Final Exam

PHYS 684: Fall 2017

December 14 (4:30 - 7:10 pm), Planetary Hall 220

1. An electron is in the spin state

|ψ〉 = c

(3i4

)(a) Determine the normalization constant c.(b) Find the expectation values of Sx , Sy and Sz in this state.

(c) Find the uncertainties σSx=√〈(Sx − 〈Sx〉)2〉, σSy

=√〈(Sy − 〈Sy〉)2〉 and σSz

=√〈(Sz − 〈Sz〉)2〉.

(d) The uncertainty principles for angular momentum operators are

σ2Lsσ2Ly≥(

1

2i〈[Lx, Ly]〉

)2

=

(1

2ii~〈Lz〉

)2

=~2

4〈Lz〉2 ⇒ σLx

σLy≥ ~

2|〈Lz〉|

and the analogous inequalities obtained by cyclic permutations of (x, y, z) indices. These conditionsshould also apply to the spin expectation values and uncertainties. Check that your results in (c)satisfy these conditions.

2. (a) Derive the commutators [Li, xj ] and [Li, pj ], where xi, pi and Li are components of the position,momentum and orbital angular momentum (L = r× p) operators respectively for i ∈ {x, y, z}.

(b) Using the above results, evaluate the commutators:

[Lz, |r|2] , [Lz, |p|2]

where |r|2 = x2 + y2 + z2 and |p|2 = p2x + p2y + p2z

(c) Show that the Hamiltonian H = p2/2m + V (r) commutes with all components of L, providedthat the potential V is a function of only the radius r = |r|. Note that if operators A and B commute,then A also commutes with any function of B - use this to handle the commutators involving V (r).

3. (a) The normalized p-wave spherical harmonics expressed in spherical coordinates are:

Y 11 (θ, φ) =

√3

8πsin θ eiφ , Y 0

1 (θ, φ) =

√3

4πcos θ , Y −11 (θ, φ) = −

√3

8πsin θ e−iφ

Rewrite them in Cartesian coordinates using:

x = r sin θ cosφ

y = r sin θ sinφ

z = r cos θ

Show that:

Y +11 − Y −11 =

√3

2π

x

r, Y +1

1 + Y −11 = i

√3

2π

y

r

1

(b) Consider a particle with the wavefunction:

ψ(x, y, z) = N(x+ y + 2z)e−αr

where r =√x2 + y2 + z2, N is a normalization constant and α is a parameter. Use the formulas from

the part (a) to calculate the probabilities P (m), m ∈ Z that a measurement of the angular momentumLz about the z-axis in the state ψ(x, y, z) would yield m~.

4. An electron is constrained to move in a two-dimensional xy plane, and a magnetic �eld B is appliedperpendicular to the plane. The Hamiltonian is:

H =(p− eA)2

2m

where

p = −i~∇ = −i~(x∂

∂x+ y

∂

∂y

)and ∇×A = Bz

Solve the Schrodinger equationHψ(x, y) = Eψ(x, y)

in the Landau gauge:A(x, y) = −xBy

and calculate the energy spectrum E and the corresponding eigenfunctions ψ(x, y):

(a) First assume that the Hamiltonian eigenfunctions take the form:

ψ(x, y) = φ(y)eikx

Substitute this form in the Schrodinger equation and derive a di�erential equation for φ(y).

(b) Show that the equation for φ(y) is equivalent to the Schrodinger equation of a linear harmonicoscillator. Identify the harmonic frequency ωc (called cyclotron frequency) and the point y = y0(called �guiding center�) about which the oscillator oscillates.

(c) Let χn(y) be the eigenfunctions of an ordinary harmonic oscillator (a particle with mass m con-�ned near y = 0 by a parabolic potential with harmonic frequency ωc). Express the full eigenfunctionsψ(x, y) using χ(y). There is no need to write the explicit formula for χ(y).

(d) What quantum numbers enumerate the stationary states? What are the energy eigenvalues Eand their degeneracy N ?

• Note that ψ(x, y) looks like a stream of current, propagating as a plane wave with wavevector kalong the x-direction, but �parabolically� con�ned near y0 in the y-direction. If the system is arectangle with edge lengths Lx and Ly, then the wavevector k must be quantized in the units of2π/Lx (with periodic boundary conditions). Argue that the minimum and maximum values of kare limited by Ly, and then count the number N of quantized values that k can take.

2

5. Consider N magnetic impurities of spin S = 12 embedded in a crystal of volume V . In the limit

of low impurity concentration n = N/V we can neglect any interaction between impurities, so theirHamiltonian in the external uniform magnetic �eld B = Bz is:

H = −γN∑i=1

BSi

where Si = xSxi + ySyi + zSzi and Sx,y,zi are the spin-projection operators of the ith impurity. Thedensity matrix of this system in thermal equilibrium is:

ρ =1

Ze−βH

where Z is the partition function, β = (kBT )−1, kB is Boltzmann constant and T is temperature.

(a) Let |σ1 · · ·σN 〉 with σi = ±1 be a simultaneous eigenstate of all impurity spins Szi :

Szi |σ1 · · ·σN 〉 =~2σi |σ1 · · ·σN 〉 , σi = ±1

Show that |σ1 · · ·σN 〉 is also an eigenstate of the total impurity spin operator:

Sz =

N∑i=1

Szi

What is the corresponding eigenvaluem~2 of Sz? What are the minimum and maximum possible values

for m among all states |σ1 · · ·σN 〉?

(b) Find the number N (m) of states |σ1 · · ·σN 〉 that correspond to the same eigenvalue m~2 of the

total impurity spin Sz. Hint: This is a combinatorial problem. Since you are given both N and m,you can determine how many spins point �up� (σi = +1) and �down� (σi = −1). You will need thebinomial formula for the number of di�erent ways to choose M out of N items:(

N

M

)=

N !

M !(N −M)!

(c) Calculate the partition function:Z = tr

(e−βH

)• Proceed by expanding the trace in the basis of |σ1 · · ·σN 〉 states. Note that the Hamiltoniandepends only on the total impurity spin Sz, so every term in the expanded trace, labeled as f(m)below, depends only on the eigenvalue m associated with the state |σ1 · · ·σN 〉. And, there areN (m) equal terms in the expanded trace for every possible value of m.

• It will be useful to �rst expressm ≡ m(M) as a function of the numberM of spins |σ1 · · ·σN 〉 thatpoint �up� (i.e. have σi = +1). Rewrite the original sum over all possible variations of σi = ±1as a sum over the number M of spins that point �up�:

tr(e−βH

)=

σi=±1∑{σ1···σN}

f(m) =

N∑M=0

N(m(M)

)f(m(M)

)• Simplify the sum and use the well-known formula:

N∑k=0

(N

k

)xk = (1 + x)N

3

(d) Show that the magnetization is:

M ≡

⟨1

V

N∑i=1

Szi

⟩= n

~2tanh

(βγ~B

2

)Show also that it saturates at low temperatures and reduces to the Curie-Weiss law at high tempera-tures:

MT→0−−−→ n

~2

, MT→∞−−−−→ n

γ~2B4kBT

• Calculate magnetization as the (mixed) ensemble average, 〈A〉 ≡ tr(ρA), with the thermal densitymatrix ρ = 1

Z e−βH . Use the same tricks as in the part (c) to compute this trace.

Useful formulas:

• Canonical commutators: [xi, pj ] = i~δij

• Pauli matrices:

σx =

(0 11 0

), σy =

(0 −ii 0

), σz =

(1 00 −1

)

4