PHYS 225
Summer 2015
Homework #4 SOLUTION (Due in class Friday, July 31)
1. Let s be the number of spots observed on the top of a single die thrown at
random. (a) How many states are possible? What is the probability of obtaining
each state? (b) Calculate
PHYS 225
Summer 2015
Homework #5 (Due in class Wednesday, August 12)
Explain your reasoning and show your work on all problems
1. Given this wave function for a particle in position space:
for x a
0,
( x) A a 2 x 2 , for a x a
for x a
0,
(a) Determine t
PHYS 225
Winter 2015
Homework #4 SOLUTION (Due in class Friday, February 20)
1. Let s be the number of spots observed on the top of a single die thrown at
random. (a) How many states are possible? What is the probability of obtaining
each state? (b) Calcu
BLACKBODY RADIATION
I.
Homework
You may refer to Section 20-4 of Tipler, posted under "pre-Class Reading" (Lect 1).
Four different rooms with no external sources of light are described below. Assume each
bar of iron is identical. You may find it helpful t
PHYS 225
Winter 2016
Homework #2.5 (Due by 4 pm Monday January 25)
SOLUTION
The original problem 6 of HW 2 on expectation values:
1. What is the expectation value S x for the state
Easiest to convert to matrix notation:
Sx Sx
10
10
3
5
z
i
2
5
z
?
0 1
Quantum cryptography QM
3
II. The Effect of an Eavesdropper
The point in QKD where an eavesdropper would act is between Step 1 and Step 2 in the previous
section. During the transmission of the qubit from Alice to Bob, you can imagine that a third
person,
QUANTUM CRYPTOGRAPHY
QM
1
I. Quantum Key Distribution (QKD)
There are several protocols that will enable two parties to distribute a secure, private key using
the principles of quantum mechanics. The protocol discussed below is called the BB84 protocol
af
225 win 16 HW 3 QU-COMP Solutions
1. McIntyre 2.8 The
n
eigenstate is
n
cos 2 ei sin 2 cos 8 ei 5 3 sin 8
The probabilities are
P y
y
2
n
1
2
i
2
cos
ei 5 3 sin 8
8
2
2
12 cos 8 iei 5 3 sin 8 12 cos 8 sin 8 sin 53 i sin 8 cos 53
1
2
cos
1
2
1
PHYS 225
Winter 2016
Homework #2 (Due in class Friday, January 22)
SOLUTION
Problems from McIntyre, Chapter 1:
1. Problem 1.10
a) The probabilities for state 1 are
P1, 1
2
4
5
i 53
2
P1, 1
2
4
5
i 53
2
i 53
1
2
x
P1, x
2
x
1
P1, y
1
2
y
P1,
TWO-STATE TIME DEPENDENCE
I.
QM
1
Im
Complex numbers
Consider the complex number z = 1 i.
A. Plot this number on the set of axes shown at right.
Re
B. Draw a vector from the origin to the plotted number.
1. What does the length of this vector represent ab
Constructing Controlled Gates
We know the CNOT gate that applies a NOT (switch, or X) gate to qubit 2 if qubit 1 is a 1, and
otherwise does nothing. It was given in class without derivation:
1
0
0
0
0
1
0
0
0
0
0
1
0
0
1
0
How did this come about?
Conside
Two-state time dependence QM
3
3. Write an expression for the inner product
x
+ (t ) as a function of time.
4. On the graph at right, plot the inner product x + (t ) at several
different instants (e.g., t = T/8, t = T/4, etc.) for this particle.
(Hint: Ex
Name (print): _SOLUTION_
Signature: _
Student ID: _
Exam 2
Phys 225
Monday, February 23, 2015
This is a closed book exam. One sheet of notes is attached (you may tear it off). Use the
space beneath each problem statement to answer; should you need extra r
Name (print): _SOLUTION_
Signature: _
Student ID: _
Exam 2
Phys 225
Monday, August 3, 2015
This is a closed book exam. One sheet of notes is attached (you may tear it off). Use the
space beneath each problem statement to answer; should you need extra room
Name (print): _SOLUTION - post_
Signature: _
Student ID: _
Exam 1
Physics 225
Wednesday, January 28, 2015
This is a closed book exam. One sheet of notes (my version) is provided. Use the space
beneath each problem statement to answer; should you need extr
HANDOUT: TWO-STATE TIME DEPENDENCE
Fill out the chart below using answers you provided on the tutorial.
For states starting in the
Does the probability
depend on time?
x
state
Write an expression for
the probability
For particles starting in the z state
TWO-STATE TIME DEPENDENCE
I.
