Problems

1. The wave function for three non-identical particles in a box of unit length with one particle in the ground state, the second in the first excited state, and the third in the second excited state is
   
   $$\psi (x_1 , x_2 , x_3 ) = \sin ( \pi x_1 ) \sin (2 \pi x_2 ) \sin (3 \pi x_3 ) .$$
   
   
   1. From this write down the wave function for three identical bosons in the above mentioned states.
   2. Do the same for three identical fermions.
   Hint: In each case there are six terms corresponding to the six permutations of $x_1$, $x_2$, and $x_3$. Exchanging any two particles leaves $\psi$ unchanged for bosons but changes the sign for fermions.
2. Two identical particles with equal energies collide nearly head-on, so that they are both deflected through an angle $\theta$, as shown in the sketch below. A physicist calculates the amplitude $\psi$ as a function of $\theta$ for this deflection to take place (using very advanced theory!), resulting in the solid curve shown in figure 19.4. However, measurements show that the actual amplitude as a function of $\theta$ (not probability!) is given by the dashed curve.
   1. What did the physicist forget to take into account? Explain.
   2. Are the particles fermions or bosons? Explain.
   Hint: If the outgoing particles (but not the incoming particles) are interchanged, how does the apparent deflection angle change?

   Figure 19.4: Incorrectly calculated and observed scattering amplitude for a headon collision between two identical particles.

   

3. Following the analysis made for the hydrogen atom, compute the ``Bohr radius'' and the ground state binding energy for an ``atom'' consisting of $Z$ protons in the nucleus and one electron.
4. Upper and lower bounds on the binding energy of the last (outermost) electron in the sodium atom may be obtained by assuming (a) that the other electrons have no effect, or (b) that the other electrons neutralize all but one proton in the nucleus. Compute the binding energy of the last electron in sodium in these two limits. (The actual binding energy of the last electron in sodium is $5.139 \mbox{ eV}$.)
5. A uranium atom ($Z = 92$) has all its electrons stripped off except the first one.
   1. What is the first electron's binding energy in electron volts?
   2. What is the ground state radius of the electron orbit in this case?
6. The energy levels of a particle in a box are given by $E_n = E_0 n^2 = E_0 , 4E_0 , 9E_0 , \ldots$ where $E_0$ is the ground state energy for the particle. Find the lowest possible total energy of a group of particles, expressed as a multiple of $E_0$, for the following particles in the box:
   1. 5 identical spin 0 particles.
   2. 5 identical spin 1/2 particles.
   3. 5 identical spin 1 particles.
   4. 5 identical spin 3/2 particles.
7. A charged particle in a 1-D box has energy levels at $E_n = E_0 n^2 = E_0 ,~ 4E_0 , ~ 9E_0 , ~ 16E_0 , ~ 25E_0 , \ldots$, where $E_0$ is the ground state energy of the particle. If the particle can absorb a photon with any of the energies $5E_0 ,~ 12E_0 ,~ 21E_0 , \ldots$, what can you infer about the initial energy of the particle? Explain.
8. The X-rays in your dentist's office are produced when an energetic beam of free electrons knocks the most tightly bound electrons ($n = 1$) completely out of the target atoms. Electrons from the next level up ($n = 2$) then drop into the $n = 1$ level.
   1. Estimate the energy in electron volts of the resulting photons for a copper target ($Z = 29$). Hint: For the inner electrons, you may ignore the effects of the other electrons to reasonable accuracy.
   2. What minimum energy must the electron beam have in this case?
9. What is the shortest ultraviolet wavelength usable in astronomy? Hint: UV photons more energetic than the binding energy of the electron in hydrogen are strongly absorbed by this gas.
10. In the naive periodic table model, the first three closed shells occur for $Z = 2,10,28$. However, the first three noble gases have $Z = 2,10,18$. Explain why this is so.