PH401 QUANTUM MECHANICS WINTER 96-97 EXAM #2

 * asterisk marks angular momentum questions which will be covered on exam #3 for 2000-2001

 

Section 1: Multiple choice. No partial credit will be given for this section. Circle only one answer for each question.

 

1) Hop|En> is a(n)

a) number, possibly complex.

b) state or vector.

c) operator.

 

*2) Which of the following is not true of all the spherical harmonics, Ylm(q,f)?

a) They are eigenstates of L2.

b) They are eigenstates of Lx, Ly, and Lz.

c) They are solution of the time-independent Schroedinger equation.

d) Yl-m(q,f) = (-1)m Ylm*(q,f).

 

3) Given the following in reference to the harmonic oscillator: a = lowering operator, a+= raising operator, and |En> = the nth eigenstate of E. a+a|En> =

a) Nop

b) aa+|En>

c) n|En>

d) all of the above.

 

4) |r1> and |r2> are eigenvectors of Rop, with eigenvalues r1 and r2. r1 is not

equal to r2. Which of the following must be true about the vectors |r1> and |r2>?

a) They are orthogonal.

b) They are degenerate.

c) |r1> + |r2> is an (un-normalized) eigenstate of Rop.

d) All of the above.

 

 

5) Which of the following is true of a Hermitian operator

a) Every Hermitian operator is equal to its Hermitian conjugate.

b) All Hermitian operators have real eigenvalues.

c) All operators corresponding to observables are Hermitian.

d) All of the above.

 

 

 

 

Section 2: Problems and questions. Partial credit will be given for this section. Show all work and justify all answers.

 

 

6) Given the following column vectors in a 2-D state space:

 

and

 

a) Show that these vectors are orthogonal.

b) Show that these vectors are normalized.

c) Show that this set of vectors is complete.

 

 

7) Given the following in reference to the harmonic oscillator: a = lowering operator, a+= raising

operator, N=a+a, and |n> = eigenstate of N with eigenvalue n,

a) Find [a,N].

b) Is a|n> an eigenstate of N? If so, what is the eigenvalue?

 

 

8) Given a particle of mass 9.1X10-31 kg in a harmonic oscillator of frequency 2.5X1011 Hz.

The particle is in a state which is a combination of energy eigenstates:

 

 

 

a) Find <E >.

b) Find <x>.

 

*9) If the angular momentum eigenstates with l =1 are used as basis states:

basis vector #1 |1 -1>

 

 

basis vector #2 |1 0>

 

 

basis vector #3 |1 1>

 

 

then the operator Lx is given by a square matrix:

 

 

 

 

 

a) Fill in the second third column, justifying each element.

b) Show that represents an eigenstate of Lx. What is the eigenvalue?

 

 

 

 

*10) a) Do simultaneous eigenstates of x and Lx exist? Justify.

b) Do simultaneous eigenstates of L+ and Lx exist? Justify