Quantum Mechanics Final Exam 2003

Final Exam, Introdu tion to Quantum Me hani s De ember 17, 2003

Work all problems. Put ea h problem on a separate page, and be sure to put your name on all the pages. Points are assigned as indi ated; it is in your interest to look over the whole exam and work what seem to you to be the easiest parts rst. Don't get too hung up on any one problem and waste all your time on that one, thereby negle ting the rest of the exam. In many of these problems you are given a tual You plug these numbers in to re eive full redit. Please don't ost yourself easy points by forgetting to do this. Also, don't set h = 1 unless I tell you expli itly in that problem that you an. Very Important Note:

numbers.

must

Good Lu k!!

Possibly useful operators

Simple Harmoni Os illator: a^ =

Spin- 12 :

  h 0 1 Sx = 2 1 0 ;

Spin-1:

0 1 0 1 0 h Sx = p  1 0 1 A ;

2 0 1 0

r





m! i x^ + p^ 2h m!

  h 0 i Sy = 2 i 0 ; 0 1 0 i 0 h iA ; Sy = p  i 0

2 0

i

0

  h 1 0 Sz = 2 0 1 0 1 0 01 Sz = h  0 0 0 A

0 0

1

Possibly useful integrals

8 p > =a > > > Z1 p < 1 dx x2n e ax2 = > 2 =a3 1 > > > : 3 p=a5 Z1 1

4

dx e Z1 0

Z

n=0

n=1 n=2

p

ax2 +bx

=

dx xn e

ax

2 =a
eb =4a

= ann+1!

b)x sin(a + b)x b) 2(a + b) b)x sin(a + b)x + 2(a + b) b) b)x os(a + b)x b) 2(a + b) Z 2 dx sin ax os ax = os2aax Z Z x sin 2ax 2 dx sin ax = 2 4a dx os2 ax = x2 + sin42aax Z sin(a b)x x sin(a + b)x a b)x os(a + b)x + x dx x sin ax sin bx = os( 2 2 2(a b) 2(a + b) 2(a b) 2(a + b) Z a b)x os(a + b)x sin(a b)x + x sin(a + b)x dx x os ax os bx = os( + + x 2 2 2(a b) 2(a + b) 2(a b) 2(a + b) Z a b)x sin(a + b)x

os(a b)x x os(a + b)x dx x sin ax os bx = sin( + x 2 2 2(a b) 2(a + b) 2(a b) 2(a + b) Z dx x sin ax os ax = sin8a22ax x os42aax Z 2 dx x sin2 ax = x4 os8a22ax x sin4a2ax Z 2 dx x os2 ax = x4 + os8a22ax + x sin4a2ax a dx sin ax sin bx = sin( 2(a Z a dx os ax os bx = sin( 2(a Z a dx sin ax os bx = os( 2(a

(25 pts) A oherent state ji of the simple harmoni os illator is one that satis es the equation a^ji = ji ; where  is a omplex number,  = jjei' . ^ 2ji, where N^ is the number operator. a. (5 pts.) Compute hjN b. (20 pts.) Show that
x in the state ji is independent of . 1.

(20 pts.) Consider three parti les, with spin quantum numbers s1 = 21 , s2 = 32 and s3 = 1. a. (5 pts.) What is the total number of possible states js1 s2 s3 ; m1 m2 m3 i? ^12 = S^1 + S^2 , with eigenstates js12m12 i. What is js12m12 i = j11i in b. (5 pts.) De ne S terms of the states js1m1 i and js2m2 i? ^ = S^1 + S^2 + S^3 , with total spin quantum number s.

. (10 pts.) De ne the total spin S What are the possible values of s? Are there any of these values whi h an be made in more than one way? Verify that the total number of states jsmi is the same as in part a. 2.

3.

(40 pts.) Consider the potential8

> 0 > > < V (x) = V0 > > > :

1

x<

 

2

2 x0 x>0

In this problem you may set h = 1. a. (10 pts.) Set V0 = 5 (so V0 = 5 in the potential). A plane wave of parti les with mass m = 2 and energy E = 4 is in ident from x = 1. Write the general solution to the S hrodinger equation in the regions 2  x  0 and x < 2 . Use all of the numeri al information that is provided.  b. (10 pts.) What are the boundary onditions that must be imposed at x = 0, x = 2 and x = 1?

. (10 pts.) Find the re e tion oeÆ ient R. d. (5 pts.) Now onsider solutions for whi h E < 0. What is the general solution to the S hrodinger equation in the two regions, after the boundary onditions at x = 0 and x = 1 are taken into a

ount? e. (5 pts.) If you imposed the boundary onditions, you would nd a trans endental equation for E whi h you would have to solve numeri ally. For general V0 , will this have at least one solution? Why or why not? (No redit for just guessing equation

orre tly!) always

(30 pts.) Suppose we have a parti le of mass m = 3 in a one-dimensional in nite square well with walls at x =  and x = . At t = 0, the wavefun tion of the parti le is given by r r x 1 (x; 0) = 3 os 2 + i 32 sin x : In this problem you may set h = 1. a. (10 pts.) Compute (x; t) for t > 0. b. (5 pts.) What is hH i, the expe tation value of the energy?

. (15 pts.) Compute hxi for general t > 0. 4.

(35 pts.) Consider a spin- 12 parti le des ribed by the Hamiltonian H^ = !1 S^x + !2 S^z ; where !1 = 3 and !2 = 4. In this problem you may set h = 1. ^ in the basis in whi h S^z is diagonal? a. (5 pts.) What is the representation of H b. (15 pts.) Find the stationary states and the orresponding energy eigenvalues. 1

. (15 pts.) Suppose that at t = 0 the parti le is in an state in whi h Sz = h 2  . What is the probability of nding Sz = 12 h for t > 0? 5.

(35 pts.) This problem on erns a parti le of spin 1. The parti le is initially in a state ve tor j i whi h is represented in the Sz basis by0the olumn 1 6.

A

i

3A 1 a. (5 pts.) Use the normalization of j i to nd A. b. (10 pts.) Compute hSy i in the state j i.

. (15 pts.) What is the operator R^ () that rotates a general state j i by an angle  about the z^ axis? Find the matrix representing R^ (=2) in the Sz basis. ^ (=2)j i, and represent the state j'i in the Sz basis. d. (5 pts.) Compute j'i = R

(15 pts.) Consider three Hermitian operators, A, B and C . Suppose that [B; C ℄ = A and [A; C ℄ = B : Show that in any state,
(AB )

C  12 haA2 + bB 2 + C 2 i ; and nd the onstants a, b and . 7.