Brief introduction to perturbation theory

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

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

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(

nE

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.

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.

=

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

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 hnmeEH

2

1

0

1

4

n

EE H

2

2

0

2

4

n

EE H

Since,

Hence,

0

2

0

1

0 EEE

2

2

2

1

0 114

nnEE 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.

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

20

r

eHH

dT

dTH

E0*0

0*0 ˆ

dT

dTr

eH

E0*0

0

12

20*0

dT

dTr

e

EE0*0

0

12

2*0

0

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 8.74348.1080

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 5.770

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