Position expectation value of 2D harmonic oscillator in magnetic field

[Mr_Allod]

Homework Statement

a. Find $\psi_{0,0}(x,y)$ for a 2D harmonic oscillator in a magnetic field.
b. Find the expectation values of $x^2 + y^2$ for the first 3 Landau levels

Relevant Equations

Hamiltonian: $H = \frac {(-i\hbar\nabla -e \vec A)}{2m} + \frac {m\omega_0^2}{2}(x^2+y^2)$
Landau Gauge: $\vec A = (-\frac {yB}{2},\frac {xB}{2},0)$

Hello there, for the above problem the wavefunctions can be shown to be:

$$\psi_{n,l}=\left[ \frac {b}{2\pi l_b^2} \frac{n!}{2^l(n+l)!}\right]^{\frac12} \exp{(-il\theta - \frac {r^2\sqrt{b}}{4l_b^2})} \left( \frac {r\sqrt{b}}{l_b}\right)^lL_n^l(\frac {r^2b}{4l_b^2})$$

Here $b = \sqrt{1 + 4\frac{\omega^2}{\omega_H^2}}$ and $l_b$ is the magnetic length $\sqrt{\frac {\hbar}{eb}}$ and $L_n^l$ are Laguerre polynomials. Also $r = \sqrt{x^2+y^2}$.

And assuming that's correct that would make the $\psi_{0,0} = \frac {1}{2\pi l_b^2}exp(\frac {-ar^2}{4l_b^2})$. Now the first thing I'm not sure about is what can I consider to be the first three Landau Levels, would it be $\psi_{(n=0)(l=0,1,2)}$ or $\psi_{(n = 0,1,2)(l=0) }$?

Also would I be correct in assuming that finding the expectation value of $r^2$ would be the equivalent of finding it for $x^2+y^2$? I would appreciate some insight into this thank you!

 

[NateD]

Yes, you are correct in assuming that the expectation value of $r^2$ is equivalent to finding it for $x^2+y^2$. The first three Landau levels are given by $\psi_{(n=0)(l=0,1,2)}$.