PHY3063 Spring 2007
R. D. Field
PHY 3063 Exam 1 Solutions
Problem 1 (20 points):
Circle true or false for the following 5 questions (1 point each):
(a) (True or False,) The laws of physics are invariant under a change in inertial frame.
(b) (True or False
PHY3063
R. D. Field
Expectation Values and Differential Operators (2)
Dynamical Variables become Differential Operators:
2
2
E op = ih
( p x ) op = ih
( p x ) op = h 2 2
x
t
x
Expectation Values of Dynamical Quantities: The average momentum is
given by
(
PHY3063 Spring 2007
Problem Set 6
PHY 3063 Problem Set #6
Due Tuesday March 20 (in class)
(Total Points = 70, Late homework = 50%)
Reading: Finish reading Tipler & Llewellyn Chapter 4 and start reading Chapter 5 and 6.
Useful Math:
x e
0
n a2 x2
dx =
1
+
PHY3063 Spring 2007
Problem Set 7
PHY 3063 Problem Set #7
Due Thursday April 5 (in class)
(Total Points = 110, Late homework = 50%)
Reading: Finish reading Tipler & Llewellyn Chapter 5 and 6.
Problem 1 (25 points):.
(a) (2 points) Show that the sum of two
PHY3063
R. D. Field
Schrdingers Equation
The Classical Hamiltonian: Classically the energy is the sum of the kinetic
energy plus the potential energy as follows (in one dimension):
p2
E = x + V ( x) ,
2m
and hence corresponding Quantum Mechanical Hamilton
PHY3063 Spring 2007
Problem Set 8
PHY 3063 Problem Set #8
Due Tuesday April 24 (in class)
(Total Points = 105)
Reading: Read Tipler & Llewellyn Chapter 7.
Problem 1 (20 points): The Pauli spin matrices are given by
0 1
0 i
1 0
x =
y =
z =
1 0
i 0
0 1
PHY3063
R. D. Field
Probability Flux
Probability Flux: Look at the time dependence of the
probability that the particle lies in the region x1 x x2
j(x1,t)
j(x2,t)
P(x1,x2,t)
x2
P ( x1 , x 2 , t ) = * ( x, t ) ( x, t ) dx .
x1
x2
x1
We see that
x
dP ( x1 ,
PHY3063
R. D. Field
The Scope of PHY3063
Physics: To devise concepts and laws that can help us
to understand the universe (i.e. nature)!
speed
seed of light in vacuum
c
Relativity Physics
Relativistic
Quantum
Physics
?
PHY3063
Quantum
Physics
Classical
Ph
PHY3063
R. D. Field
The Infinite Square Well (1)
One Dimensional Box
Particle in a One-Dimensional Box: Consider the
solution of
h 2 d 2 ( x )
+ V ( x ) ( x ) = E ( x ) ,
2m dx 2
where
0
( x, t ) = ( x )e iEt / h ,
for the case V(x) = if x 0 and V(x) =
PHY3063
R. D. Field
The Infinite Square Well (2)
Normalized Wavefunctions: Now we require that
n
( x ) n ( x ) dx = 1 ,
and hence
L
4 LA 2
4 A sin ( nx / L ) dx =
n
0
2
2
n
sin
2
I multiplied the
wavefunctions by the phase
e-i/2 to make them real!
d = 2
PHY3063
R. D. Field
Euclidean Geometry in Empty Space
Experimental observation!
The homogeniety and isotropy of euclidean space can be express by
three invariance principles (i.e. symmetries of empty space):
Leads to linear
momentum conservation!
