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10  1
5.73
Lecture
#10
updated 9/18/02 8:55 AM
Matrix Mechanics
should have read CDTL pages 94121
read CTDL pages 121144 ASAP
Last time:
NumerovCooley Integration of 1D Schr. Eqn. Defined on a Grid.
2sided boundary conditions
nonlinear system  iterate to eigenenergies (NewtonRaphson)
So far focussed on
ψ
(x) and Schr. Eq. as differential equation.
Variety of methods
{E
i
,
ψ
i
(x)}
↔
V(x)
Often we want to evaluate integrals of the form
ψφ
*
() ()
x
x dx
a
ii
=
∫
overlap of special
ψ
on standard
functions {
φ
}
a
is “mixing coefficient”
{
φ
} is complete set of “basis
functions”
OR
φ
i
*
ˆx
n
φ
j
dx
≡
x
n
()
ij
∫
expectation values
transition moments
There are going to be elegant tricks for evaluating these integrals and relating one
integral to others that are already known.
Also “selection” rules for knowing
automatically which integrals are zero:
symmetry, commutation rules
Today: begin matrix mechanics  deal with matrices composed of these integrals 
focus on manipulating these matrices rather than solving a differential
equation  find eigenvalues and eigenvectors of matrices instead
(COMPUTER “DIAGONALIZATION”)
* Perturbation Theory:
tricks to find approximate eigenvalues of
infinite matrices
* WignerEckart Theorem and 3j coefficients: use symmetry to identify and inter
relate values of nonzero integrals
*
Density Matrices:
information about state of system as separate from
measurement operators
H
I
G
H
L
I
G
H
T
S
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View Full Document 10  2
5.73
Lecture
#10
updated 9/18/02 8:55 AM
First Goal:
Dirac notation as convenient NOTATIONAL simplification
It is actually a new abstract picture
(vector spaces) — but we will stress the utility (
ψ
↔

⟩
relationships)
rather than the philosophy!
Find equivalent matrix form of standard
ψ
(x) concepts and methods.
1.
Orthonormality
2.
completeness
ψ
(x) is an arbitrary function
A.
Always possible to expand
ψ
(x) uniquely in a COMPLETE BASIS SET {
φ
}
ψ
i
*
ψ
j
dx
=δ
ij
∫
ψ
(x)
=
a
i
φ
i
i
∑
a
i
=φ
i
*
ψ
dx
∫
mixing coefficient — how to get it?
Always possible to expand
in { } since we can write
in terms of { }.
So simplify the question we are asking to
What are the b
j
Multiply by
j
*
ˆ
ˆ
?
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This note was uploaded on 11/28/2011 for the course CHEM 5.74 taught by Professor Robertfield during the Spring '04 term at MIT.
 Spring '04
 RobertField

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