Complex numbers
QM
1
Im
Consider the complex number z = 1 i.
A. Plot this number on the set of axes shown at right.
Re
B. Draw a vector from the origin to the plotted number.
1. What does the length of this vector represent ab
CLASSICAL AND QUANTUM SPIN
I.
QM
1
Review of magnets
A. A bar magnet is placed in the uniform magnetic field shown at
right.
1. Draw a vector indicating the direction of the magnetic
dipole moment of the magnet.
N
1
2. Determine the direction of the net f
PHYS 225 Exam 1 Cheat Sheet
Classical Magnetic Dipoles in a B Field: B
U B
Fz z
Bz
z
Quantum Spin- Components: S x or S y or S z , 1.055 1034 J s 6.582 1016 eV s
2
Spin z Orthonormal Basis:
Euler Formula: ei cos i sin
z
in book ,
z
in book , 1,
Polarization of Light
I. Classical Description:
Light is an electromagnetic wave consisting of electric and magnetic fields that are mutually
perpendicular and both perpendicular to the direction of motion of the light.
Lets assume a plane wave propagatin
PHYS 225 Final Exam Cheat Sheet page 1
Photon Polarization States: Basis: x , y ,
General Linear Pol. State: cos x sin y
Circular Basis States (Right and Left Circular): R
1
2
x
i y ,
L
1
2
x
i y
Euler Formula: ei cos i sin
1.055 1034 J s 6.582 1016
QUANTUM CRYPTOGRAPHY: HANDOUT I
QM
1
Alice sends
0 Z
1.
2.
0 Z
1Z
1X
0 X
0 Z
1X
1Z
1Z
0 X
0 X
X
X
X
Z
X
Z
Z
X
Z
X
Z
X
Basis Alice sent:
( Z or X )
3.
0 X
Element of Alices
key: ( 0 or 1 )
4.
Basis Bob
measures:
5.
Possible States
after Bobs
measurement:
6
QUANTUM CRYPTOGRAPHY: HANDOUT 2
QM
1
0 Z
0 X
0 Z
1Z
1X
0 X
0 Z
1X
1Z
1Z
0 X
0 X
Basis Alice sent:
Z
X
Z
Z
X
X
Z
X
Z
Z
X
X
Element of Alices
key:
0
0
0
1
1
0
0
1
1
1
0
0
Basis Eve
measures:
X
Z
Z
X
X
Z
X
Z
X
X
Z
X
X
X
X
Z
X
Z
Z
X
Z
X
Z
X
Alice sends
State
Quantum Mechanics 225
Mid Term Examination 2009 SOLUTIONS
Name (Please Print):
Total Points: 100
Savage
(last)
(rst)
You should attempt all questions. There is a total of 100 points.
Problem 1
Problem 2
Problem 3
:
:
:
1
25 Points
40 Points
35 Points
Mid
1x01-Part01: The Innite Square Well
1x01-Part02: Problem 2.3
1x01-Part03: Some trajectories of a particle in a box
according to Newton's laws of classical mechanics (A),
and according to the Schr?dinger equation of
quantum mechanics (B-F). (From wikipedia
QUANTUM CRYPTOGRAPHY: HANDOUT I
QM
1
Alice sends
1.
2.
Basis Alice sent:
( Z or X )
3.
Element of Alices
key: ( 0 or 1 )
4.
Basis Bob
measures:
5.
Possible States
after Bobs
measurement:
(ket, or kets)
6.
0 Z
0 X
0 Z
1Z
1X
0 X
0 Z
1X
1Z
1Z
0 X
0 X
X
X
X
Z
1
The CNOT Quantum Gate
Submitted by: Ohad Levinkron
In this document we shall provide an explanation about the CNOT quantum logic gate, followed by a brief illustration
of the idea using NMR spectroscopy.
= [1] Motivation
In quantum computation we perfor
Physics
225
Lecture 18:
NMR & Review
HW #4 is Due Now (Semi Alphabetical)
HW #5 Will Be Posted Soon, Due March 4
Exam 2 Monday
Will Include Cheat Sheet as Before
Same Format
No Use of Graphic / Storage Capabilities
Topics & Equations Coming Soon
1
Revi
BLACKBODY RADIATION
I. Homework
You may find it useful to refer to Section 20-4 of Tipler, posted under "pre-Class Reading" (Lect 1).
Four different rooms with no external sources of light are described below. Assume each bar of iron
is identical. You may
Modern Applications of Quantum Mechanics
Comparing the scattering force in Fig. lo. 12 with the Maxwellian velocity distribution in
Fig. 168. we note that the range of velocities that are affected by optical molasses is very small.
In a typical experiment