Physics 517
Homework Set #1
Due in class 10/8/13
Autumn 2013
300 pts
1. (150 pts) Variations on Sakurai, 1.2, 1.9 and 1.11.:
a. Suppose a 2x2 matrix X is written as
X = a0 + a
where
1 =
0
1
1
0
,
2 =
i
0
0
i
,
3 =
1
0
0
1
,
and hence
tr(i ) = 0 ,
tr(i j )
Physics 518
Homework Set #7
Winter 2014
Due in class 3/5/14
300 pts
1. (75 pts) Factoring out the center of mass motion: Denote the coordinate
and momentum operators two particles interacting via a potential V (r1 r2 )
by r1 , r2 and p1 , p2 , and the cor
Physics 517
Homework Set #5
Autumn 2012
Due in class 11/6/12
300 pts
1. (60 pts) Sakurai, 2.7: Consider a free-particle wave packet in one dimension. At t = 0 it satises the mimimum uncertainty relation
(x)2 (p)2 =
h
2
4
(t = 0).
In addition we know
x = p
Physics 517
Homework Set #6
Due in class 11/19/13
Autumn 2013
300 pts
1. (50 pts) Sakurai, 2.12.
a. Write down the wave function (in coordinate space) for the state formally
given by
ipa
|0
exp
h
where p is the momentum operator and a is some number with
Physics 517
Homework Set #5
Due in class 11/12/13
Autumn 2013
300 pts
1. (75 pts) SHO via Shooting Method: We want to numerically nd the
eigenvalues of the SHO Hamiltonian by solving
x2
+ = E
2
2
(1)
where all variables are taken to be unitless and so can
Physics 517
Homework Set #7
Autumn 2013
Due in class 12/3/13
(Note that this is after the Thanksgiving break!)
300 pts
1. (100 pts) SUSY QM: Consider a 1d Schrdinger equation
o
1
+ V (x) = E
2
with potential
V =
a2
a0 (a0 1)
0
+
2
2 cos2 (x)
where x runs
Physics 517
Homework Set #6
Autumn 2013
Due in class 11/19/13
300 pts
1. (50 pts) Sakurai, 2.12;
a. Write down the wave function (in coordinate space) for the state formally
given by
ipa
|0
exp
h
where p is the momentum operator and a is some number with
Physics 517
Homework Set #5
Due in class 11/12/13
Autumn 2013
300 pts
1. (75 pts) SHO via Shooting Method: We want to numerically nd the
eigenvalues of the SHO Hamiltonian by solving
x2
+ = E
2
2
(1)
where all variables are taken to be unitless and so can
Physics 517
Homework Set #7
Due in class 12/3/13
Autumn 2013
(Note that this is after the Thanksgiving break!)
300 pts
1. (100 pts) SUSY QM: Consider a 1d Schrdinger equation
o
1
+ V (x) = E
2
with potential
V =
a2
a0 (a0 1)
0
+
2
2 cos2 (x)
where x runs
Physics 518
Homework Set #6
Winter 2014
Due in class 2/26/14
300 pts
1. (75 pts) Sakurai, 3.20: We are to add angular momenta j1 = 1 and j2 = 1
to form j = 2, 1, and 0 states. Using either the ladder operator method or the
recursion relation, express all
Physics 518
Homework Set #5
Winter 2014
Due in class 2/19/14
300 pts
1. (50 pts) Variation on Sakurai, 3.4: Consider a spin 1 particle. Evaluate
the matrix elements of
Sz (Sz + )(Sz )
h
h
and
Sx (Sx + )(Sx ).
h
h
3
What does your result imply for the expe
Physics 517
Homework Set #1
Due in class 10/8/13
Autumn 2013
300 pts
1. (150 pts) Variations on Sakurai, 1.2, 1.9 and 1.11.:
a. Suppose a 2x2 matrix X is written as
X = a0 + a
where
1 =
0
1
1
0
,
2 =
i
0
0
i
,
3 =
1
0
0
1
,
and hence
tr(i ) = 0 ,
tr(i j )
Physics 518
Homework Set #4
Due in class 2/12/14
Winter 2014
300 pts
1. (100 pts) Derive an explicit form for the general rotation matrix
Rn () = exp(i T ) ,
n
where n is a unit vector giving the axis of rotation, and is the [real] rotation
angle. You can
Physics 517
Homework Set #3
Due in class 10/22/13
Autumn 2013
300 pts
1. (70 pts) Delta functions live under integral signs, and two expressions
D1 (x) and D2 (x) involving delta functiosn are said to be equal if
f (x)D1 (x)dx =
f (x)D2 (x)dx,
for every r
Physics 517
Homework Set #3
Autumn 2013
Due in class 10/22/13
300 pts
1. (70 pts) Delta functions live under integral signs, and two expressions
D1 (x) and D2 (x) involving delta functiosn are said to be equal if
f (x)D1 (x)dx =
f (x)D2 (x)dx,
for every r
Physics 517
Homework Set #4
Autumn 2013
Due in class 11/5/13
300 pts
1. (60 pts) Sakurai, 2.7: Consider a free-particle wave packet in one dimension. At t = 0 it satises the mimimum uncertainty relation
(x)2 (p)2 =
h
2
4
(t = 0).
In addition we know
x = p
Physics 517
Homework Set #2
Due in class 10/15/13
Autumn 2013
300 pts
1. (60 pts) Derivation of Schwarz inequality (Sakurai, 1.18 a and b):
a. First argue that
( | + |) (| + | ) 0
for any complex number . Chose in such a way that this inequality reduces
t
Physics 518
Homework Set #6
Due in class 2/26/14
Winter 2014
300 pts
1. (75 pts) Sakurai, 3.20: We are to add angular momenta j1 = 1 and j2 = 1
to form j = 2, 1, and 0 states. Using either the ladder operator method or the
recursion relation, express all
Physics 518
Homework Set #5
Due in class 2/19/14
Winter 2014
300 pts
1. (50 pts) Sakurai, 3.4: Consider a spin 1 particle. Evaluate the matrix
elements of
Sz (Sz + )(Sz )
h
h
and
Sx (Sx + )(Sx ).
h
h
2. (50 pts) Sakurai 3.12: An angular momentum eigenstat
Physics 518
Homework Set #7
Due in class 3/5/14
Winter 2014
300 pts
1. (75 pts) Factoring out the center of mass motion: Denote the coordinate
and momentum operators two particles interacting via a potential V (r1 r2 )
by r1 , r2 and p1 , p2 , and the cor
Physics 517
Homework Set #2
Due in class 10/15/13
Autumn 2013
300 pts
1. (60 pts) Derivation of Schwarz inequality (Sakurai, 1.18 a and b):
a. First argue that
( | + |) (| + | ) 0
for any complex number . Chose in such a way that this inequality reduces
t