Exercises

1. A particle of mass $m$ is placed in a finite spherical well:
   
   $$V(r) = \left\{ \begin{array}{lcl} -V_0&\mbox{\hspace{1cm}}&\mbox{for $r\leq a$}\\ 0&&\mbox{for $r>a$} \end{array}\right. ,$$
   
   
   with $V_0>0$ and $a>0$. Find the ground-state by solving the radial equation with $l=0$. Show that there is no ground-state if $V_0 a^2< \pi^2 \hbar^2/8 m$.

2. Consider a particle of mass $m$ in the three-dimensional harmonic oscillator potential $V(r)=(1/2) m \omega^2 r^2$. Solve the problem by separation of variables in spherical polar coordinates, and, hence, determine the energy eigenvalues of the system.

3. The normalized wavefunction for the ground-state of a hydrogen-like atom (neutral hydrogen, ${\rm He}^+$, ${\rm Li}^{++}$, etc.) with nuclear charge $Z e$ has the form
   
   $$\psi = A \exp(-\beta r),$$
   
   
   where $A$ and $\beta$ are constants, and $r$ is the distance between the nucleus and the electron. Show the following:
   1. $A^2=\beta^3/\pi$.
   2. $\beta = Z/a_0$, where $a_0=(\hbar^2/m_e) (4\pi \epsilon_0/e^2)$.
   3. The energy is $E=-Z^2 E_0$ where $E_0 = (m_e/2 \hbar^2) (e^2/4\pi \epsilon_0)^2$.
   4. The expectation values of the potential and kinetic energies are $2 E$ and $-E$, respectively.
   5. The expectation value of $r$ is $(3/2) (a_0/Z)$.
   6. The most probable value of $r$ is $a_0/Z$.

4. An atom of tritium is in its ground-state. Suddenly the nucleus decays into a helium nucleus, via the emission of a fast electron which leaves the atom without perturbing the extranuclear electron, Find the probability that the resulting ${\rm He}^+$ ion will be left in an $n=1$, $l=0$ state. Find the probability that it will be left in a $n=2$, $l=0$ state. What is the probability that the ion will be left in an $l>0$ state?

5. Calculate the wavelengths of the photons emitted from the $n=2$, $l=1$ to $n=1$, $l=0$ transition in hydrogen, deuterium, and positronium.

6. To conserve linear momentum, an atom emitting a photon must recoil, which means that not all of the energy made available in the downward jump goes to the photon. Find a hydrogen atom's recoil energy when it emits a photon in an $n=2$ to $n=1$ transition. What fraction of the transition energy is the recoil energy?