Lecture like 22 or something Exam Prep

Topics to be covered for exam 2

• Propagating waves + steps
  • Probability Current
  • Reflection + Transmission
• Operators
  • Hermiticity + Measurement
  • Eigenstate Expansions
  • Matrix QM
• Higher Dimensions
  • Separation of variables
  • Cartesian and spherical coordinates (maybe not on the test?)
  • Spherical $n,l,m_l$ angular momentum
• Spin
• Identical Particles
  • Symmetrical and Anti-Symmetrical

Practice Test Overview

1. Spherical Harmonics

• We have a spherical Hamiltonian with derivatives in r, $\theta$ and $\varphi$

$$H_{sp}\Psi \to \psi =R(r)\Theta (\theta )\Phi (\varphi )$$

Solving the spherical potential $V(r)$ leads to a specific set of $\theta$ and $\varphi$ solutions, these are the spherical harmonics.

$$|L^2|=l(l+1){\hslash }^2$$

For the hydrogen atom, coulomb potential, l must be less then n, which is the principle quantum number, solution of energy eigenstate.

$$L_z=m_l\hslash$$

and

$$-l<m_l<l$$ $$\iint y_{l,m}{y}_{l^{\prime },m^{\prime }}\sin \theta d\theta d\varphi$$ $$={\delta }_{l,l^{\prime }}{\delta }_{m,m^{\prime }}$$

The integral over spherical harmonics is 1 because they are independently normalized. They also form an orthogonal basis set, and a complete set. You can describe any function that is a function of $\theta$ and $\varphi$ as a sum of spherical harmonics times a constant.

Back to the question:

$$⟨L^2⟩=c_{3,2}^2{L}_{3,2}^2+c_{2,1}^2{L}_{2,1}^2$$ $$\frac{1}{2}(3(3+1){\hslash }^2)+\frac{1}{2}(2(2+1){\hslash }^2)$$

Spin is added to total angular momentum, but not orbital angular momentum. $L$ is orbital angular momentum, $S$ is spin angular momentum

2. Finding the expectation value of $L_z$

Set up the same way you set up for $L^2$

The relevant integer is 2 and 1.

3. Positronium is the bound state of an electron

$$E=13.6ev\frac{Z_{eff}^2}{h^2}$$ $$=-E_{ry}\frac{z^2}{h^2}$$

there’s a term for mass somewhere in there to some power. Use effective mass

$$\mu =\frac{m_1{m}_2}{m_1+m_2}$$

We have two masses that are the same, therefore the effective mass $\mu$ is 1/2 of an electron mass

$$-E_{ay}\frac{z^2}{h^2}{\mu }^{?}$$

If it were hydrogen, it would be A. It is not hydrogen, it needs to be scaled to the proper mass.

$$\Delta E_{Hyd}=-13.6\left(\frac{1}{n_{up}^2}-\frac{1}{n_{low}^2}\right)$$

4. The wavelength question

$$E=\frac{hc}{\lambda }$$ $$\Delta E=\frac{hc}{{\lambda }_1}-\frac{hc}{{\lambda }_2}$$ $$hc\approx 1240ev-nm$$

5. Two wavefunctions with different momentum If two eigenfunctions have two different eigenfunction solutions, are they orthogonal. This orthogonality relationship means that they are 0 if multiplied together. This goes to 0.

$$p_{op}{\psi }_b=b{\psi }_b$$

0!!!

6. The positions question

… help

Free Response Section

1. a Dont forget to normalize. Drink your ovaltine

In terms of a and constants

1. b

$$⟨q⟩=\frac{\int {\psi }^{\ast }{q}_{op}\psi dv}{\int {\psi }^{\ast }\psi dv}$$ $$d{v}_{sphere}=r^2\sin \theta drd\theta d\varphi$$

2. a The general form of the wavefunction…

$$K_1=E-V_1,K_2=E$$ $$k_1=\sqrt{\frac{2m}{{\hslash }^2}{K}_1}$$ $$k_2=\sqrt{\frac{2m}{\hslash }{K}_2}$$

2. b Continuity and smoothity relationships at the boundary

3. calculate a commutator

4. What is the energy $E_r$ of a transmitted positron

The transmitted energy for a positron is equal to the incident energy

There is current inside the barrier. Its gonna be the quantum tunneling thing.

You need the relation here.