**Problem 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

**Problem 2:**

(a) Show that the eigenvalues of a general 2×2 matrix A can be expressed as

**
**where D is the determinant of A and T is the trace of A (sum of diagonal
elements).

Show that the matrix

with M >> m has two eigenvalues, with one much larger than the other.

(b) Show that the most general form of a 2×2 unitary matrix U with unit determinant can be parameterized as

**Problem 3:**

A quantum system with two states has the Hamiltonian matrix (ε_{1} <
ε_{2})

(a) What are the two energies E_{±} of the system?

(b) It is of advantage to parameterize the eigenstates as

with real α, φ. Show that this state is normalized.

Show that φ = γ/2 for complex off-diagonal matrix element ν = |ν|e^{iγ},
and find the values of α for the two eigenstates.

**Problem 4:**

Use the uncertainty principle,
ΔxΔp ≥ ħ/2, to estimate the ground state energy of a particle in a
one-dimensional well of the form

(a) U(x) = U_{0}, -a/2 < x < a/2, U(x) = infinite for
all other values of x,

(b) U(x) = k|x|.

**Problem 5:**

(a) Find the relativistic energy (in MeV) of an electron with a deBroglie
wavelength of 0.0012 nm.

(b) Estimate the minimum value for the speed v_{max} of an electron in
a hydrogen atom, confined within a distance of about 0.1 nm?