Assignment 8, solutions

Problem 1:

Using the definition of angular momentum
L = r × p,
(a)  show that [Lx, Ly] = iħ Lz,   [Ly, Lz] = iħ Lx,   [Lz, Lx] = iħ Ly.
(b)  Using this result show that L2 commutes with Lz.


Problem 2:

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 3:

Consider the Hamiltonian of a spinless particle of charge qe in the presence of a static and uniform magnetic field B = B k.
H = (1/(2m)) (p - qeA(r,t))2.
By using the gauge in which A = 0, demonstrate that the Hamiltonian can be expressed as
H = p2/(2m) - (qe/(2m)) LB + (qe2B2/(8m))(x2 + y2).
Note that the second term corresponds to the linear coupling between the external field and the magnetic moment.


Problem 4:

An electron is a rest in magnetic field B = B0 k + B1(cos(ωt) i + sin(ωt) j).  B0, B1, and ω are constants.  At t = 0 the electron is in the |+> eigenstate of Sz.  Let μ = -γS be the magnetic moment of the electron and  ω0 = γB0, ω1 = γB1 be constants.
(a)  Construct the Hamiltonian matrix for this system and write down the time-dependent Schroedinger equation,



  = H(t)  



in matrix form in the {|+>, |->} basis.

(b)  Convert this equation into a "Schroedinger equation" with a time independent "Hamiltonian" by choosing new expansion coefficients
a(t) = exp(iωt/2)α(t), b(t) = exp(-iωt/2)β(t). 
(Hint:  Given ∂α/∂t and ∂β/∂t find ∂a/∂t and ∂b/∂t.)
(c)  Find the eigenvalues and eigenvectors if the "Hamiltonian" in part (b).
Hint:  Write

H' = A  

  cosθ     sinθ    
  sinθ   -cosθ   


The eigenvalues of the matrix are λ = ±1, and the corresponding eigenvectors are
+> =  cos(θ/2)|+> + sin(θ/2)|->,  |ψ-> =  -sin(θ/2)|+> + cos(θ/2)|->.]
(d)  The Schroedinger equation now implies that
U(t,0)|ψ+> = |ψ+>exp(-iAλ+t/ħ), and U(t,0)|ψ-> = |ψ->exp(-iAλ-t/ħ).
Find the probability of finding the electron in the |-> eigenstate of Sz as a function of time.


Problem 5:

For any two quantum-mechanical operators A and B, the uncertainty principle says that
<(ΔA)2><(ΔB)2>  ≥ ¼|<[A,B]>|2.  Consider a spin ½ particle.  Show that for the spin operators Sx and Sy the eigenstate |+> of the Sz operator is a minimum uncertainty state.