Problem Set 10
8.04 Spring 2013
Solutions
Thursday, May 9
Problem 1. (5 points) Zero Point Energy in a Lattice
Requiring go 1 means that the barriers are very strong. In this case, transmission across
the barriers is very small so the electrons can be app
Physics 143a
Problem Set 1
Due Friday, September 6, 2013
1. Problem 1.1
2. Problem 1.3
3. Problem 1.5
4. Problem 1.7
5. Problem 1.13
6. Problem 1.14
Reading assignment:
Thursday, September 5: Read Chapter 1.
Please send me an email (townsend@hmc.edu) befo
Physics 143a
Problem Set 2
Due Friday, September 13, 2013
1. Problem 2.1
2. Problem 2.5
3. Problem 2.6
4. Problem 2.12
5. Problem 2.13
6. Problem 2.21
Reading assignments:
Reminder: For the Tuesday and Thursday meetings, please read the material before
co
Physics 143a
Problem Set 3
Due Friday, September 20, 2013
1. Problem 3.2
2. Problem 3.5
3. (a) Problem 3.10
(b) Problem 3.13
4. (a) Problem 3.15
(b) Problem 3.16
5. Problem 3.17
6. Problem 3.20 Be sure to set up the calculation of the fraction in part (a)
Physics 143a
Problem Set 4
Due Friday, September 27, 2013
1. Problem 4.4
2. Problem 4.5
3. Problem 4.8
4. Problem 4.12 Note: You can also determine the eigenstates of Sx by using the results
of Problem 3.19 with = / 2 .
5. Problem 4.13
6. Problem 4.14
Rea
Physics 143a
Problem Set 6
1. Problem 5.21
2. Problem 5.23
3. Problem 6.1
4. Problem 6.2
5. Problem 6.4
6. (a) Problem 6.11
(b) Problem 6.12
Reading:
Tuesday, October 15: Sections 6.1 through 6.6
Thursday, October 17: Sections 6.7 through 6.9
Due Friday,
Physics 143a
Problem Set 7
Due Friday, October 25, 2013
1. Problem 6.13
2. Problem 6.14
3. Problem 6.15
4. Problem 6.17
5. Problem 6.21
6. Problem 6.23
Reading:
Tuesday, October 22: The reading material has been posted to the course website. It
consists o
Physics 143a
Problem Set 8
1. (a) Problem 7.1
(b) Problem 7.3
2. Problem 7.4
3. Problem 7.7
4. Problem 7.14
5. Problem 7.16
6. (a) Problem 7.20
(b) Problem 7.21
Reading:
Tuesday, October 29: Sections 7.1 through 7.5
Thursday, October 31: Sections 7.6 thro
Physics 143a
Problem Set 9
1. Problem 9.5
2. Problem 9.10
3. Problem 9.12
4. Problem 9.13
5. Problem 9.20
6. Problem 9.23
Reading:
Tuesday, November 5: Sections 9.1-9.5
Thursday, November 7: Sections 9.6-9.10
Due Friday, November 8, 2013
Physics 143a
Problem Set 5
Due Friday, October 4, 2013
1. Carbon dioxide is a linear molecule (OCO) that likes to pick up an extra electron and
become a negative ion. Suppose that the electron would have energy EO if it were
attached to either oxygen atom
Problem Set 2
8.04 Spring 2013
Solutions
February 21, 2013
Problem 1. (10 points) Wave-riding Mechanics
(a) (4 points) Given a dispersion relation (k), the phase velocity vp is dened as
vp
,
k
(1)
and is the rate at which a single plane wave (i.e. a si
Problem Set 4
8.04 Spring 2013
Solutions
March 05, 2013
Problem 1. (10 points) Simultaneous Eigenstates
We do not in fact know the states momentum precisely. As an example, consider the ground
state of a particle in a box of width L:
for x < 0
0
2
0 (x) =
Problem Set 6
8.04 Spring 2013
Solutions
April 2, 2013
Problem 2. (10 points) Finding Meaning in the Phase of the Wavefunction
(a) (3 points) We calculate the expectation value of x in the usual way:
(
x)new = dxnew
(x)
xnew (x) = dxx(eiqx/n (x) eiqx
Problem Set 7
8.04 Spring 2013
Solutions
April 09, 2013
Problem 1. (15 points) Mathematical Preliminaries: Facts about Unitary Operators
(a) (3 points) Suppose u is an eigenfunction of U with eigenvalue u, ie
U u = uu
We thus have that
(U u |U u ) = (u
Problem Set 3
8.04 Spring 2013
Solutions
February 26, 2013
Problem 1. (20 points) Mathematical Preliminaries: Linear Operators
is linear, we need to show that
(a) (6 points) To show that an operator O
(af (x) + bg(x) = aO
f (x) + bO
g(x).
O
(1)
To
Problem Set 5
8.04 Spring 2013
Solutions
March 12, 2013
Problem 1. (10 points) The Probability Current
We wish to prove that
dPab
= J(a, t) J(b, t).
(1)
dt
Since Pab (t) is the probability of nding the particle in the range a < x < b at time t it is
mathe
Problem Set 1
8.04 Spring 2013
Solutions
February 13, 2013
Problem 1. (15 points) Radiative collapse of a classical atom
(a) (5 points) We begin by assuming that the orbit is circular. This seems like circular1
logic, but is actually a fairly common te
Problem Set 8
8.04 Spring 2013
Solutions
April 17, 2013
Problem 1. (15 points) Superposition State of a Free Particle in 3D
(a) (4 points) Recall from lecture that the energy eigenstates of a free particle in 3D are
given by
= A exp i kk kr = A exp [i
Problem Set 9
8.04 Spring 2013
Solutions
Thursday, May 2
Problem 1. (10 points) Coulomb Potential Superposition States
(a) (2 points) Our wavefunction is given by
= C 100 + 4i210 2 2221
(1)
To normalize, we compute (|) and set it to 1:
(|) =
C 1
Physics 143a
Problem Set 10
Due Friday, November 15, 2013
1. Problem 10.2
2. Problem 10.4
3. Problem 10.7
4. Problem 10.8
5. Problem 10.17
6. Problem 10.19
Reading:
Tuesday, November 12: Sections 10.1 and 10.2
Thursday, November 14: Sections 10.3 through