Homework 9

1. An electron in the Hydrogen potential $V(r)=-{e^2\over r}$ is in the state $\psi(\vec{r})=C e^{-\alpha r}$. Find the value of $C$ that properly normalizes the state. What is the probability that the electron be found in the ground state of Hydrogen?

2. An electron is in the $\psi_{210}$ state of hydrogen. Find its wave function in momentum space.

3. A spin ${1\over 2}$ particle is in an eigenstate of $S_y$ with eigenvalue $+{\hbar\over 2}$ at time $t=0$. At that time it is placed in a constant magnetic field $B$ in the $z$ direction. The spin is allowed to precess for a time $T$. At that instant, the magnetic field is very quickly switched to the $x$ direction. After another time interval $T$, a measurement of the $y$ component of the spin is made. What is the probability that the value $-{\hbar\over 2}$ will be found?

4. Consider a system of spin ${1\over 2}$. What are the eigenstates and eigenvalues of the operator $S_x+S_y$? Suppose a measurement of this quantity is made, and the system is found to be in the eigenstate with the larger eigenvalue. What is the probability that a subsequent measurement of $S_y$ yields ${\hbar\over 2}$?

5. Let's define the u axis to be in the x-z plane, between the positive x and z axes and at an angle of 30 degrees to the x axis. Given an unpolarized spin ${1\over 2}$ beam of intensity $I$ going into the following Stern-Gerlach apparati, what intensity comes out?
   
   $$I\rightarrow \left\{\matrix{+ \cr - \vert}\right\}_z\rightarrow \left\{\matrix{+ \cr - \vert}\right\}_x\rightarrow ?$$
   
   
   
   
   $$I\rightarrow \left\{\matrix{+ \cr - \vert}\right\}_z\rightarrow \left\{\matrix{+ \vert\cr - }\right\}_u\rightarrow ?$$
   
   
   
   
   $$I\rightarrow \left\{\matrix{+ \cr - \vert}\right\}_z\rightarr... ...ghtarrow \left\{\matrix{+ \vert\cr - }\right\}_z\rightarrow ?$$
   
   
   
   
   $$I\rightarrow \left\{\matrix{+ \cr - \vert}\right\}_z\rightarr... ...ghtarrow \left\{\matrix{+ \vert\cr - }\right\}_z\rightarrow ?$$
   
   
   
   
   $$I\rightarrow \left\{\matrix{+ \cr - \vert}\right\}_z\rightarr... ...ghtarrow \left\{\matrix{+ \vert\cr - }\right\}_x\rightarrow ?$$