Problem 1:

An operator A has two normalized eigenstates ψ1 and ψ2, with eigenvalues a1 and a2, respectively.  An operator B, has two normalized eigenstates, φ1 and φ2, with eigenvalues b1 and b2, respectively.   The eigenstates are related by
ψ1 = (3φ1 + 4φ2)/5,    ψ2 = (4φ1 - 3φ2)/5.
(a)  Observable A is measured, and the value a1 is obtained.  What is the state of the system immediately after this measurement?
(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 a1?

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

subject to the constraint aa* + bb* = 1, where * denotes complex conjugation.

Problem 3:

A quantum system with two states has the Hamiltonian matrix (ε1 < ε2)

(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, 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) = U0,  -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 vmax of an electron in a hydrogen atom, confined within a distance of about 0.1 nm?