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
PURPOSE
To investigate the addition property of vectors and to analyse vectors in terms of their
components along two perpendicular directions.
APPARATUS
Force table with removable central pin, 4 pulleys, cardboard underlay to protect workbench,
plastic r
Purpose:
To discover the concepts of relative motion and frames of reference.
Apparatus:
A motion sensor with leads, two dynamic carts, a screen (an optic frame with a slit
diaphragm), a long track.
Theory: See manual. Equations used in evaluation:
1. vs
PURPOSE
To use the properties of a conical pendulum to analyze centripetal force and acceleration.
APPARATUS
Metal ball attached to string, table clamp, long rod, clamp (test-tube holder) with two 50
g mass pieces to clamp the string, extra metal ball, st
Purpose: To study motion by analysis of graphs of displacement, x, velocity, v, and a vs.
time, t.
Apparatus: There is a track on two supports with two end stops, a magnetic cart, a string
with a wire hook, pulley, string wire hook, 5, 10, 20 g mass piece
PURPOSE
To test Hookes Law of Force, to determine the force constant of a spring, and to investigate
force constants of spring combinations.
APPARATUS
Track on two supports, 2 end-stops, two meter sticks (one taped to the track), 3 springs: two
long and o
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
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 ([email protected]) 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
PURPOSE
To study friction in various situations.
APPARATUS
Track on two supports, two end stops, laboratory jack, motion sensor, pulley, friction
block (block of wood with cloth on two surfaces) collision cart (without piston), roll of
cloth, string with
PURPOSE
To investigate the buoyant force and use it to determine density.
APPARATUS
Force sensor, 50 g mass piece, string, plastic beaker, scale (mass-meter), laboratory jack, table
clamp, long rod, rod connector, short rod, various objects and various li
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
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
PURPOSE
To study rotational inertia and acceleration.
APPARATUS
Base for rotation experiments, smart pulley, string (attached to base), solid plastic disc,
mass pieces (1,2,2,5,10,20,20,200,200,200,200 g pieces),vernier callipers, ruler
THEORY
See manual.
PURPOSE
To investigate the relationship between mass, force and acceleration.
APPARATUS
Track, two end stops, dynamics cart (without spring-loaded piston), motion sensor with
lead, rectangular metal bar, weighing scale, string (with loops at both ends), s
PURPOSE
To investigate the relationship between work and energy
APPARATUS
Track, 2 end stops, cart (i.e. dynamics cart without spring-loaded piston), force sensor,
rectangular metal bar, smart pulley, table clamp, string (with loops at both ends), mass
ha
PURPOSE
To study dynamic phenomena using the concept of momentum
APPARATUS
Track, 2 end stops, 2 motion sensors, two collision carts (with magnetics bumpers), one
dynamics cart (with spring-loaded pistol) rectangular metal bar, and scale, two rolls of
clo
PURPOSE
To investigate projectile motion and the components of motion in two dimensions.
APPARATUS
Mini-launcher, table clamp, two photogates with leads, steel ball, container for ball while
not in use, loading cylinder, laboratory jack, timing pad plus e