Homework 7

1. A particle is in the state $\psi=R(r)\left(\sqrt{1\over 3}Y_{21}+i\sqrt{1\over 3}Y_{20}-\sqrt{1\over 3}Y_{22}\right)$. Find the expected values of $L^2$, $L_z$, $L_x$, and $L_y$.

2. A particle is in the state $\psi=R(r)\left(\sqrt{1\over 3}Y_{11}+i\sqrt{2\over 3}Y_{10}\right)$. If a measurement of the $x$ component of angular momentum is made, what are the possilbe outcomes and what are the probabilities of each?

3. Calculate the matrix elements $\langle Y_{\ell m_1}\vert L_x\vert Y_{\ell m_2}\rangle$ and $\langle Y_{\ell m_1}\vert L_x^2\vert Y_{\ell m_2}\rangle$

4. The Hamiltonian for a rotor with axial symmetry is $H={L_x^2+L_y^2\over 2I_1}+{L_z^2\over 2I_2}$ where the $I$ are constant moments of inertia. Determine and plot the eigenvalues of $H$ for dumbbell-like case that $I_1»I_2$.

5. Prove that $\langle L_x^2\rangle=\langle L_y^2\rangle=0$ is only possible for $\ell=0$.

6. Write the spherical harmonics for $\ell\leq 2$ in terms of the Cartesian coordinates $x$, $y$, and $z$.

7. A particle in a spherically symmetric potential has the wave-function $\psi(x,y,z)=C(xy+yz+zx)e^{-\alpha r^2}$. A measurement of $L^2$ is made. What are the possible results and the probabilities of each? If the measurement of $L^2$ yields $6\hbar^2$, what are the possible measured values of $L_z$ and what are the corresponding probabilities?

8. The deuteron, a bound state of a proton and neutron with $\ell=0$, has a binding energy of -2.18 MeV. Assume that the potential is a spherical well with potential of $-V_0$ for $r<2.8$ Fermis and zero potential outside. Find the approximate value of $V_0$ using numerical techniques.