Physics 215B
Final
Winter, 2000

Open book, open notes.

Problem 1 [15 points]

A particle of mass M moves in one dimension acted on by the potential

$$V(X)=\left\{\begin{array}{cl} \infty & {\rm for}\ X\le 0\, ,\\ \frac{1}{2}KX^2 & {\rm for}\ X>0\, . \end{array} \right .$$

A second particle, of mass m, with m<<M and coordinates y moves between the origin and X (the second particle cannot go to the right of the first one).

[a.] Use the Born-Oppenheimer approximation to obtain the effective potential seen by the heavy particle; the light particle is in its lowest state. Make a sketch of this potential. [Hint: Don't be fancy - this problem does not involve Berry's phase.]

[b.] Find the approximate low lying spectrum of the heavy particle.

Problem 2 [10 points]

The differential cross section, in the Born approximation, for a particle scattering off some potential $V({\vec r})$ is known,

$$\frac{\partial\sigma_1}{\partial\Omega}(\theta,\phi)\, .$$

What is the differential cross section

$$\frac{\partial\sigma_2}{\partial\Omega}(\theta,\phi)$$

for scattering this particle off two such potentials located at ${\vec r}_1$ and ${\vec r}_2$, (the potential, now, is $V({\vec r} -{\vec r}_1)+V({\vec r}-{\vec r}_2)$)?

Problem 3 [10 points]

An atom in an excited state decays into the ground state by emitting a photon (neglect complications due to photon polarization). The matrix element for this process is taken to be constant,

$$\langle {\rm ground\ state},\ {\vec k}\vert H_I\vert{\rm excited\ state}\rangle=M\, .$$

The difference in energies of the excited and ground states is $\Delta$. For a photon of momentum $p=\hbar k$ the energy is $E=\hbar\omega=\hbar ck$(c is the velocity of light).

[a.] Derive an expression for the density of states of the photon, i.e. what is the analog for the photon of the massive particle density of states

$$d^3k=\rho(E)dE\, d\Omega=\frac{mk}{\hbar^2}dE\, d\Omega\, ?$$

[b.] What is the lifetime of the excited state?

Problem 4 [15 points]

A particle of mass m=1 is moving in a one dimensional potential V(x). $V(x)=\infty$ for |x|>1 and has some shape for $-1\le x\le +1$. We know the following moments of V(x):

$$\begin{eqnarray*}\int_{-1}^1\, dx V(x)&=&2\hbar^2\, ,\\ \int_{-1}^1dx\, x V(x)&... ...x\, x^3V(x)&=&0\, ,\\ \int_{-1}^1dx\, x^4V(x)&=&4\hbar^2\, .\\ \end{eqnarray*}$$

Using the above information, guess a reasonable approximate wave function and obtain an upper bound for the ground state energy of this system.