Download - 124994 Midterm Exam Fall 2009

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PHYS M1410: Quantum Mechanics I.

Midterm Examination

PHYS M1410: Quantum Mechanics (I)November 20, 2009 Friday

Problem 1. (16 points) (a) Study the hermiticity of the operators:

, = ,d

X D P i Ddx

. What are the complex conjugate of these operators? (b) Show

that [ , ] X P i . (c) Show that the translation operator /( ) iPaT a e is unitary and is defined

to be such that ( ) ( ) ( )T a x x a . (d) Consider the following two matrices:

1 0 1 1 1 2

= 0 0 0 , 1 0 1

1 0 1 2 1 1

. Can they be simultaneously diagonized? If so, give the

reason why, then find the eigenvectors common to both, and verify that under a unitary

transformation to this basis, both matrices are diagonized.

Problem 2. (12 points) (a) Show that the time derivative of the expectation value of any

observable = | | is given by ,d i

Hdt t

, where H is the

Hamiltonian of the quantum system. (b) Using the result of (a), show that ifH is not explicitly

time-dependent, the total energy of the system is conserved. (c) If2

( )2P H V X

m , show that

d X

dtand

d P

dthave respective form reminiscent of Hamilton equation of classical

mechanics.

Problem 3. (12 points) In a double-slit experiment with a source of monoenergetic electrons,

detectors are placed along a vertical screen parallel to the y-axis to monitor the diffraction

pattern of the electrons emitted from the two slits. (a) If the intensity of the electron beam is so

low such that one is doing the experiment with only one electron at a time, can one predict the

vertical location of a given single electron on the screen? Why? (b) Can we predict the

interference pattern if we wait until many electrons have arrived? Why? (c) If a light source is

used to determine which of the slits the electron went through, can the interference pattern be

observed on the screen after we wait until many electrons have arrived? Why? [Please give

your reasons why in (a), (b) and (c) in terms of interpretations by quantum mechanics]..

Problem 4. (20 points) Consider the following operators on a Hilbert space 3V ( )C


2/3


0 1 0 0 0 1 0 01 1 1

1 0 1 , 0 , 0 0 02 2 2

0 1 0 0 0 0 0 1

x y z

i

L L i i L

i

(a) What are the possible

values one can obtain if xL is measured? Why? (b) Take the state in which 1xL . In this

state what are2

, , andy y L L L , where2 2 2 2

x y z L L L L . (c) Find the normalized

eigenstates and eigenvalues of yL in the zL basis. (d) If the particle is in the state with

1zL , and yL is measured, what are the possible outcomes and their possibilities? (e)

Consider the state

2 / 3

= 1/ 3

2 / 3

in thez

L basis. Ifx

L is measured in this state and a result

1 is obtained, what is the state after the measurement? How probable was the result? Ifz

L

is measured, what are the possible outcomes and their probabilities?

Problem 5. (24 points) Consider a free particle with2/ 2 H P m . (a) Find its eigenvalues

and eigenfunctions in the X basis. (b) Find the propagator ( , ; ', ') ( ) 'U x t x t x U t x (c)

consider as an initial wave function the wave packet2 2

0 / / 2 2 1/ 4( ,0) /( )ip x x x e e . Calculate

(0) , (0) , (0), (0) X P X P . (d) Find ( , )x t and probability density at time t. (e) Find .

( ) , ( ) , ( ), ( )X t P t X t P t . (f) For a macroscopic particle with 132g and 4 10 cmm ,

estimate how long is it for ( ) 1mmX t ?

Problem 6. (16 points) (a) Consider the one-dimensional potential barrier as shown in the

figure. Write down the wave function that is

in physical Hilbert space in each region of

the potential for V2


3/3


state, the probability current density is a constant in coordinate (independent of x). (d)

Calculate the probability current density in Region I and III, as well as using the results of (c),

show that R+T=1, where R is the reflection coefficient and Tis the transmission coefficient.

Problem 7. (20 points) Consider a Harmonic oscillator with

2 2 2

+2 2

P m X

H

. (a) Show that

( 1/ 2) H a a , where 1 1

, and2 2 2 2

m ma X i P a X i P

m m

. Show

also that [ , ] 1a a . (b) Given that 1 , and 1 1a n n n a n n n , show that

( 1/ 2) H n n n . (c) Find the normalized ground state wavefunction by projecting

0 0a on the X basis. (d) Suppose at t=0, a particle starts out in

1 1 1(0) 0 1 2

2 2 2 . Find (0)X and the average energy of the system at t=0. (e)

Find ( ) , ( )t X t , and the average energy at time t.

Useful Gaussian Integrals:

2

2

2 2

1/ 2

1/ 2

2

1/ 2

/ 4

for 0,

1,

2

.

x

x

x x

e dx

x e dx

e dx e

Download - 124994 Midterm Exam Fall 2009

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