{a^2}{2m^2} \pm \sqrt{\frac{a^2}{4m^2}-\frac {k^2}{m^2}}}$$

First of all I am uncertain if all four solutions are physical further more, if $$\frac{a^2}{4m^2}<\frac{k^2}{m^2}$$ discriminant has an imaginary part, hence the motion is damped. But I recall, that the Lorenz-Force does no work, can it still damp? And if $$\frac{a^2}{4m^2}<\frac{k^2}{m^2}$$ there should be no damping, is this physical? Or did I make a mistake?

## 3 Answers

To decouple your equations, you can perform the change of variable $$z=x+iy$$ so that $$m\ddot z=kz-ia\dot z$$ Solutions are exponential, $$z=e^{rt}$$ with $$mr^2=k-iar$$. Finally, $$x$$ and $$y$$ are given by the real and imaginary parts of $$z(t)$$.

This is a system of linear differential equations with constant coefficients, which can be solved by many approaches. Perhaps, it is worth pointing out that, since the order of the system is $$4$$, one may potentially have four linearly independent solutions (not two as the OP suggests.) However, the ansatz $$x,y\propto e^{i\omega t}$$ is a possible method of solution.

Apart from this small clarifications, this is a homework problem. I strongly suggest that you do some reading (beyond wikipedia) on linear ODE with constant coefficients and systems of such equations - it is part of the obligatory background for a physicist.

The Ansatz is valid. The real requirement is that both $$x$$ and $$y$$ have to be real. Clearly you cannot achieve that at all times if the frequencies are different. Second, the reality condition also imposes relations between A and B (and C and D).

The fact that you got four solutions is of course a consequence that you assumed you have four independent variables, so by itself it does not necessarily mean the solution is wrong (the outer $$\pm$$ is more about phase than actual frequency). However what does trouble, is that your solution does not reproduce the trivial answer when you put $$B=0$$. You definitely made a mistake somewhere. Also the dimensionality is inconsistent: you essentially add $$a^2/m^2$$ with $$a/m$$ inside the larger square root. This also indicates there is a mistake somewhere.

In general when you come up with an expression it is always good to perform these simple sanity checks:

1. check limiting cases where you already know what the answer should be
2. check if dimensions between different terms match