Lecture 1: We covered Shankar 1.1 and 1.2
Some additional remarks: We included the axiom 1 |V
lem below.
= |V . This is needed in the prob-
Show that |0 = 0|V for any vector |V .
Solution: There are many ways of proving such relations. One can assume the
Physics 827
Problem Set 7
Due Friday 11/20/2009
7.1 The student Misty Fide comes to you and says, I am confused by tunneling. Consider the rectangular potential barrier of height V0 between a/2 and a/2. When E < V0
we know that the wave function in the cl
Physics 827
Problem Set 6
Due Friday 11/13/2009
6.1) Consider the two delta-function potential,
V (x) = va (x + a) + va (x a)
where v < 0.
(a) Find the transcendental equation that determines the ground state (Use the parity
of the ground state wave funct
Physics 827
Problem Set 5
Due Friday 11/06/2009
The rst two problems almost made it to the midterm
h
5.1 Consider the state eip0 X/ |p where |p is an eigenstate of the momentum operator P , X is the position operator and p0 is real. Show that this is also
Physics 827
Problem Set 4
Due Friday 10/23/2009
4.1. A system is described by the Hamiltonian H and we consider an observable
denoted by Sy . The two operators are given by
H=
0
0
and Sy =
h
2
0 i
i0
.
(a) What is the dimensionality of the Hilbert space t
Problem Set 3 Solutions
3.1 (a) The mass of a baseball is roughly 0.14 kg .1 Given the speed of 40m/s2 we
obtain for the de Broglie wavelength
=
h
6.62 1034 J.s
=
1.2 1034 m .
mv
0.14kg 40m/s
Please do not just give the answer as a number without giving
FALL 2009
PHYSICS 827
HW2 Solutions
2.1) (7 points) Find the eigenvalues and eigenvectors of
5 12
43
.
Are the eigenvectors orthogonal? Find a row vector that is orthogonal to the eigenvector corresponding to the largest eigenvalue. Can you nd any interes
PHYSICS 827 HW1
Due Wednesday 09/30 by 4:59PM in Mr. Nick Harmons mailbox
1.1 Shankar Exercise 1.3.4 (p17) (10 points)
Solution: We have
|V + W |2 =
V + W |V + W
(1)
=
V |V + W |W + V |W + W |V
(2)
=
V |V + W |W + 2 [ V |W ]
(3)
| V |W |2 = ( [ V |W ])2 +
Lecture 3: We covered Shankar 1.6 ( page 26) and 1.9
We check that () = . Let
|V
= |V
and |V
= |V
.
By the denition of adjoint operators we have
V | = V | and
V | = V | .
Thus we obtain
V | = V | = V | .
We compare this with
|V
= |V
= ()|V
and its adjoint
Physics 827
Problem Set 8
Due Monday 11/30/2009
8.1 Shankar 10.2.1 (page 258) and 10.2.3 (page 260) (5+3 points) In 10.2.3 (2)
just re-express the rst three states in terms of spherical polar coordinates.
The degeneracy question needs a simple combinatori