# Energy of Free-electron Gas - Landau Levels in 3D

so i am looking into Landau Diamagnetism and am reading Dupre's paper. I am slightly confused at where he has got a term in his value of E from.

He states that:

$$E=(n+1/2)\hbar\omega+\hbar^2k_z^2/2m$$

and i can easily get to the first part, but am confused where the second term $$\hbar^2k_z^2...$$ comes from as it always seems to disappear.

the hamiltonian i am using is

$$H=(-\hbar^2/2m)*d^2/dy^2+\hbar^2k_z^2/2m+m\omega_c/2(y-\hbar k_x/m\omega_c)$$

however i may have the wrong indices (it could be $$k_x$$ or $$k_z$$ switched im not certain)

You can think about it classically. If you have a charged particle moving in a straight line and you put a magnetic field $$\mathbf{B}$$ in the $$z$$-direction, its motion will change but not the velocity in the $$z$$-direction (because Lorentz force is zero in that direction $$\mathbf{v}\times\mathbf{B}$$).
The same happens with the electrons in your Fermi gas (free electron model if you wish), the magnetic field is confining the motion in a plane perpendicular to $$\mathbf{B}$$ but not in the $$z$$-direction. So basically you have some quantized motion in the $$xy$$-plane but you have the equivalent to a 1D free particle in the $$z$$-direction. The energy of a free particle you can write it as you wish $$E_z=\frac{p_z^2}{2m}=\frac{\hbar^2k_z^2}{2m}$$ but $$(E_z,p_z,k_z)$$ are just free parameters that can go from $$0$$ to $$\infty$$.