Phys 323, Sample Problems 2 (A. Polychronakos)

1) The Hamiltonian for a spin-half particle is

H = 2a (Sx + Sy)

where a is a positive constant and Sx , Sy are the x and y components of the spin. Initially (at time t=0) the particle is in the state

|ψ> = (1/√2) (|↑>+|↓>)

where up and down arrows denote eigenstates of the z component of the spin.

a) Calculate the expectation value of the energy <E> and of the three components of the spin <Sx>, <Sy> and <Sz> at t=0.

b) What is the probability that a measurement of the x component of the spin at t=0 will give the value +ħ/2 ?

c) Find the eigenstates and eigenvalues of the Hamiltonian.

d) Find the state |ψ(t)> at a later time t.

e) What is the probability that a measurement of the x component of the spin at time t will give the value +ħ/2 ?

2. A particle of mass m is in a one-dimensional harmonic oscillator potential perturbed with a small additional term. Its Hamiltonian is given by

H = Ho + c (a† + a) , Ho = ħω (a† a + ½)

where c is a small positive coefficient and a, a† are the harmonic oscillator “ladder” operators defined:

a = √(mω/2) (x+ip/(mω)) , a† = √(mω/2) (x-ip/(mω)) , [ a , a† ] = 1

a) Use first-order perturbation theory to calculate the correction to the ground state energy and first excited energy of the problem. (Hint: use the properties of a and a† .)

b) Express the Hamiltonian in terms of the usual position and momentum operators x and p.

c) Shift the origin of the coordinate x, thus redefining x by a constant, to show that the perturbed Hamiltonian can be mapped to a familiar Hamiltonian; justify the results of part (b) using this form of the Hamiltonian.

3. A particle of mass m is in a one-dimensional finite square well potential of the form:

V(x) = 0 for |x|<a

Vo for |x|>a

with a and Vo positive constants.

a) Discuss the possible type of energy eigenstates that the particle can have for the three ranges of energy: E < 0, 0 < E < Vo and E > Vo .

b) The parity operator Π is defined as: Πψ(x) = ψ(-x). Show that this operator commutes with the Hamiltonian of the particle. What does that mean for the energy eigenstate wavefunctions?

c) Show that there is always at least one bound state of energy Eo < Vo . (Hint: the ground state has no nodes; use also the result in (b).)

d) Find a condition for Vo so that there is at least one more bound state. (Hint: the next excited state must have one node; use also the result in (b).)

Scroll down for the answers.

Answers:

1)

a) <E> = aħ, <Sx> = ħ/2, <Sy> = 0, <Sz> = 0

b) P(Sx = ħ/2) = 1

c) E1 = aħ√2, |1> = (1/√2) |↑> + (1+i)/2 |↓> and E2 = -aħ√2, |2> = (1/√2) |↑> - (1+i)/2

d) |ψ(t)> = [1/√2 cos(a√2 t) - (1+i)/2 sin(a√2 t)] |↑>

+ [1/√2 cos(a√2 t)+(1-i)/2 sin(a√2 t)] |↓>

e) P(Sx = ħ/2) = cos²(a√2 t)

2)

a) ΔEo = ΔE1 = 0

b) H = p²/2m + ½ mω² x² + c √(2mω) x

c) In terms of x͂ = x + c√(2/mω³), H = p²/2m + ½ mω² x͂² - c²/ω

So it is the harmonic oscillator with an extra energy quadratic in c.

3)

a) E < 0 : No states; 0 < E < Vo : discrete energy levels, bound states; E > Vo : continuous energy levels, scattering states

b) [Π,H]=0 since V(x) = V(-x); all energy eigenfunctions will be either even or odd functions.

c) Ground state has no nodes, so it must be an even function. Take it from there.

d) Next state has one node so it must be an odd function. Condition for its existence:

Vo > π²ħ²/(8ma²)

Now that you got a good scare, I must tell you that the problems in the final will be easier than these!