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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)

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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 $\infty$.

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