Assignment 6

Problem 1:

Calculate the probability current density for of a freely moving particle whose quantum state is described by a de Broglie plane wave.

Problem 2:

Find <x> and ∆x for the nth stationary state of a free particle in one dimension restricted to the interval 0 < x < a.  Show that as n --> ∞ these become the classical values.

Problem 3:

Consider a system of N particles with only 3 possible energy levels separated by ε.  Let the ground state energy be zero.  The system occupies a fixed volume V and is in thermal equilibrium with a reservoir at temperature T.  Ignore interactions between particles and assume Boltzmann statistics apply.

(a)  What is the partition function for a single particle in the system?
(b)  What is the average energy per particle?
(c)  What is the probability that the 2ε level is occupied in the high-temperature limit kBT >> ε?
Explain your answer on physical grounds.
(d)  What is the average energy per particle in the high-temperature limit kBT >> ε?
(e)  At what energy is the ground state 1.1 times as likely to be occupied as the 2ε level?
(f)  Find the heat capacity CV of the system, analyze the low-T (kBT >> ε) and high-T (kBT >> ε) limit, and sketch CV as a function of T.

Problem 4:

A river of width D flows on the northern hemisphere at a geographical latitude φ toward the north with a certain flow speed v0.  By which amount is the right bank higher that the left one?
First apply the equation of motion in a non-inertial frame to the problem at hand, and then use D = 2 km, v0 = 5 km/h, and φ = 45o to find the super-elevation of the river for the given parameters.

Problem 5:

Two equal mass particles are connected by a spring and are executing simple harmonic motion when they are placed in a horizontal, frictionless open pipe rotating at a constant angular velocity Ω about a vertical axis through its midpoint. The midpoint of the spring is initially at the midpoint of the pipe, but neither the particles nor the spring are attached to the pipe.
(a)  Find the maximum angular velocity Ω for which the particles stay in the pipe.
(b)  Assume (a) is satisfied and the particles stay in the pipe.  Find their positions as a function of time.