Physics Asked by xhtxht on January 1, 2021
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:
$$
E=(n+1/2)hbaromega+hbar^2k_z^2/2m
$$
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
$$
H=(-hbar^2/2m)*d^2/dy^2+hbar^2k_z^2/2m+momega_c/2(y-hbar k_x/momega_c)
$$
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 $mathbf{v}timesmathbf{B}$).
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 $infty$.
Answered by Mauricio on January 1, 2021
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