7/27/2019 khyvbjlk;

1/1

PHYS 2912 (Advanced) Quantum Physics Assignment 1

Due Friday 11 October 2013

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

|1

= i

|+

+ 2

|, (1)

|2 = 3|+x ei/3|x , (2)

For each of these two states, answer the following questions:

(a) Normalise this state vector.

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

(c) Calculate the expectation value Sz for this state. How does this quantity relate to youranswer to part (b)?

(d) What are the possible results of a measurement of the spin component Sx, and with whatprobabilities do they occur?

(e) Calculate the expectation valueSx

and the uncertainty Sx for this state.

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

| =

2

3|+ 1

3ei/3| .

Answer the following questions:

(a) Identify a vector spatial n and the associated spin operator Sn (written as a matrix) for whichthe state | is an eigenvector with eigenvalue +/2.

(b) Find the normalised state that is the other eigenvector of this same spin operator, with eigen-value /2.

(c) Calculate the inner product between these two eigenvectors.

3. Any unit vector n in real three-dimensional space can be described using polar coordinates with apolar angle , in the range 0 , and an azimuthal angle , in the range 0 < 2. Thevector can be expressed in Cartesian coordinates as

n = (sin cos , sin sin , cos ) . (3)

(a) Calculate the corresponding spin operator Sn in matrix notation.

(b) Calculate the normalised eigenvector of this spin operator with eigenvalue +/2.

(c) Argue that the set of all possible normalised states of a spin-1/2 particle are in one-to-onecorrespondence with the points on the surface of a sphere. This parametrisation of states ofspin-1/2 particles is known as the Bloch sphere. Hint: you will need to use the fact that the

overall phase of a quantum state is irrelevant to its description.4. (a) Determine the matrix representation of the rotation operator R( k) using the states |+ and

| as a basis. Using your matrix representation, verify that this rotation matrix is a unitarymatrix.

(b) Repeat part (a) for the rotation operator R( i).

(c) Calculate the matrix representation for the rotation operator that describes a rotation by

about a unit vector pointing in the (i+k)/

2 direction, i.e., pointing in a direction 45 betweenboth the +x and +z axes.

1