In class 24

24.1

An electron with mass $m_0$ is confined in a spherical shell with radius R. Determine eigenvalues and eigenfunctions of system in a magnetic field $\mathbf{B}=B_0\hat{\mathbf{z}}$. Recall that Hamiltonian of system without external magnetic field is $\frac{L^2}{2m_0{R}^2}$

Write down Time independent Schrodinger equation write down potential component? magnetic potential? use Lz instead of big L someone said the eigenfunctions and i missed it eigenfunctions (wave functions) corresponding to eigenvalues (energy levels)

$$R(r{)}_{n,l}=r^l{F}_n(r)e^{-\sqrt{2m_{0}E/{\hslash }^2}r}$$

24.2

Electron with mass $m_0$ confined to move in a circle in the x-y plane with radius R. System is in magnetic field given by $\mathbf{B}=B_0\hat{\mathbf{z}}$. Ignore kinetic energy, only magnetic interaction energy.

At t=0, wave function has form $\psi (\varphi )=A\cos \varphi$

Determine what A is:

$$1={\int }_0^{2\pi }(A\cos \varphi {)}^{2}d\varphi$$ $$\frac{1}{A^2}={\int }_0^{2\pi }{\cos }^2\varphi d\varphi$$ $$\frac{1}{A^2}=\pi$$ $$A=\sqrt{\frac{1}{\pi }}$$

24.3

Electron with mass $m_0$ confined to move in x-y plane with radius r. System is placed in a magnetic field $\mathbf{B}=B_0\hat{\mathbf{z}}$. Consider only magnetic interaction energy.

The wavefunction is given by

$$\Psi (\varphi ,t)=\frac{1}{\sqrt{2}}[{\phi }_1{e}^{-i{E}_{1}t/\hslash }+{\phi }_2{e}^{i{E}_{2}t/\hslash }]$$

Where

$${\phi }_1=\frac{1}{\sqrt{2\pi }}{e}^{i\varphi }\&{\phi }_2=\frac{1}{\sqrt{2\pi }}{e}^{-i\varphi }$$

a. Determine matrix of energy operator: b. Determine expectation of the energy

24.4

Determine the energy spectrum of a H-atom placed in a magnetic field

$$B=B_0\hat{\mathbf{z}}$$

Consider only $n=2,l=1$ state.

$$H=H_0-\frac{e}{2m_0}\mathbf{L}\cdot \mathbf{B}=H_0+\frac{e}{2m_0}{B}_0{L}_z$$ $$H_0=-\frac{{\hslash }^2}{2m_0}{\nabla }^2+V(r)$$