# Hamiltonian in Landau gauge

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1. Dec 18, 2015

### shinobi20

1. The problem statement, all variables and given/known data
Define n=(x + iy)/(2)½L and ñ=(x - iy)/(2)½L.
Also, ∂n = L(∂x - i ∂y)/(2)½ and ∂ñ = L(∂x + i ∂y)/(2)½.
with ∂n=∂/∂n, ∂x=∂/∂x, ∂y=∂/∂y, and L being the magnetic length.
a=(1/2)ñ+∂n and a=(1/2)n -∂ñ
a and a are the lowering and raising operators of quantum mechanics.

Show that H=ħωc(aa + ½)

2. Relevant equations
L=ħc/eB, ωc=eB/mc (cyclotron frequency), e for the charge of the electron
H = Px2/2m + ( Py2 + eBx/c )2/2m

3. The attempt at a solution
I have tried to find x,y,∂x,∂y in terms of n,ñ,∂n,∂ñ. But I ended up getting only some if the right terms to come out but not all, is my first step wrong? Any suggestions?

2. Dec 19, 2015

### blue_leaf77

Should the exponent "2" of $P_y$ be there?

3. Dec 19, 2015

### shinobi20

Sorry, it was a typo. Do you have any suggestions?

4. Dec 19, 2015

### blue_leaf77

You should post your initial attempt before we can discuss further. In particular, how the old variables look like in terms of the new ones?

5. Dec 19, 2015

### shinobi20

This is what I've done so far. My problem is that everything is there except for the ½. I wrote ∂ for ∂n and ∂(bar) for ∂ñ.

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6. Dec 20, 2015

### blue_leaf77

According to this link https://en.wikipedia.org/wiki/Landau_quantization, the Gauge you should be using is the symmetric gauge and hence the original Hamiltonian should be different than that you are using. For instance, in Landau gauge, the operator ${y}$ is not present.

7. Dec 20, 2015

### shinobi20

Why can't I show it using the Landau gauge? The choice is just for simplification of computation right?

8. Dec 20, 2015

### blue_leaf77

$x$ and $y$ appear symmetrically in the gauge transformation, but they do not in the original Hamiltonian.

9. Dec 20, 2015

### shinobi20

Oh I see that, then I'll try it again using the symmetric gauge. Thanks!