Intermediate Quantum Mechanics
Spring 2017
Homework 5
1) An electron in the state of spin-up in the x-direction | i is placed in a region with a
uniform magnetic field, B, in the z-direction. After a period of time equal to T /8 = /4
where = eB/m:
a) what
Intermediate Quantum Mechanics
Spring 2017
Solutions to Homework 8
1) Calculate the following commutators:
x, L
y]
[L
+, L
z]
[L
z]
[L2 , L
, L
z]
[L
+, L
]
[L
Solution
x = ypz zpy
L
y = zpx xpz
L
[
xi , xj ] = 0
+ = L
x + iL
y
L
[
pi , pj ] = 0
Intermediate Quantum Mechanics
Spring 2017
Solutions to Homework 3
1) Show that the vector specifying the state of an electron that is spin up in the n
direction is given by:
!
cos 2
|n i =
sin 2
where
nx = sin
ny = 0
nz = cos .
Hint: you might want to u
Intermediate Quantum Mechanics
Solutions to Homework 4
1) A particle is in the state |i with probability distribution in x given by the Gaussian:
1/2
1
2
2
(x)(x) =
ex /2x
2
2x
where (x) is the representation of |i in the position, x, basis.
a) Find an e
Intermediate Quantum Mechanics
Spring 2017
Homework 6
1) If the initial wave packet has a small enough , then we can have:
1
(t)
=
4
h
m
> c
This would seem to violate relativity since initially the probability of finding the particle say
a lightyear away
Intermediate Quantum Mechanics
Spring 2016
Solutions Homework 8
1) In class we discussed expressions for the ground state energy and the Bohr radius of the
hydrogen atom in terms of the constants: h
, c, and m. We can form muonic hydrogen by
binding a mu
Intermediate Quantum Mechanics
Spring 2017
Solutions to Homework 1
1) For the following objects at room temperature, T 300 K, determine the approximate
distance scale at which effects due to quantum mechanics would become manifest.
a soccer ball: m 0.1 k
Intermediate Quantum Mechanics
Spring 2016
Solutions to Homework 2
1) Prove that the trace of an operator, the sum of the diagonal elements of the matrix,
is independent of the basis chosen to represent the operator. In other words, for the
unprimed basis
Intermediate Quantum Mechanics
Spring 2017
Solutions Homework 10
1) For the spin system of two electrons we have for the total angular momentum:
Jz = S1z + S2z
J2 =
J+ = S1+ + S2+
J = S1 + S2
S1 + S2 S1 + S2 = Jz2 + h
Jz + J+ J
h
h
i
and
S2i = i
S1i =
2
Intermediate Quantum Mechanics
Spring 2017
Solutions Homework 7
a
a
1) Evaluate the commutation relations [H,
] and [H,
].
Solution
a
[H,
] = [(
h
a a
+h
/2), a
] = h
[
a a
, a
] = h
(
a a
a
a
a
a
)
h
h
i
i
= h
a
a
a
[
a, a
] + a
a
a
= h
19. If the potential energy of a particle is given by U = x2 - 4xy, find the
magnitude and direction of the force on the particle when it is located at the
point (2, 5).
0 Please explain your reasoning clearly - it is your responsibility to convince me that you
understand the material.
- Points will be deducted for incorrect and/or incomplete reasoning, or for evidence of
misconceptions, even if your nal answer is correct
0 Please explain your reasoning clearly it is your responsibility to convince me that you
understand the material.
0 Points will be deducted for incorrect and/or incomplete reasoning, or for evidence of
misconceptions, even if your nal answer is correct.
A constant force F is applied to the wheel shown below through the rope. The wheel
rolls without slipping on the horizontal surface [which is not possible without
friction). The wheel has a mass M, radius R and rotational inertia (1/3]MR2 about its
axis o
- Please show all work
. Please box your nal answers
0 Points will be deducted for incorrect working or misconceptions even if
nal answer is correct
A solid sphere of radius r and mass In (rotational inertia 2/5 MR2) rotating with
angular velocity we is p
0 Please explain your reasoning clearly it is your responsibility to
convince me that you understand the material.
. Points will be deducted for incorrect and / or incomplete reasoning, or
for evidence of misconceptions. even if your nal answer is correct
Exam
Name_
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
1) A uniform sign is supported against a wall at point P as shown in the figure. If the sign
1)
is a square 1.20 m on a side and its mass is
2015 Final Exam
Name_
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
1) If the mass of the earth and all objects on it were suddenly doubled, but the size remained the
same, the acceleration due to g
Exam
Name_
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
^
^
1) Object A has a position as a function of time given by r A(t) = (3.00 m/s)t i + (1.00 m/s2 )t2 j. Object B
1)
^
^
has a position as a
Exam
Name_
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
1) In the figure, point P is at rest when it is on the x-axis. The linear speed of point P
1)
when it reaches the y-axis is closest to
A) 0.7
Test 2
Name_
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
1) In order to get an object moving, you must push harder on it than it pushes back on you.
A) True
B) False
1)
2) A block is on a friction
Exam
Name_
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
1) The acceleration of an object as a function of time is given by a(t) = (3.00 m/s3 )t, where t is in
seconds. If the object is at rest at t
Exam
Name_
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
1) A uniform solid sphere is rolling without slipping along a horizontal surface with a speed of 6.50
m/s when it starts up a ramp that makes