Assignment 8

Problem 1:

Assume that the highest energy (Fermi energy) of an electron inside a block of metal is 5 eV and that the work function of the metal, i.e. the additional energy that is necessary to remove an electron from the metal, is 3 eV.
(a)  Estimate the distance through which the wave function of an electron at the Fermi level penetrates the barrier responsible for the work function, assuming that the width of the 3 eV barrier is much greater than the penetration distance.  (In this estimate you can assume that the barrier is a simple step function.)
(b)  Estimate the transmission coefficient (i.e., provide a numerical value for it) for an electron at the Fermi level if the 3 eV barrier is now assumed have a width of 20 Å.
Note: the probability of transmission through a barrier of width a is given by
T = 4E(U0 - E) / [(U02sinh2[(2m(U0 - E)/ħ2)½ a] + 4E(U0 - E)],
or
T = (4E/U0)(1 - E/U0) / [sinh2[(2m(U0 - E)/ħ2)½ a] + (4E/U0)(1 - E/U0)].

Problem 2:

Consider the wave function in one dimension ψ(x) = C exp(-a|x|), where C and a are positive real numbers. 
(a)  Normalize ψ(x).
(b)  Evaluate ∂Ψ(x)/∂x| - ∂Ψ(x)/∂x| = ∫(∂2/∂x2)Ψ(x)dx as ε --> 0.
(c)  Write down the expression for (∂2/∂x2)Ψ(x).
(d)  Calculate p2ψ(x).  Does your answer give the correct sign for <p2>?

Problem 3:

The wave function ψ(r) of a spinless particle is ψ(r) = Nz2exp(-r2/b2), where b is a real constant and N is a normalization constant.
(a)  If L2 is measured, what results can be obtained and with what probabilities?
(b)  If Lz is measured, what results can be obtained and with what probabilities?
(c)  Is ψ(r) an eigenfunction of L2 or Lz?

Problem 4:

Consider a spin ½ particle in the presence of a uniform static magnetic field B = B0i.  Suppose that at t = 0 the spin state of the particle is the |->z eigenket of Sz.
(a)  State briefly how you would prepare such a state.
(b)  Suppose at time t > 0 we measure the z-component of the spin.  What values can be obtained and with what probabilities?
(c)  Evaluate the mean value of that measurement and comment on the physics of your result.

Problem 5:

An electron is contained in a one dimensional potential well, having a potential energy of 0 when between x = 0 and x = 8 nm, and a potential energy of ∞ for all other values of x. 
(a)  Write Schroedinger's equation for this problem, obtain well-behaved solutions, and determine the energy eigenvalues.
(b)  Obtain normalized wave functions, which will give unit probability of the electron existing in all of space.
(c)  Find the probability that the electron in its lowest energy state will exist in the space between x = 2 nm and x = 4 nm.