Homework 3 Solutions
AS.171.303: Quantum Mechanics I
Due: Tuesday, October 8
1. (a) From equation (3.28), we have the following representations for the rotation generators,
0
x 1
J
20
0
Jy i
20
10
Jz 0 0
00
10
0 1
10
i 0
0 i
i
0
0
0
1
Computing [Jx , Jy ]
Homework 4 Solutions
AS.171.303: Quantum Mechanics I
Due: Tuesday, October 15
1. The particles exiting the Stern-Gerlach device are all in the state
1
3
| = j = , mx = +
2
2
=
31
,+
22
x
The possible values for the measurement of the z -component are simp
Homework 7 Solutions
AS.171.303: Quantum Mechanics I
Due: Friday, November 8
1. (a) In general, the probability of nding a particle between points x0 and x1 is
x1
P=
x0
dx | (x)|2
which in this case is
P=
c
1
a2
dx e(xc)
2 /a2
However we know that this in
Homework 2 Solutions
AS.171.303: Quantum Mechanics I
Due: Tuesday, October 1
1. (a) We can nd the matrix representation form of | x simply by inserting the matrix
form of | z into the provided equation,
1
1
1
|x = |+z |z =
2
2
2
1
1
0
2
0
1
=
1
2
1
2
(b
Introduction to Quantum Mechanics 171.303
Final Exam 12/20/05 9-12
Check the attached formula pages. Start each problem on a fresh page and give detailed
reasoning. Please write your name on each page and ask your proctor for clarification if
the text is
Introduction to Quantum Mechanics 171.303
Final Exam 12/18/07 9-12
Check the attached formula pages. Start each problem on a fresh page and give detailed
reasoning. Please write your name on each page and ask your proctor for clarification if
the text is
Introduction to Quantum Mechanics 171.303
Final Exam 12/20/06 9-12
Check the attached formula pages. Start each problem on a fresh page and give detailed
reasoning. Please write your name on each page and ask your proctor for clarification if
the text is
Final
171.303 Fall 2010
(open book and notes)
1. (a) We know |3/2, +3/2 = | + z 1 | + z 2 | + z 3 . Using the lowering operator,
h
S = S1 + S2 + S3 , and S |j, m = j (j + 1) m(m 1) |j, m 1 , we
nd
1
31
| , + = [| 1 |+ 2 |+ 3 + |+ 1 | 2 |+ 3 + |+ 1 |+ 2 |
Final
171.303 Fall 2010
(open book and notes)
1. (20 points) Three spin-1/2 particles are released in a pure quantum state of total
spin quantum number of s = 3/2 and total z -component of angular momentum
m = +1/2, labeled |3/2, +1/2 .
(a) Write this sta
Final Exam, Introduction to Quantum Mechanics
December 15, 2004
Work all problems. Put each problem on a separate page, and be sole to put your
name 011 all the pages. Points are assigned as indicated; it is in your interest to look over
the whole exam an
Introduction to Quantum Mechanics 171.303
Midterm Exam 11/3/05 1-2
Check the attached formula pages. Start each problem on a fresh page and please give
detailed reasoning. Please ask your proctor for clarification if the text is unclear.
Problem 1 (30 poi
Introduction to Quantum Mechanics 171.303
Midterm Exam 11/1/07 1-2
Check the attached formula pages. Start each problem on a fresh page and give detailed
reasoning. Please ask your proctor for clarification if the text is unclear.
Problem 1 (25 points)
A
Introduction to Quantum Mechanics 171.303
Midterm Exam 11/4/08 1:30-2:30
Check the attached formula pages. Start each problem on a fresh page and give detailed
reasoning. Where possible discuss appropriate limits and the physical implications of
your resu
171.303 Quantum Mechanics Midterm Exam
October 21, 2014 1:30 - 2:20
Start each problem on a fresh page. Explain your reasoning clearly and
completely for each problem. You may use your copy of Townsends textbook.
No other sources are allowed. You should c
Section 1 Notes
AS.171.303: Quantum Mechanics I
Tuesday, September 17
When discussing the spin or angular momentum of particles, we often only measure a
particular component of those vectors, such as the z -component. The Stern-Gerlach experiment tells us
Section 2 Notes
AS.171.303: Quantum Mechanics I
Friday, September 27
As a simple example for understanding rotations, lets consider vectors in R2 . Well
represent all of these as 2-component column vectors with only real entries. We have two
linearly-inde
Introduction to Quantum Mechanics 171.304
Final Exam 5/16/06 9-12
Check the attached formula pages. Start each problem on a fresh page and give detailed
reasoning. Please write your name on each page and ask your proctor for clarification if
the text is u
Homework 6 Solutions
AS.171.303: Quantum Mechanics I
Due: Tuesday, November 5
1. (a) There are three possible values for ml (+1, 0, 1) and two possible values for ms
(+ 1 , 1 ), which means there are six total combinations,
2
2
11
|1, +1 | , +
22
11
|1, +
Homework 5 Solutions
AS.171.303: Quantum Mechanics I
Due: Tuesday, October 29
1. This is a subtle problem, and turns out to be very important when considering more
complicated systems with time-dependent Hamiltonians. For a completely general
system, [H (
Homework 1 Solutions
AS.171.303: Quantum Mechanics I
Due: Tuesday, September 24
1. (a) Lets rst determine the units of each parameter. In SI units, we have
[ re ] = m
[me ] = kg
[e] = C
[c] = m/s
[40 ] =
s2 C2
kg m3
Now we can look at the units of both si
Midterm Solutions
AS.171.303: Quantum Mechanics I
Tuesday, October 22, 2013
1. (a) Using the matrix representations of Sx , Sy , and Sz , we can calculate the three
expectation values,
1
3i 4
Sx =
5
2
1
3i 4
Sy =
5
2
1
3i 4
Sz =
5
2
1
5
01
10
3i
4
=0
1
Midterm Exam
AS.171.303: Quantum Mechanics I
Tuesday, October 22, 2013
This is a closed book exam. Please be sure to show ALL essential steps of your work.
You will have 60 minutes to complete this exam.
1. A spin- 1 particle is in the state
2
| =
1
5
3i
Midterm
171.303 Quantum Mechanics
Fall 2009
(open book and notes)
1. (30 points) A spin-2 particle is prepared in the state |2, +2 , where S 2 |2, +2 =
2
2 (2 + 1) |2, +2 and Sz |2, +2 = 2 |2, +2 . What is the probability that, when
h
h
measured, it has z
Midterm
171.303 Quantum Mechanics
Fall 2009
Solutions
1. (30 points) To nd the probability P = |x 2, 0|2, +2 z |2 , we should nd |2, 0 x in the z
basis. To do so, we need to nd the matrix representation of Sx and nd the eigenvector
associated with the m =
Midterm SOLUTIONS
171.303 Fall 2010
(open book and notes)
1. (15 points) Note A is proportional to x , B is x + I (the identity matrix), and C
is proportional to z . Thus, A and B commute.
(a) A and B .
(b) A: 3 and 3
B : 2 and 0
C : 2 and 2
2. (35 points
Midterm
171.303 Fall 2010
(open book and notes)
1. (15 points) Three quantum operators A, B , and C are represented by the matrices:
A=
03
30
11
11
, B=
,C=
20
0 2
(a) Which of these operators have the same eigenstates?
(b) Find the eigenvalues of each ma