Assignment 8

Problem 1:

The quantum numbers l1 and l2 of the orbital momenta of particle A and particle B are 1 and 2, respectively.  Find the 15 possible ‘kets' in the coupled representation (notation |l1,l2;L,ML>) where L represents the quantum number of the total orbital momentum.

Problem 2:

A pair of magnetic ions with individual spins s1 and s2 interact through the scaled Hamiltonian H = s1s2.  Let s(s + 1) be the eigenvalue of s2, where s = s1 + s2 is the total angular momentum.  Note ħ = 1.
(a)  Show that the total angular momentum is a conserved quantity.
(b)  Find the ground state energy E0 and the energy of the highest state Emax.
(c)  Now suppose a magnetic field is applied and the new Hamiltonian is H', where
H' = H - b(s1z + s2z).  What is the residual symmetry of the new Hamiltonian H', and what are the associated "good quantum numbers"?

Problem 3:

Consider two distinguishable spin ˝ systems that can be repeatedly prepared in the state
|ψ> = 2(|++> + |-->) = 2(|+>(1)|+>(2) + |->(1)|->(2)).
Observer 1 measure the X(1) = σx(1) or Y(1) = σy(1), i.e. the x- or y-component of the spin in units of ħ/2, of system 1, and observer 2 measures X(2) or Y(2) of system 2.
(a)  Show that the product operators X(1)Y(2) and Y(1)X(2) commute and therefore form a complete set of commuting operators for the system.
(b)  What are the possible results of measuring X(1), X(2), Y(1), or Y(2) for the given |ψ>?
(c)  What are the possible outcomes of measuring X(1)Y(2) or Y(1)X(2)?

Problem 4:

A system has a wave function ψ(x,y,z) = N*(x + y + z)*exp(-r22) with α real.  If Lz and L2 are measured, what are the probabilities of finding 0 and 2ħ2?

Problem 5: 

Some organic molecules have a triplet (S = 1) excited state that is located at an energy Δ above the singlet (S = 0) ground state.  Consider an ensemble of N such molecules where N is of the order of Avogadro's number
(a)  Find the average magnetic moment <μ> per molecule in the presence of a magnetic field B.  Assume Boltzmann statistics.  You may also assume that Δ is large compared to the field-induced level splittings.
(b) Show that the magnetic susceptibility χ = N d<μ>/dB is approximately independent of Δ when kBT >> Δ.