Time Independent Perturbation Theory, 1st order correction, 2nd order correction

The presentation is about how to solve the new energy levels and wave functions when the simple Hamiltonian is added by another term due to external effect (can be due to external field) .
The intended reader of this presentation were physics students. The author already assumed that the reader knows dirac braket notation.

1. Time-Independent
Perturbation Theory
Prepared by: James Salveo L. Olarve
Graduate Student


January 19, 2010

2. Introduction
The presentation is about how to solve the new energy
levels and wave functions when the simple Hamiltonian
is added by another term due to external effect (can be
due to external field) .
The intended reader of this presentation were physics
students. The author already assumed that the reader
knows dirac braket notation.

3. ThePerturbHamiltonian
The Hamiltonian of a quantum mechanical system is written
Here, is a simple Hamiltonian whose eigenvalues and
eigenstates are known exactly.
We shall deal only with nondegenerate systems; thus to
each discrete eigenvalue there corresponds one and
only one eigenfunction
And will be the additional term (can be due to
external field)
EE
o
n
o
n
E

4. Task:
To find how these eigenkets and eigenenergies change if a
small term (an external field, for example) is added to the
Hamiltonian, so:
So on adding

5. Assumption:
In perturbation theory we assume that is
sufficiently small that the leading corrections are
the same order of magnitude as itself, and
the true energies can be better and better
approximated by a successive series of
corrections, each of order compared with
the previous one.

6. Strategy:
Expand the true wave function and corresponding eigenenergy as series in
It is more convenient to introduce dimensionless parameter λ
The series expansion
match the two sides term by term in powers of λ (taking λ=1).
Zeroth Term:
First Order Correction:
Second Order Correction:
ooo
nEnEnEnHnH
nnn
2''2''2
++=+

7. Matching the terms linear in on both λ sides (taking λ =1)
Taking the inner product of both sides with
Since it is normalized
The Hamiltonian is a Hermittian Operator
Eigenvalue Equation
1==
oo
oo
nn d
*
oo
HH=
oooo
nEnHn =
First Order Correction

8. First Order Correction
So,

We find first order correction

for energy
1D continuous spectrum
oooooooo
nEnnEnnHnnEn
nnn
''''
+=+
oooooooo
nnEnnEnHnnnE
nnn
''''
+=+
dzHE
n oo
YY=ò
¥
¥-
'*'
''''
nnEnnEnnEnHn
nnn
oooooooo
-+=
oo
nHnE
n
''
=

9. First Order Correction
Solving for the 1
st
order change in the wave function
here
oooooooo
nEmnEmnHmnHm
nn
'
''
''
''
'
+=+
0==
-
ood
oo
nm
nm
oooo
mEmH
m
=
oooooooo
nmEnmEnHmnmE
nnm
''
'
''
+=+
oo
oo
o
mnEE
nHm
nm
-
=
'
'
oo
ooo
oo
mn
EE
nHmm
nnmm
-
==
'
''

10. Second Order Correction
Taking the inner product with yields
has no component in the direction, we can therefore drop the term
Now,
o
n
oooooooo
nEnnEnnEnnHnnHn
nnn
2''2''2
++=+
ooo
nEnEnEnHnH
nnn
2''2''2
++=+
oooooooo
nnEnnEnnEnHnnnE
nnnn
2''2''2
++=+
'
n
o
n
''
nnE
n
o
oo
oo
oo
ooo
oo
mn
nm
mn
nm
n
EE
nHm
EE
nHmm
HnnHnE
-
S=
-
S==
¹¹
2
''
'''2

11. Second Order Correction
ooo
nEnEnEnHnH
nnn
2''2''2
++=+
oooooooo
nEmnEmnEmnHmnHm
nnn
2''2''2
++=+
oooooooo
nmEnmEnmEnHmnmE
nnnm
2''2''2
++=+
Solving the 2
nd
order change for the wave function
22''''
nmEnmEnmEnHm
mnn
oooooo
-=-
oo
oo
o
mn
n
EE
nmEnHm
nm
-
-
=
''''
2
oo
oo
oo
oo
oo
o
o
mnmnmn EE
nHm
EE
nHn
EE
nHm
nm
--
-
-
=
''''
2

12. Second Order Correction
( )
2
'''
2
2
oo
oo
oo
o
o
mn
mn EE
nHm
EE
nHm
nm
-
-
-
=
( )
2
'''
2
2
oo
o
oo
mn
mn EE
nH
EE
nH
n
-
-
-
=

13. Finally,
The Eigenenergy
The Wave Function
...
2'0
+++=
nnnn EEEE
...
2
'
'0
+
-
S++=
¹
oo
oo
oo
mn
nm
nn
EE
nHm
nHnEE
...
2'
+++= nnnn
o
( )
...
2
''''
2
+
-
-
-
+
-
+=
oo
o
oooo
o
o
mn
mnmn EE
nH
EE
nH
EE
nH
nn