Invarian
PHY3063
R. D. Field
The Infinite Square Well (3)
Average Value of px: The average value of px for the nth state is
dn ( x )
< p x > n = n ( x )( p x ) op n ( x ) dx = ih n ( x )
dx
dx
n
2i h n
2i h
=
sin( nx / L ) cos( nx / L ) dx =
sin( ) cos( ) d = 0
PHY3063
R. D. Field
Galilean Transformation
y
Consider two frames of reference the
O-frame (label events according to
t,x,y,z) and the O'-frame (label events
according to t',x',y',z') moving at a
constant velocity V, with respect to each
other at let the
PHY3063
R. D. Field
The Infinite Square Well (4)
One Dimensional Box
Particle in a One-Dimensional Box: Consider the
solution of
h 2 d 2 ( x )
+ V ( x ) ( x ) = E ( x ) ,
2m dx 2
where
-L/2
( x, t ) = ( x )e iEt / h ,
for the case V(x) = if x -L/2 and V
PHY3063
R. D. Field
Postulates of Classical Physics
Consider two frames of reference the
O-frame (label events according to
t,x,y,z) and the O'-frame (label events
according to t',x',y',z') moving at a
constant velocity V, with respect to
each other at le
PHY3063
R. D. Field
The Infinite Square Well (5)
Case II: Another set of solutions comes from taking A' = 0 and sin(kL/2) =
0, which implies that kL/2 = n- with n- = 1, 2, 3, . Thus,
n ( x ) = 2 B ' sin( 2n x / L )
and
h 2 k 2 2 h 2 2 ( n ) 2
2h 2 2
=
=
PHY3063
R. D. Field
Conservation of Linear Momentum (Classical)
y'
y
y'
y
V
V
Before Collision
After Collision
m2
m1
M1
O O'
O O'
x
x'
x
z'
z
M2
x'
z'
z
Consider two frames of reference the O-frame (label events according to
t,x,y,z) and the O'-frame (lab
PHY3063
R. D. Field
Useful Math
Trigonometric Relations:
sin( A B ) = sin A cos B cos A sin B
cos( A B ) = cos A cos B m sin A sin B
2 cos A cos B = cos( A + B ) + cos( A B )
2 sin A sin B = cos( A + B ) cos( A B )
2 sin A cos B = sin( A + B ) + sin( A B
PHY3063 Spring 2007
Problem Set 5
PHY 3063 Problem Set #5
Due Thursday February 22 (in class)
(Total Points = 60, Late homework = 50%)
Reading: Finish reading Tipler & Llewellyn Chapter 3 and start reading Chapter 4.
Problem 1 (5 points): A metal has a wo
PHY3063
R. D. Field
Expectation Values and Differential Operators (1)
One Space Dimension: To make things easier we will start with just one
space dimension x so that
2
( x, t ) dx = ( x, t ) dx
is the probability of finding the particle at time t betwee
PHY3063 Spring 2007
Problem Set 4
PHY 3063 Problem Set #4
Due Thursday February 15 (in class)
(Total Points = 70, Late homework = 50%)
Reading: Continue reading Tipler & Llewellyn Chapter 3.
Problem 1 (10 points): Consider a two dimensional conducting squ
PHY3063
R. D. Field
Double Slit Interference
The simplest way to produce a
phase shift a difference in the
path length between the two
wave sources, S1 and S2 is with
a double slit. The point P is
located on a screen that is a
distance L away from the sli
PHY3063 Spring 2006
R. D. Field
PHY 3063 Exam 1 Name_
Problem 1 (25 points):
A rod with is at rest in the O-frame (parallel to the
y'
y
Cart at rest in
x-axis) with the left end of the rod (L) at x = 0 and
the O'-frame.
right end (R) at x = R and a cart w
PHY3063
R. D. Field
Double Slit Intensity Pattern
We form the superposition of the two
waves at the point P on the screen as
follows.
Double Slit
r1
S1
d
S2
1 = Aei ( kr1 t )
r2
2 = Aei ( kr2 t )
tot = 1 + 2
r =d sin
and thus
tot = 1 + 2 = Aei ( kr1t ) +
PHY3063 Spring 2006
R. D. Field
PHY 3063 Exam 2 Solutions
Problem 1 (25 points): Consider an atom consisting of a proton and a muon. A proton has an
electric charge +e and rest mass energy Mpc2 940 MeV. A muon is an elementary particle with
charge e simil
PHY3063
R. D. Field
The Davidson-Germer Experiment
Bragg X-Ray Scattering (photons):
Consider the scattering (or reflection) of Xrays from two adjacent planes of atoms (called
Bragg planes). To a good approximation the
path length difference between ray 1
PHY3063 Spring 2006
R. D. Field
PHY 3063 Exam 2 Name_
Problem 1 (25 points): Consider an atom consisting of a proton and a muon. A proton has an
electric charge +e and rest mass energy Mpc2 940 MeV. A muon is an elementary particle with
charge e similar t
PHY3063 Spring 2006
R. D. Field
PHY 3063 Final Exam Solutions
Problem 1 (35 points): Consider an particle with mass m
Infinite Square Well
confined within an infinite square well defined by
V(x) = 0 for 0 < x < L,
V = +infinity
V = +infinity
V(x) = + othe
PHY3063
R. D. Field
Wave of what?
Inescapable Consequence of Wave Packets
Wave Packet
De Broglie postulated that the pilot wavex
packet governs the motion of the electron.
However, the electron is a point like
electron
elementary particle (i.e. radius zer
PHY3063 Spring 2006
R. D. Field
PHY 3063 Final Exam Name_
Problem 1 (35 points): Consider an particle with mass m
Infinite Square Well
confined within an infinite square well defined by
V(x) = 0 for 0 < x < L,
V = +infinity
V = +infinity
V(x) = + otherwis