Download Physics 3MM3, Problem sheet 10 1. Consider a free particle of mass

Physics 3MM3, Problem sheet 10
1. Consider a free particle of mass µ constrained to move on a ring of radius a.
(a) Show that the Hamiltonian of this system is Ĥ = L̂2z /2I where the z-axis
is through the centre of the ring and is perpendicular to its plane, and I is the
moment of inertia of the particle with respect to the z-axis.
(b) Find the energy eigenfunctions for the system and write down a general
expression for the solution of the time-dependent Schrödinger equation.
2. (a) If a spin-1/2 particle is in the up spin state along z, what is the probability
that if its spin along the y-direction is measured it will be found to be pointing in
the “up” direction along y?
(b) Calculate the expectation values of the components of Ŝ, i.e. {Ŝx , Ŝy , Ŝz },
(z)
for a spin-1/2 particle in the state χ+ , i.e. spin-up along z
(z)
(c) Ditto in the state χ− , i.e. spin-down along z.
3. The component of the spin vector S along an arbitrary direction (θ, φ) specified by the unit vector n is Sn = n.S. Since the cartesian components of n are
{sin θ cos φ, sin θ sin φ, cos θ}, we therefore find that the operator representing Ŝn
can be written in terms of {Ŝx , Ŝy , Ŝz } as
Ŝn =Ŝx sin θ cos φ + Ŝy sin θ sin φ + Ŝz cos θ
h̄
cos θ
sin θ e−iφ
=
2 sin θ eiφ − cos θ
(1)
(2)
where in the second step we substituted the matrices for {Ŝx , Ŝy , Ŝz } for a spin-1/2
particle in the z-basis. Diagonalizing the matrix for Ŝn we find the eigenvalues
±h̄/2 corresponding to the eigenvectors
sin 2θ
cos 2θ
(n)
(n)
χ− =
(3)
χ+ =
sin 2θ eiφ
− cos 2θ eiφ
(for details on this procedure see the handout on “Spin about an arbitrary direction”
on the course website).
(a) If a spin-1/2 particle is in the up spin state along z, what is the probability
that if its spin along n is measured it will be found to be pointing in the “up”
direction?
(b) If a spin-1/2 particle is in the up spin state along n, what is the probability
that if its spin along z is measured it will be found to be pointing in the up
direction?
(c) Calculate the expectation values of the cartesian components of Ŝ, i.e.
1
(n)
{Ŝx , Ŝy , Ŝz }, for a spin-1/2 particle in the state χ+ , i.e. spin-up along n
(n)
(d) Ditto in the state χ− , i.e. spin-down along n.
4. An electron with no orbital angular momentum is subjected to a uniform magnetic field B oriented in the z-direction. At time t = 0 the electron is in an
eigenstate of Ŝx with eigenvalue +h̄/2.
(a) Using the time-dependent Schrödinger equation, show that at time t the
state is described by
−iωt/2 1
e
√
(4)
eiωt/2
2
and give an expression for ω.
(b) At time t = π/ω the state will again be an eigenvector of Ŝx , which one?
(c) When will the system return to its original state?
(d) Show that at a general time t the state is an eigenstate of the operator
Ŝφ ≡ Ŝx cos φ + Ŝy sin φ
where φ = ωt.
2
(5)