Physics 224   Fall 2002

		Discoveries and Inventions of Modern Physics 		due 11:00 am Tuesday Nov. 19

PROBLEM SET 7

November 14 Colloquium: ``Fluorescence correlation spectroscopy in living cells: detection of single protein molecules and of internal protein dynamics''
Enrico Gratton, University of Illinois at Urbana-Champaign
3:30 pm, 101 Rowland Hall

1.

AC Josephson Effect When a static DC voltage V is applied across a Josephson junction, an AC current results. To see how this comes about, notice that an electron pair experiences a potential energy difference qV on passing across the junction, where q=-2e. We can say that a pair on one side is at potential -eV and a pair on the other side is at +eV. Thus the equations of motion become

$$i\hbar\frac{\partial\psi_1}{\partial t}=\hbar T\psi_2-eV\psi_... ...i\hbar\frac{\partial\psi_2}{\partial t}=\hbar T\psi_1+eV\psi_2$$ (1)

where $\psi_1$is the superconducting order parameter on side 1:

$$\psi_1=\sqrt{n_1}e^{i\theta_1}$$ (2)

n₁ is the density of superconducting pairs on side 1. Similarly

$$\psi_2=\sqrt{n_2}e^{i\theta_2}$$ (3)

Assume that the superconductors are identical. Find the current density J as a function of time and of the phase difference $\delta(0)$. $\delta(0)=\theta_2-\theta_1$ is the phase difference at V=0. What is the angular frequency $\omega$at which the current oscillates when a voltage V is applied?

2.

Eisberg and Resnick Problem 13.18.

3.

Eisberg and Resnick Problem 13.24.

4.

Eisberg and Resnick Problem 13.30 (n-p-n transistor).