Phys 312 - Assignment 7 - PLEDGED - Due 18 Mar 99

1.
The ground state of hydrogen is described by the wavefunction

$$\psi \left( \mathbf{r},t\right) = \frac{1}{\sqrt{\pi }}\left( \frac{1}{a_{0}}\right) ^{\frac{3}{2}}\,e^{-r/a_{0}},$$

where a₀ is the Bohr radius.

(a) Compute the charge density

$$\rho \left( \mathbf{r},t\right) =e\left\vert \psi \left( \mathbf{r},t\right) \right\vert ^{2}$$

and plot $4\pi r^{2}\rho$ as a function of the dimensionless variable r/a_o. Does the plot peak at r = a₀?

(b) Compute the average value of r. Is it equal to a₀?

(c) Does the electron have a definite value of r? Does it have a definite speed? Does it have a definite angular momentum? Does it have a definite energy? If the answer to any of these questions is yes, give the value of the quantity.

(d) Compute the electric field due to this charge density everywhere in space. (Hint: use Gauss's theorem,) Add to it the field of the proton. Plot the total (radial) field as a function of r/a₀ for r > a₀/2 to check that it decreases exponentially.

(e) Compute the potential by integrating the field from r to infinity.

2.
Read the short section on impurity levels on page 368 of Kittel and Kroemer and look up the dielectric constants of Si and Ge, as well as the effective masses for the valence and conduction bands. Then:

(a) Compute the "effective Bohr radius" for donor and acceptor ground states. The easy way to do this is to scale the usual Bohr radius by the appropriate factors.

(b) Compute the ionization energies and compare with Table 13.2.

3.
Compute the bias voltage of a germanium p-n junction at 77 K, at room temperature, and at 500 K. Comment on the effect of increased temperature on the performance of such a junction.

4.
Short questions:

(a) Using just the lattice parameter, compute the Fermi energy for Aluminum and compare with the value given, for instance, in Tipler. (Hint: how many valence electrons per atom are there in Al?) If there is a discrepancy, it indicates that the effective mass of electrons in Al is different from the free electron mass.

(b) Using the measured resistivities, given by Tipler on page 722, compare the resistance (or, if you wish, the conductance) of an Al wire to that of a Cu wire having the same length and cross section.

(c) Compare the resistance (or, if you wish, the conductance) of an Al wire to that of a Cu wire having the same length and weight.

(d) You should have found from (b) and (c) that, at least by some criteria, Al is a better conductor than Cu. It is also much cheaper. Why is then Cu used for most electrical wiring?

(e) Why is the filament of an incandescent light bulb made of Tungsten, rather than steel or Copper?

(f) Why has Silicon become the standard semiconductor material for most purposes? What are possible alternative materials?

(g) Why is Silicon not suitable for a Light Emitting Diode? What materials are used for this purpose?

(h) In a p-n junction at zero external bias, what is the sign of the net electrical charge on the p side? On the n side?