For the state \[ \left|{\Psi}\right\rangle = \sqrt{\frac{7}{10}} |2, 1, 0\rangle + \sqrt{\frac{1}{10}} |3, 2, 1\rangle + i\sqrt{\frac{2}{10}} |3, 1, 1\rangle\]

Calculate

• \(\mathcal{P}(L_z=\hbar)\)

• \(\langle L_z\rangle\)

  Then, if you have time, continue with these calculations:

• \(\mathcal{P}(L^2=2\hbar^2)\)
• \(\langle L^2\rangle\)
• \(\mathcal{P}(E=-13.6eV/3^2=-1.51eV)\)
• \(\langle E\rangle\)
• What measurements can be degenerate on the Hydrogen atom?
• What is the time development of this state?
• What is the probability of finding the particle in the region \(0<\theta <\pi/6\), \(\pi/3< \phi < \pi/2\), and \(r_1< r <r_2\)?