Lecture 11 - the WKB method
Any wave function in coordinate space (1-dim) can be written as
(x) = A(x)eiS (x)/ .
When plugged into Schrdinger equation, we found the two following relations
o
1 2S
1 A S
A
A=
m x x
2m x2
2
1 2 2A
1
S
S=
V
A 2m x2
2m x
(1)
Graduate Quantum II
Multi-Fermion Symmetry
Spring 2013
1
Two Particle State
Bose-Einstein statistics: completely symmetric under the exchange of two particles
Fermi-Dirac statistics: () minus sign under exchange of any two particles
Subset of possible two
One Dimensional Solid (Crystal)
Lecture Notes 2/5/2013
Consider N atoms in a row.To make life easier we make following approximation:
a)some electrons are bound to the nucleus and some are free. In particular,
say we have q electrons per atom are free, i.
Graduate Quantum II
Identical Particles
Spring 2013
1
Identical Particles
Before, we discussed entanglement where two particles were created in a well-dened total spin state. Now
we need to address particles that were always entangled (how created doesnt
Additional Literature
Introduction to Quantum Mechanics, David J. Griths (Pearson) (Undergrad text)
Dance of the Photons, Anton Zeilinger (Farrar, Straus and Giroux) (Theoretically a General Audience book)
Quantum Mechanics by D.H. McIntyre (Pearson) (Und
Quantum Mechanics II Lecture 2
January 24, 2013
Density Matrix
Assume we have an ensemble of Ntot particles, each of which can be in
one of n states. Then the density matrix of that system can be expressed in
one of two ways:
N
1 tot
=
|i i |
(1)
Ntot i=1
Lecture Notes: 22/01/2013
Density Matrix:
If Ni particles are in the state |i then the density matrix for such mixed
state is,
pi |i i |
=
and pi =
Ni
,
N
i
pi = 1
The polarization is dened as,
Pj = i pi i |j |i , j = 0, 1, 2, 3. We always have P0 = 1 and
Tensor Operators
Vectors and tensors are defined by how they transform under rotation
Classical case: Momentum for example -
= (! , ! , ! )
Under the rotation , each component transforms:
!
! =
!" !
!
:
rotational
M
Classical Limits
We want to explore the classical limits of Quantum Mechanics (when and it what sense does a QM
system resemble classical counterparts). In particular, we want to discuss when and how QM systems
exhibi
Variational Method for the Helium Ground State
The Hamiltonian for Helium is given by
H=
1
2
P 2 + P2
2m 1
Z e2
1
1
+
r1
r2
+
e2
,
r12
(1)
where r12 = |~1 ~2 |. To obtain an estimate function for the ground state
r
r
consider neglecting the r12 . The rema
Quantum Mechanics Lectures
1
Time Independent Perturbation Theory
We start with the Hamiltonian H = H0 + Hp where we assume that we know the
solution of the Schrdinger equation for the unperturbed Hamiltonian H0
o
H0 |n = En |n
We assume that the perturbe
Connecting Path Integral Formalism with Schrodinger Formalism
March 5, 2013
The full time dependent Schrodinger equation is
i
2 2
(x, t) + V (x)(x, t)
(x, t) =
t
2m x2
2
P
(As a brief aside, the shortest notation for the Hamiltonian in x-space is x| H =
2/21/2013
The same Hamiltonian is used in both CM and QM. Only difference in QM is that the Hamiltonian is
converted to an operator.
|( ) = !"#/ |( = 0)
For a very short time (infinitesimal) period t, the exponential
Lecture 10
From last discussion, QM has both wave and uid aspects, is there a connection
betwen these 2?
Consider a state in the r representation (plane wave) to be
(r) = A(r)eiS (r)/ ,
(1)
were A(r) = A and S (r) = S are taken to be real functions. Cons
Quantum Mechanics II Lecture 1
January 17, 2013
State Vectors
As we learned last semester, the state of a particle in a system is represented by a state vector | in some Hilbert space. For each observable
associated with the system there is a hermitian op