Homework Assignment 1 Updated
Physics 501
Due September 20, 2010 by noon
1. Prove the following two Dirac delta function identities.
(ax) = (x)/a
(f (x) =
i
(x ai )
df /dxx=ai
f (x) (x) dx = f (0)
2. As we saw in class, the delta function is dened
Lecture 26
The Hydrogen Atom
December 8, 2010
Lecture 26
Review of Spherically Symmetric Potentials
"
#
2
h
2m
2
+ V (r ) E Elm(r, , ) = 0
Elm(r, , ) = REl (r )Ylm(, )
"
2
h 1d
2m r 2 dr
r
2
d
dr
+
REl (r ) =
Let
"
2
#
2
l(l + 1)
h
+ V (r ) E REl (r ) = 0
Lecture 25
Spherical Potential Well
December 6, 2010
Lecture 25
Review: Spherical Harmonics
V (r )
For spherically symmetric potential:
m
r Elm
= Elm(r, , ) = REl (r )Yl (, )
m
Yl (, ) =
lm
1/2
1 (2l + 1)!(l + m)!
m
m
im
m
Yl (, ) = (1) l
e (sin )
2 l!
Lecture 22
Rotations in Thrree Dimensions
November 22, 2010
Lecture 22
Rotations in Three Dimensions
Now we need to consider rotations
about three axes: x, y and z .
Just as Lz generates rotations about the z axis, Lx
and Ly will generate rotations about
Lecture 21
Rotations in Two Dimensions
November 17, 2010
Lecture 21
Spatial Translations in Two Dimensions
px generates translations in the x direction
py generates translations in the y direction
Translate in y and in x.
The order doesnt matter since [x,
Lecture 20
Parity and Time Reversal
November 15, 2009
Lecture 20
Time Translation
 (t + ) = U [T ( )]  (t) =
H
i
I H  (t)
h
is the generator of time translations.
i
U [T ( )] = I H
h
[H, H ] = 0
also
[H, H ] = 0
i
h
i
h
time translational invariance
H
Lecture 19
Transformations
November 10, 2010
Lecture 19
Transformations
We are now going to study transformations and
their relationship with symmetries, invariance of the
equation of motion under the transformation and
conservation laws.
There are two ca
Lecture 18
The Harmonic Oscillator Revisited
November 8, 2010
Lecture 18
Harmonic Oscillator Again
Review of wave function solution
i (x, t) =
h
t
1
h
2 2
+ m 2x2 (x, t)
2m x2
2
energy eigenstates
2 2
h
1
+ m 2x2 E
2m x2
2
uE (x) = 0
En = h (n + 1/2)
uE
Homework Assignment 6
Physics 501
Due Nov 11, 2010 by 3 pm
1. a) Solve the radial timeindependent Schrodinger equation for the twodimensional isotropic
harmonic oscillator.
h
2
2m
1d
d2
+
2
d
d
+
1
m2 2
lh
+ m 2 2 E REml () = 0
2
2m
2
[Hint: Find the a
Homework Assignment 5
Physics 501
Due October 28, 2010 by 3 pm
1. Consider a particle of mass m in an innite onedimensional potential well V (x) given by
V =0
for
x a
V =
for
x > a
a) Find x for the nth energy eigenstate.
b) Show that as n this becom
Homework Assignment 4
Physics 501
Due October 18, 2010
1. Consider the vector space of complexvalued functions of one real variable,  with x =
(x). Show that the normalization  is the same in both the xbasis and in the k basis.
That is, show that

Homework Assignment 3
Physics 501
Due October 11, 2010
1. Starting with the set of basis vectors
3
I = 0
0
0
II = 1
2
0
III = 2
5
nd an orthonormal basis using the GramSchmidt procedure.
1
2
2. a) Prove that, for an n dimensional space, the set of all
Revised Homework Assignment 2
Physics 501
Due September 30, 2010
1. For the state specied by:
show that
2
x2 = x + a2
x =a
p = i
h
and using the momentum operator
show that
1/4 (xa)2 /4 2
1
x
e
2 2
(x) =
p2 =
p =0
d
dx
h
2
2
4x
2. Show that the freepart
Lecture 27
More on the Hydrogen Atom
December 13, 2010
Lecture 27
Size of the Hydrogen Atom
general form of energy eigenstate wave function
unlm(r, , ) = Rnl (r )Ylm(, )
unlm(r, , )
r
na0
l "nX1
l
ak
k=0
r
na0
k #
e
for given n:
Ylm(, )
lmax = n 1
for l