## Results of the quantization of the Schrödinger equation for free electrons a magnetic field in 2 and 3 dimensions.

2-D Schrödinger equation

3-D Schrödinger equation

Eigenfunction solutions

Energy eigenvalues

$$E=\hbar \omega_c\left( \nu+ \frac{1}{2}\right) \pm \frac{g\mu_B}{2}B_z$$ $$E=\hbar \omega_c\left( \nu+ \frac{1}{2}\pm \frac{g}{4}\right)$$ $$\nu = 0,1,2,\cdots\qquad\omega_c=\frac{|eB_z|}{m}$$$$E=\hbar \omega_c\left( \nu+ \frac{1}{2}\right) + \frac{\hbar^2}{2m} k_{z}^2 \pm \frac{g\mu_B}{2}B_z$$ $$E=\hbar \omega_c\left( \nu+ \frac{1}{2}\pm \frac{g}{4}\right) + \frac{\hbar^2}{2m} k_{z}^2$$ $$\nu = 0,1,2,\cdots\qquad\omega_c=\frac{|eB_z|}{m}$$

Density of states

Energy spectral density
at zero temperature

Fermi energy EF

Internal energy
at zero temperature

Magnetization
at zero temperature