5.73 Lecture #10
10 - 1
Matrix Mechanics
should have read CDTL pages 94-121
read CTDL pages 121-144 ASAP
Last time:
Numerov-Cooley Integration of 1-D Schr. Eqn. Defined on a Grid.
2-sided boundary conditions
nonlinear system - iterate to eigenenergies (Newton-Raphson)
So far focussed on ψ(x) and Schr. Eq. as differential equation.
{Ei, ψi(x)} ↔ V(x)
Variety of methods
Often we want to evaluate integrals of the form
overlap of special
ψ on standard
functions {φ}
expectation values
transition moments
∫
ψ * ( x) φi ( x) dx = ai
OR
a is “mixing coefficient”
{φ} is complete set of “basis
functions”
∫ φ*i x̂nφ jdx ≡ (xn )ij
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”)
H
I
G * Perturbation Theory: tricks to find approximate eigenvalues of infinite matrices
H
L * Wigner-Eckart Theorem and 3-j coefficients: use symmetry to identify and interrelate values of nonzero integrals
I
G
H * Density Matrices: information about state of system as separate from
measurement operators
T
S
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5.73 Lecture #10
First Goal:
10 - 2
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
∫ ψ*i ψ j dx = δij
2. completeness
ψ(x) is an arbitrary function
(expand ψ)
A. Always possible to expand ψ(x) uniquely in a COMPLETE BASIS SET {φ}
ψ(x) =
∑ a iφi (x)
i
*
(expand Bψ)
B.
mixing coefficient — how to get it?
*
left multiply by φ j
∫
a i = ∫ φ*i ψdx
ˆ ψ in {φ} since we can write ψ in terms of {φ}.
Always possible to expand B
ˆ φ = ∑ b φ What are the b ? Multiply by ∫ φ*
So simplify the question we are asking to B
j
j
i
j j
j
{ }
b j = ∫ φ*j B̂φ i dx ≡ B ji
B̂φ i = ∑ B ji φ j
j
note counter-intuitive pattern of
indices. We will return to this.
* The effect of any operator on ψi is to give a linear combination of ψj’s.
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5.73 Lecture #10
10 - 3
3. Products of Operators
(ÂB̂)φi = Â(B̂φi ) = Â∑ Bjiφ j
j
can move numbers (but not operators) around freely
= ∑ B ji Âφ j = ∑ ∑ B ji A kjφ k
j
note repeated index
j k
(
)
= ∑ A kjB ji φ k = ∑ (AB)ki φ k
j,k
k
* Thus product of 2 operators follows the rules of matrix multiplication:
ÂB̂ acts like A B
Recall rules for matrix multiplication:




 

 indices are
A
 row,column
must match # of columns on left to # of rows on right
order
matters!
(N × N )
(1 × N )
⊗
⊗
" row vector"
( N × 1)
column
vector
(N × N )
( N × 1)
→
→
(N × N )
(1 × 1)
a matrix
a number
→
(N × N )
a matrix!
" column vector"
⊗
(1 × N )
row
vector
Need a notation that accomplishes all of this memorably and compactly.
updated 9/18/02 8:55 AM
5.73 Lecture #10
10 - 4
Dirac’s bra and ket notation
Heisenberg’s matrix mechanics
ket
 a1 
a
is a column matrix, i.e. a vector  2 
 M 
 aN 
contains all of the “mixing coefficients” for ψ expressed in some basis set.
[These are projections onto unit vectors in N-dimensional vector space.]
Must be clear what state is being expanded in what basis
[i
ai
]
ψ(x) = ∑ ∫ φ*i ψdx φ i (x)
 ∫ φ1*ψdx 
* ψ expressed in φ basis
 *

* a column of complex #s
φ
ψdx
ψ =∫ 2

* nothing here is a function of x
M


 φ* ψdx
∫ N
 φ ← bookkeeping device (RARE)
OR, a pure state in its own basis
 0
 1
φ 2 =  0
one 1, all others 0
 M
 0 φ
bra
is a row matrix
( b 1,b 2 … b N )
{ }
contains all mixing coefficients for ψ* in φ* basis set
ψ * (x ) =
∑[∫ φiψ * dx ]φ*i (x )
(This is * of ψ (x) above)
i
The * stuff is needed to make sure ψ ψ = 1 even though φ i ψ is complex.
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5.73 Lecture #10
10 - 5
The symbol ⟨a|b⟩, a bra–ket, is defined in the sense of product of
(1 × N) ⊗ (N × 1) matrices → a 1 × 1 matrix: a number!
Box Normalization in both ψ and ⟨ | ⟩ pictures
1 = ∫ ψ * ψdx
(
i
)
ψ = ∑ ∫ φ*i ψdx φ i
(
expand both in
φ basis
)
ψ* = ∑ φ jψ * dx φ*j
j
∫
1 = ψ * ψdx =
i, j
∑ ∫ φ*j ψdx
c.c.
2
δij
p
1=
∑  ∫ φj ψ * dx ( ∫ φ*i ψdx )∫ φ*j φidx
forces 2 sums to
collapse into 1
j
real, positive #’s
We have proved that sum of |mixing coefficients|2 = 1. These are called
“mixing fractions” or “fractional character”.
now in
picture


 *

φ
ψ
dx


 1


ψ ψ =  φ1ψ * dx
φ2 ψ * dx L   φ*2 ψdx 
 144444424444443  

M
row vector: “bra”

