QUANTUM MECHANICS-upper level
In the harmonic oscillator problem, the normalized wave
functions for the ground and first excited states are ψ0 and ψ1.
Using these functions, at some point t, a wave function u = Aψ0 +
Bψ1 is constructed, where A and B are real numbers.
(a) Show that the average value of x in the u state is generally
non-zero.
(b) What condition A and B must satisfy if we want the function u
to be normalized?
(c)...
Use the classical model to derive a formula for the electron’s
kenetic energy as a function of its orbital distance, assuming the
electron is in a circular orbit. In other words, derive a formula
that you could use to calculate the kinetic energy of the electron
if you knew its orbital distance.
What would be the formula for the electron's kinetic energy K in
terms of the orbital distance d.
An
oscillator is describe by the expression x(t) = .25 m cos[(1.3
rad/s)t - 1.8 rad].
A) what is its period of oscillation?
B) what is its velocity at t = 1.9 seconds?
Thermodynamics and Statistical Mechanics
problem:
(a) Derive the Maxwell speed distributions in
one and two dimensions.
(b) What is most likely speed in each case?
(c) What is average speed in each case?
(d) What is root-mean-square speed in each
case?
Derive an expression for the most probable translational energy
for an ideal gas. Compare
your results to the mean translational energy for the same
gas.
Derive the density of states for free electrons as a function of
energy E in 1) one-dimension, 2) twodimension, and 3)
three-dimension (N=total number of electrons, m=electron mass,
V=volume of solid) (Hint: First, derive the total number of
electrons as a function of k)