Brief introduction to perturbation theory

2. IO
:
Perturbation theory is an extremely important
method of seeing how a quantum system will be
affected by a small change in the potential. It
allows us to get good approximations for
system where the Eigen values cannot be easily
determined. In real life not many Hamiltonians
are easily solvable. Most of the real life
situations require some approximation methods
to solve their Hamiltonians. Perturbation
theory is one of them.
INTRODUCTION

3. SUBJECT DESCRIPTION
Perturbation means “small disturbance”. The
Hamiltonian of a system is nothing but the total
energy of that system. Some external factors can
also affect the energy of a system and its behavior.
To analyze a system’s energy in a case when we
don’t know the exact way of solution then we can
study the effect of external factor (perturbation)
on the Hamiltonian. Perturbation can be applied to
following two types of systems:
Time Dependent
Time Independent






1
2

4. The perturbation treatment of
degenerate & non degenerate energy
level differs. When application of
perturbation is restricted to non
degenerate energy levels then it is
known as Non Degenerate Perturbation
Theory. It is represented as

Particularly represents unperturbed non
degenerate level with energy

NON DEGENERATE PERTURBATION
THEORY )0(
n
 )0(
n
E

5. We have to split the Hamiltonian into two parts. One is a
Hamiltonian whose solution we know exactly and the other part
is the perturbation term. For example we can use the
Hamiltonian of Hydrogen atom to solve the problem of helium .
Ĥ and Ψ have to be very compatible to each other in following
equation:
ĤΨ = EΨ
A specific ψ will have specific value of Hamiltonian operator.
When difference is not very large we apply it as such.Later we
introduce modification using perturbation method.
APPLICATION OF PERTURBATION
THEORY TO HELIUM ATOM
Now we consider the Helium atom. According to
SCHRӦDINGER EQUATION.

6. ᴪ = Eᴪ
+2e
r1
-e
-e
(x1, y1,z1)

(x2,y2,z2 )
r12
r2
In this approximation where we have used incorrect Hamiltonian and
incorrect Ψ by considering that the repulsive energy term is operating
between two energy terms is insignificant and can be ignored. This
approximation is zero order perturbation. On solving the Schrӧdinger
equation by this approach we will realize that the solution can be generated
by breaking the original equation into two parts where each part is
equivalent to the equation that we obtained in case of Hydrogen like atom.

7. =
In this case,



………Eq(1)
=

………Eq(2) E 22
1
420
1 /8 hnmeE 
Equation (1) and (2) are same as those obtained for Hydrogen
like atom where Z= 2.Since this kind of equations have already
being solved and we know the Eigen energy value for such a
system is obtained as: 22
2
420
2
/8 hnmeE 

8. n1,2 = 1,2 ,3………
We also know that energy of H – like atom in ground state
is


i.e ; n = 1
Therefore, the above equations can be written as

&




2242
/2 hnmeE
H
 2
1
0
1
4
n
E
E
H 2
2
0
2
4
n
E
E
H


9. Since,

Hence,
0
2
0
1
0
EEE  






2
2
2
1
0 11
4
nn
EE
H
If suppose Helium atom is present in ground state
then




eV8.1086.13*8 
This is much lower than actual energy which is -79 eV.

10. In the first order approach



We will now adjust the Hamiltonian by inserting
the repulsion energy term and making
it(Hamiltonian) actual. But Ψ would still be as
ᴪᵒ, the correct wave function cannot be
guessed. In first order perturbation it is
assumed that term is small enough and it may
be taken as a minor modification or
perturbation of Hamiltonian Thus in the
evaluation of energy of Helium correct
Hamiltonian with incorrect wave function will
be implemented.

FIRST ORDER PERTURBATION 






12
2
0
r
e
HH

11. 








dT
dTH
E
0*0
0*0ˆ 














dT
dT
r
e
H
E
0*0
0
12
2
0*0 dT
dT
r
e
EE
0*0
0
12
2
*0
0



















12. The energy so obtained is called perturbation
energy.
The Perturbation energy gives average
potential energy of repulsion between
electron 1 and 2 over space.
The value of perturbation energy for Helium
atom in ground state on calculation comes
out to be 34eV. (n1 = n2 = 1)
Therefore,

This is now comparatively closer to actual
energy that is -79eV.
eVE 8.74348.108
0


13. Now,

If we use any well behaved approximate wave
function in evaluating the value of
expectation energy then the calculated
value of energy will always be algebraically
equal to or greater than the actual energy.
By the application of variation theorem we
obtain,

Which is very close to the actual value.
)( heoremVariationTEE
actualcal
 eVE 5.77
0


14. Principles of Physical Chemistry – Puri,
Sharma & Pathania
Quantum Chemistry – Levine
Quantum Chemistry – R.K Prasad
Atkin’s Physical Chemistry – Peter
Atkin & Julio De Paula

REFERENCES