 1
424
3
column


 vector “ket” 
∫
=
∑∫
∫
∫
∫
2
φ*j ψdx
same result
j
[CTDL talks about dual vector spaces — best to walk before you run. Always
translate ⟨ ⟩ into ψ picture to be sure you understand the notation.]
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5.73 Lecture #10
10 - 6
Any symbol ⟨ ⟩ is a complex number.
Any symbol | ⟩ ⟨ | is a square matrix.
again ψ ψ = ( ψ φ1
 φ1 ψ 
ψ φ2 …) φ2 ψ 


M


= ∑ ψ φi φi ψ = ψ ψ = 1
i
1 unit matrix
 1
1 0 L
 0 

what is φ1 φ1 = (1 0 … 0)  =  0 0

 M 
M
O


 
 0
1

 1 0

what is ∑ φ i φ i = 

0
1


i


O

a shorthand for
specifying only
the important
part of an
infinite matrix
unit or identity
matrix = 1
“completeness” or “closure” involves insertion of 1 between any two symbols.
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5.73 Lecture #10
10 - 7
Use 1 to evaluate matrix elements of product of 2 operators, AB (we know how
to do this in ψ picture).
square matrix
i-th
 0
 :
 
φi A φ j = (0 …1… 0)(A) 1
j-th position – picks
 :
out j-th column of A
 
i-th
 0
 A1 j 
= (0 …1… 0) A 2 j  = A ij


 M 
φi AB φ j = ∑ φi A φ k φ k B φ j
k
1
= ∑ A ik Bkj = (AB)ij a number
k
In Heisenberg picture, how do we get exact equivalent of ψ(x)?
basis set δ(x,x0) for all x0 – this is a complete basis (eigenbasis
for ^
x , eigenvalue x0) - perfect localization at any x0
⟨x ψ ⟩ is the same thing as ψ(x)
↑
 i.e.,

∫
δ ( x, x′ ) * ψ ( x′ ) dx′ = ψ ( x)
x is continuously variable ↔ δ(x)
overlap of state vector ψ with δ(x) – a complex number. ψ(x) is a complex
function of a real variable.
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5.73 Lecture #10
10 - 8
other ψ ↔ ⟨  ⟩ relationships
1. All observable quantities are represented by a Hemitian operator (Why –
because expectation values are always real). Definition of Hermitian operator.
A ij = A*ji
A = A†
or
2. Change of basis set
Aφ ↔ A u
Easy to prove that if all
expectation values of A
are real, then A = A† and
vice-versa
{φ} to {u}
Aijφ ≡ φi A φ j = φi 1A1 φ j
=
∑φ
k,l
uk u k A u l ul φ j
i
S*ki ≡ S†ik
↑
u
1st index is u
=
∑
Slj
← φ S is Frequently used
to denote “overlap”
↑
integral
u
j-th column of
S
φ
(
)
Sik≤ AuklSlj = S ≤ Au S
k,l
φ
A = S ≤ Au S
φ
ij
≡ Aij
a special kind of
transformation (unitary)
(different from usual
T–1AT “similarity”
transformation)
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5.73 Lecture #10
10 - 9
What kind of matrix is S?
Sl j = u l |φ j
[
]
Sl* j = u l |φ j * = φ j |u l ≡ Sj≤l
≤ means take complex conjugate and interchange indices.
Using the definitions of S and S ≤ :
∑
SljS†jk = u l | φj φj | u k
SljS†jk =
j
∑
u l | φj φj | u k = u l | 1 | u k
j
= u l | u k = δ lk = 1 lk
∴ SS† = 1
OR
S† = S–1 “Unitary”
a very special and
convenient property.
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5.73 Lecture #10
10 - 10
Unitary transformations preserve both normalization and orthogonality.
Aφ = S†A uS
SAφS† = SS†A uSS† = A u
A u = SAφS†
Take matrix element of both sides of equation:
A uij = u i | A | u j = (SAφS† ) ij
=
∑
S ik φ k | A | φl Sl† j
k ,l
∴ uj =
∑
φ l S l≤j
j - th column of S ≤
l
φ → u via S† ,S : u j is j- th column of S†
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5.73 Lecture #10
10 - 11
Thus,
j-th
uj
≤
 0  S1 j 
 M  S≤ 
   2j 
=  1 =  M 
 M  M 
   ≤ 
 0  Snj 
φ
Alternatively,
A φpq = φ p|A|φq = (S ≤ Au S) pq
=
∑
≤
Spm
um |A|u n Snq
mn
∴ φq =
∑
u n Snq
q - th column of S
n
φq
q-th
 0
 S1q 
 M
S 
 
2q
=  1 =  
 M 
 M
S 
 
 nq  u
 0 φ
updated 9/18/02 8:55 AM
5.73 Lecture #10
10 - 12
Commutation Rules
*
[Â, B̂] = ÂB̂ − B̂Â
e.g. [ x̂, p̂] = ih
means ( xp - px )φ = x
h dφ h 
dφ
− φ+x 

i dx i
dx 
= ihφ
*
If  and B̂ are Hermitian, is ÂB̂ Hermitian?
Hermitian A and B
( AB)ij = ∑ A ik Bkj = ∑ A*ki B*jk = ∑ B*jk A*ki = (BA )*ji
k
but this is not what we need to say AB is Hermitian: That would be:
(AB)ij = (AB)*ji
AB is Hermitian only if [A,B] = 0
However,
1
[AB + BA] is Hermitian if A and B are Hermitian.
2
This is the foolproof way to construct a new Hermitian operator out of
simpler Hermitian operators.
Standard prescription for the Correspondence Principle.
updated 9/18/02 8:55 AM