Consider a 2 by 2 matrix M with matrix elements m_{11},
m_{12}, m_{21}, and m_{22}.

(a) What relationships between these matrix elements must exist for the matrix
M to be a unitary matrix, M = U?

(b) Show that the determinant of M = U has magnitude 1.

An operator A has two normalized eigenstates ψ_{1} and ψ_{2},
with eigenvalues a_{1} and a_{2}, respectively. An operator
B, has two
normalized eigenstates, φ_{1} and φ_{2}, with eigenvalues b_{1}
and b_{2}, respectively. The eigenstates are related by**
**ψ

(a) Observable A is measured, and the value a

(b) If B is measured immediately afterwards, what are the possible results, and what are their probabilities?

(c) If the result of the measurement of B is not recorded and right after the measurement of B, A is measured again, what is the probability of getting a

A pencil is placed with its point down on a flat surface. Assume that the pencil point is infinitely "sharp" and the the surface is perfectly flat. Ignore any extraneous effect such as vibration, air currents etc. The uncertainty principle says that it is not possible to fix the pencil exactly vertically AND exactly at rest. As a result, no matter how well it is positioned it will fall over in a finite time. Using the uncertainty principle, estimate how long the pencil will take to fall if it is positioned as well as can be done. Make reasonable estimates for the pencil mass, length, etc.

(a) What is the wavelength of a 10 eV electron and what is the
energy of a photon with this same wavelength?

(b) Light with a wavelength of 300 nm strikes a metal whose work function
is 2.2 eV. What is the shortest de Broglie wavelength for the electrons that
are produced as photoelectrons?

(c) A surface is irradiated with monochromatic light whose
wavelength can be varied. Above a wavelength of 500 nm, no photoelectrons are
emitted from the surface. With an unknown wavelength, a stopping potential
of 3 V is necessary to eliminate the photoelectric current. What is the unknown
wavelength?

Consider a two-state system governed by the Hamiltonian H with energy eigenstates |E_{1}> and |E_{2}>,
where H|E_{1}> = E_{1}|E_{1}> and H|E_{2}> = E_{2}
|E_{2}>. Consider also two other states,

|x> = (|E_{1}> + |E_{2}>)/√2, and |y> = (|E_{1}>
- |E_{2}>)/√2.

At time t = 0 the system is in state |x>. At what
subsequent times is the probability of finding the system in state |y> the
largest, and what is that probability?