Physics 125a Midterm Solutions Due November 4, 2015
Problem 1
A quantum mechanical observable is represented by the Hermitian operator O. It has an orthonormal basis of eigenstates | , where O| = | . Suppose a system is prepared in a state
| =
|1 + | 1 +
Physics 125a Final Due Friday December 9, 2016
Instructions
You have up to three hours to do the exam once you start. You can use your notes, my notes on
the web, your problem sets (and solutions) and the text Shankar. Mathematica will not be useful
and s
Physics 125a
Problem Set 3 Solutions
Problem 1
q
~
Using X = 2m
(a + a ) and P = i m~
(a a), along with a|ni = n|n 1i,
2
a |ni = n + 1|n + 1i, we see that
r
r
~
~
hXi = hn|X|ni =
hn|(a + a )|ni =
nhn|n 1i + n + 1hn|n + 1i = 0, (1)
2m
2m
By orthogonality
Ph125: Problem Set 1 Solutions
October 13, 2016
Problem 1
Note that,
|3i =
2 2
0 2
=2
0 0
1 0
+
2
2
2 2
= 2|1i + |2i .
Therefore, vector |3i is a linear combination of vectors |1i and |2i. This indicates that |1i, |2i and
|3i are not linearly independent.
Ph125a Homework 2 Solutions
Fall 2016
Problem 1
0 i are orthogonal states with unit norm, representing the particles K 0 and K
0 (these
|K 0 i and |K
particles are called neutral kaons). We approximate the dynamics of the two kaons as a two-level
system
0v
1; 034.5
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. ,mmmc
/ 0
0
W , L .
cfw_.M b .r. b
. . E
Q l . a K ,.
A , (a a _ c n _ w U. .
_._m In. , , _ /
AN. . , ,4 .
,Z_ A A, .w W: U _L :9 w
._ _ _ A m . m . _ . _
- 5 u H m I w m
lllllillllllllllilllllllll ll I hbo
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en
cfw
Physics 125a Midterm Due November 2, 2016
Instructions
You have up to three hours to do the exam once you start. You can use your notes, my notes on
the web, your problem sets (and solutions) and the text Shankar. Mathematica will not be useful
and so is
Physics 125a
Problem Set 3, Due Wed. Nov 26, 2016
Problem 1
Find hXi, hP i, P and X for a particle of mass m in a one dimensional
harmonic oscillator, V (X) = (1/2)m 2 X 2 , in energy eigenstate |ni. (In this
class our particles are always non relativisti
Physics 125a
Problem Set 5, Due Wed. Nov 23, 2016
Problem 1
(a) Two identical bosons are found to be in the states |i and i. Write
down the normalized state vector describing the system when h|i 6=
0.
(b) A particle moves in a potential V (x) = V0 sin(2x/
Physics 125a
Problem Set 3, Due Wed. Oct. 21, 2015
Problem 1
We are given the Hamiltonian H = g = g1 1 + g2 2 + g3 3 . The basis vectors of the Hilbert Space are |1 and
|2 which are eigenstates of 3 with eigenvalues +1 and 1 respectively.
(a) We seek all
Ph125: Problem Set 5 Solutions
Chia-Hsien Shen
November 21, 2015
Problem 1
(a) The system is in the ground state of old SHO. The ground state is annihilated by its annihilation operator a,
a|0old = 0.
(1)
However, it is not true if we replace a by the new
Lecture 15. Canonical Transformations
(Dated: November 17, 2015.)
I.
REVIEW FROM LAST TIME AND SUMMARY OF THIS LECTURE
Last time, we discussed:
1. Examples of converting the Lagrangian into Hamiltonian dynamics.
2. Liouvilles Theorem: Phase Space Volume i
Physics 125a
Problem Set 4, Due Wed. Nov. 11, 2015
Problem 1
Using X =
that
2m (a + a ) and P =
X = n|X|n =
2m
m
i
(a
2
a), along with a|n =
n|(a + a )|n =
2m
n|n 1 , a |n =
n n|n 1 +
n + 1 n|n + 1
n + 1|n + 1 , we see
= 0,
(1)
By orthogonality of |n st
Physics 125a
Problem Set 2, Due Wed. Oct. 14, 2015
Problem 1
|K 0 and |K 0 are two orthogonal states with unit norm that correspond to the particles K 0 and K 0 . These
particles decay into other particles so if we approximate their physics as a two state
Physics 125a
Problem Set 6, Due Wed. Nov 25, 2015
Problem 1
(a) Two identical bosons are found to be in the states | and . Write
down the normalized state vector describing the system when | =
0.
(b) A particle moves in a potential V (x) = V0 sin(2x/a). T
Physics 125a
Problem Set 4, Due Wed. Nov 9, 2015
Problem 1
A particle of mass m moves in one dimension under the influence of a harmonic oscillator potential
H=
P2
m 2 2
+
X
2m
2
(1)
The particle is in the ground state. Suddenly, at time t = 0 the value o
Physics 125a
Problem Set 2, Due Wed. Oct. 12, 2016
Problem 1
|K 0 i and |K0 i are two orthogonal states with unit norm that correspond
to the particles K 0 and K0 . These particles decay into other particles so if
we approximate their physics as a two sta
Physics 125a - Problem Set 1 - due October 5, 2016.
Problem 1 (5 points)
Consider three elements from the vector space (over the real numbers) of real 2 2 matrices,
0 0
2
2
2 2
, |2i =
, |3i =
(1)
|1i =
1 0
2 2
0 2
Are they linearly independent? Justify y