(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider a free electron in a constant magnetic field [tex]\vec{B}=B\hat{z}[/tex] and a perpendicular electric field [tex]\vec{E}=\varepsilon\hat{y}[/tex]. Find the energy eigenvalues and eigenfunctions in terms of harmonic oscillator eigenfunctions

Hint: Use Landau gauge [tex]\vec{A}=-By\hat{x}[/tex]

What I actually don't understand is at the end... read on

2. Relevant equations

The Hamiltonian of a charged particle in an external em field is

[tex]H=\frac{1}{2m}\left(\vec{p}-\frac{q}{c}\vec{A}\right)^2+q\phi[/tex]

The hint says to use [tex]\vec{A}=-By\hat{x}[/tex] and since [tex]\vec{E}=-\nabla\phi[/tex] I can make [tex]\phi=-\varepsilon y[/tex]

3. The attempt at a solution

Plug in expressions for A and [tex]\phi[/tex] into H, which gives

[tex]H=\frac{1}{2m}\left(p_x^2+p_y^2+p_z^2+\left(\frac{qB}{c}\right)^2y^2+\frac{qB}{c}p_x y\right)-q\varepsilon y[/tex]

I know I just have to play around with this expression to make it look like a harmonic oscillator, but I have no idea how.... In another problem that I solved, I used an A potential with both an x and a y components, but I didn't have the scalar potential in that case. It's that term that ruins everything!!

Any ideas??

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# Homework Help: Eigenfunction of electron in E and B fields

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