Assignment 10

Problem 1:

A particle in a spherical potential is known to be in an eigenstate of L2 and Lz with eigenvalues l(l + 1)ħ2 and mħ, respectively.  Prove that the expectation values between |l m> states  satisfy
<Lx> = <Ly> = 0,   <Lx2> = <Ly2> = (l(l + 1)ħ2 - m2ħ2)/2.

Problem 2:

A space scientist proposes to measure the Gravitational constant G by locating a solid gold sphere of mass 8*104 kg and radius 1 m in a spaceship.  A hole is to be drilled through the diameter of the sphere and a small gold ball of mass 80 gram is released from rest at its surface so that it oscillates back and forth within the tube passing through the large sphere.
(a)  Calculate the period of the oscillatory motion of the particle, assuming that there is no friction present.
(b)  If air friction is present so that the particle returns to a radial distance of 99 cm rather than to 100 cm at the end of one full oscillation, obtain an estimate of the frictional force (assumed to be of form Ff = -bv), and the change in period caused by the frictional force.

Problem 3:

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,

iħ 

  ∂α/∂t  
  ∂β/∂t  

  = H(t)  

  α  
  β  

 .


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 4:

(a)  Find the magnetic field at any point in the x-y plane for y > 0 due to a wire of length l carrying a current I from x = 0 to x = l along the x-axis.
(b)  Use the result of (a) to find the self inductance of a square current loop of side l.  You may leave your result as a definite integral, but the limits must be specified and the integrand must be in terms of the dimensions of the loop, constants, and the variables being integrated.
(c)  A conducting square current loop of side l with sides parallel to the x- and y-axis has resistance R and self inductance L.  It moves at a speed v in the +x-direction.  The loop passes through a magnetic field given by B = B0 k, -l/2< x < l/2, B = 0 elsewhere.  Find the current in the wire as a function of time assuming that I = 0 a long time before the loop reaches the magnetic field.

Problem 5:

At t = 0 the x-component of the spin of a spin ½ particle is measured and found to be ħ/2.  At t = 0 the particle is therefore in the |+>x eigenstate of the Sx operator.  The particle is confined to a region with a uniform magnetic field B = B0k, its Hamiltonian is H = ω0Sz.  The eigenstates of H are |+> and |->,
H|+> = (ħω0/2)|+> and H|-> = -(ħω0/2)|->.
|+>x can be written as a linear combination of eigenstates of H.
(a)  Find the probability of measuring Sx = ħ/2 at t = T.
(b)  What is the mean value of Sx, <Sx>, at t = T?
(c)  Find the probability of measuring Sz = ħ/2 at t = T.