PHY4604–Introduction to Quantum Mechanics
Fall 2004
Problem Set 11
Dec. 1,2004
Due: Dec. 8, 2004
Griffiths Sec. 4-4
1. Drill.
(a) Consider a system of spin 1/2. What are the eigenvalues and eigenvectors
of the operator Ŝx + Ŝy ? Suppose a measurement of the observable corresponding to this operator is made, and the system is found to be in the
state corresponding to the larger eigenvalue. What is the probability that
a measurement of Ŝz in this state yields h̄/2?
(b) Consider a system represented by the normalized state vector
Ã
cos α
sin αeiβ
!
.
(1)
What is the probability that a measurement of Ŝy equals −h̄/2?
2. Magnetic Precession. An electron is in an eigenstate of Ŝx , with eigenvalue
+h̄/2 at time t = 0. At that time it is placed in a magnetic field B = B ẑ in which
it is allowed to precess for a time t0 . At that time the field is instantaneously
rotated so that it points along the ŷ direction, B = B ŷ. After another time
interval t0 a measurement of Sx is carried out. What is the probability that h̄/2
will be obtained?
3. Cooper pairing. A Cooper pair is a pair of electrons in a superconductor
bound to one another by a net attractive interaction, usually produced by the
interactions of the electrons with the lattice vibrations in the solid. In a classic
superconductor the electrons are in a spin singlet (total S = 0, relative orbital
angular momentum “s” state (` = 0). Identify all other possibilities for“pairing”
two electrons given 1) electrons are spin-1/2; 2) Pauli principle holds ; 3) pair is
in a state of definite relative orbital angular momentum. In each case, specify
the number of independent components of the pair wave function.
1