Quantum Well #
A state in which the direction of electron movement is confined.
Boundary Conditions #
At \(x=0\), \(\varphi(0)=0\)
At \(x=L\), \(\varphi(L)=0\)
$$\varphi (x)=\Big(\frac{2}{L}\Big)^{\frac{1}{2}}\sin\Big(\frac{n\pi}{L}x\Big)$$$$P_r=|\varphi (x)|^2=\Big(\frac{2}{L}\Big)\sin^2\Big(\frac{n\pi}{L}x\Big)$$$$E_n=\frac{\pi^2\hslash^2}{2mL^2}\cdotp n^2$$3-Dimensional Case #
$$\varphi (x,y,z)=\Big(\frac{2}{L}\Big)^\frac{3}{2}\cdotp\sin \Big(\frac{n_x\pi}{L}x\Big)\cdotp\sin \Big(\frac{n_y\pi}{L}y\Big)\cdotp\sin \Big(\frac{n_z\pi}{L}z\Big)$$$$E_{n_x,n_y,n_z}=\frac{\hslash ^2}{2m}\Big(\frac{\pi}{L}\Big)^2(n_x^2+n_y^2+n_z^2)$$The maximum energy value E is called the Fermi energy.