Physics 501
Final Exam
December 19, 2011
Name:
Ground rules:
You can look at one textbook (e.g., Shankar), your lecture notes (or any other notes
prepared by you in advance), and the homework assignments and your personal solutions.
Do not consult other
DIRECT PRODUCT SPACES
Note that the terms direct product and outer product are essentially synonymous.
Vectors in a direct product spaces
Let |i (with i = 1, . . . , n1 ) form an orthonormal basis of V1 dimensions = n1 ).
Let | (with a = 1, . . . , n2 ) f
Physics 501
Homework Set 1
Fall, 2012
Due date: Monday, Sept. 17.
Reading: Shankar Ch. 1 up through p. 43.
1) (5 points) Shankar Ex. 1.4.2, p. 18.
2) (15 points.) Consider the set G of all possible polynomials in x and y having maximum
overall order of 2.
Physics 501
Homework Set 2
Fall, 2012
Due date: Monday, Sept. 24.
Reading: Finish Ch. 1
1) (6 points.) Shankar Ex. 1.8.10, p. 46 (simultaneous diagonalization of two matrices).
2) (4 points.) A two-dimensional space is spanned by the orthormal basis vecto
Physics 501
Homework Set 3
Fall, 2012
Due date: Wednesday Oct. 3.
Reading: Shankar Chs. 2, 3, and 4.1-2
1) (8 points.)
Consider two point charges of identical mass m conned to a 2D plane with an external
1
2
potential 2 (x2 + yi ) aecting both particles (
Physics 501
Homework Set 4
Fall, 2012
Due date: Wednesday Oct. 10.
Reading: Finish Shankar Ch. 4. Also, start reading Ch. 5.
1) (10 points.)
Referring to Ex. 4.2.1 on p. 129:
a) Obtain eigenvalues and normalized eigenvectors for all three operators Lx , L
Physics 501
Homework Set 5
Fall, 2012
Due date: Wednesday Oct. 17.
Reading: Shankar Ch. 5. You can skip Sec 5.4. Also read Ch. 6.
1) (4 points) Do Shankar Ex. 5.2.1 on p. 163.
2) (4 points) Do Shankar Ex. 5.2.4 on p. 164.
3) (6 points) Do Shankar Ex. 5.2.
Physics 501
Homework Set 6
Fall, 2012
Due date: Wednesday Oct. 24.
Reading: Shankar Ch. 7. Also look at Ch. 9.
Note: The take-home midterm will be handed out in class on Wednesday Oct. 31 and
will be due in my oce at 11am on Friday Nov. 2. See web site fo
Physics 501
Homework Set 7
Fall, 2012
Due date: Wednesday, Nov. 14.
Reading: Shankar Ch. 10.
1) (4 points.) Show that the potential V (x, y ) = + for |x| > L/2 and V (x, y ) =
1
m 2 y 2 for |x| < L/2 (harmonic oscillator in a strip) is separable, and writ
Physics 501
Homework Set 8
Fall, 2012
Due date: Monday, Nov. 26.
Reading: Shankar Ch. 11 and 12.1-4.
1) (3 points.) If | is a simultaneous eigenvector of Hermitian operators A and B that
anticommute, AB +BA=0, what can you say about the eigenvalues a and
Physics 501
Homework Set 9
Fall, 2012
Due date: Monday, Dec. 3.
Reading: Shankar Chs. 12 and 13
Reminder: Make-up class on Thursday, 11/29, 10:20-11:40am in Hill 116.
1) (3 points.) Fill in the missing parts and keep for your future reference. (Derivation
Physics 501
Homework Set 10
Fall, 2012
Due date: Wednesday, Dec. 12. (Last day of class.)
Reading: Shankar Chs. 14 and 15
Note: This homework set is a bit extra-long and is worth 40 points.
1) (6 points.) Some spinor algebra:
a) Shankar 14.3.2, p. 384.
b)