1.   Explain briefly (less than 50 words each) what is meant by each of the following.

(a) An observable.

(b) Spin precession.

(c) A quantum dot.

(12 marks)

2.   Consider a standard Stern-Gerlach experiment, with a beam of spin- 1/2 particles prepared in the state

|ψ〉= 3|+〉_ 1|_〉.

(a) Normalise this state vector.

(b) What are the possible results of a measurement of the spin component Sz , and with what probabilities do they occur?

(c) What are the possible results of a measurement of the spin component Sx , and with what probabilities do they occur? Hint: the eigenstates for Sx, expressed in terms of the eigenstates of Sz, are:

|+〉x  =  (|+〉+ |_〉) ,

|_〉x  =  (|+〉_ |_〉) ,

(d) Suppose that the Sz  measurement was performed, with the result Sz   =  _3/2. Subsequent to that result, a second measurement is performed to measure the spin component Sx . What are the possible results of that measurement, and with what probabilities do they occur?

(e) If the beam of spin- 1/2 particles was instead prepared in the state

|ψ〉= 3|+〉_ i|_〉,

would your answer to part (b) change? Why or why not?

(12 marks)

3.   Consider spin- 1/2 particles prepared in the initial state

|ψ(0)〉= 1^2 |+〉+ 1^2 |_〉,

and precessing in a magnetic field B aligned in the z-direction according to the Hamilto- nian

H = _µ . B = ω0 Sz ,

where ω0  = eB0 /me .

(a) Calculate |ψ(t)〉, the state at some later time t.

(b) Calculate the expectation value〈Sx〉as a function of time.  Draw a plot of your answer. Hint: the Sx operator in matrix notation is Sx  =  (1(0) 0(1) ).

(c) Show that state |ψh〉= 1^2 |+〉_ 1^2 |_〉describes a spin- 1/2 particle aligned in the opposite direction to one described by |ψ(0)〉.

(d) At what time t will the spin- 1/2 particle, initially in the state |ψ(0)〉, evolve to the state |ψh〉?

(12 marks)

4.   An electron is confined to a one-dimensional quantum well of width L = 0.100 nm.

(a) If the height of the potential can be considered extremely large, calculate the ener- gies of the three lowest energy eigenstates.

Hint: The formula for the energies of an infinite well can be obtained from the ki- netic energy expression E = p2 /(2m), where the momentum takes quantized values pn  = 2π3/λn for the allowed wavelengths λn  = 2L/n.

(b) Draw a diagram to show the probability density functions (the functions |ψ(x)|2 ) for the ground state and first excited state.

(c) Draw these same functions for the case where the height of the potential is not very large (but still higher than the energy of the second excited state), clearly marking the difference in the form of the functions compared with part (c).

(d) Will the energies of the finite well be higher or lower than the corresponding ener- gies of the infinite well? Justify your answer.

(12 marks)