Perturbation

The postulates of quantum mechanics have been successfully used for deriving exact solutions to Schrodinger equation for problems like  A particle in 1 Dimensional box  Harmonic oscillator  Rigid rotator  Hydrogen atom • However for a multielectron system, the SWE cannot be solved exactly due to inter-electronic repulsion terms.
The SWE is solved by method of seperation of variables.

• However, the inter-electronic repulsion term cannot be solved because the variables cannot be seperated and the SWE cannot be solved. • Approximate methods have helped to generate solutions for such and even more complex real quantum systems. • Approximate methods have been developed for solving Schrodinger equation to find wave function and energy of the complex system under consideration. • Two widely used approximate methods are, 1. Perturbation theory 2. Variation method
Perturbation theory is an approximate method that describes a complex quantum system in terms of a simpler system for which the exact solution is known. • Perturbation theory has been categorized into, i. Time independent perturbation theory, proposed by Erwin Schrodinger, where the perturbation Hamiltonian is static. ii. Time dependent perturbation theory, proposed by Paul Dirac, which studies the effect of time dependent perturbation on a time independent Hamiltonian H0.
PERTURBATION THEOREM
FIRST ORDER PERTURBATION THEORY
FIRST ORDER ENERGY CORRECTION
FIRST ORDER WAVE FUNCTION CORRECTION
APPLICATIONS OF PERTURBATION METHOD
SIGNIFICANCE OF PERTURBATION METHOD

1. TUMKUR UNIVERSITY DEPARTMENT OF STUDIES AND RESEARCH IN CHEMISTRY UNIVERSITY COLLEGE OF SCIENCE, TUMAKURU. 2019-2020 SEMINAR TOPIC PERTURBATION METHOD- FIRST ORDER CORRECTIONS Presentation by, BHAVANA R I M.Sc., II Sem

2. CONTENT INTRODUCTION PERTURBATION THEOREM TIME INDEPENDENT NON-DEGENERATE PERTURBATION FIRST ORDER PERTURBATION THEORY FIRST ORDER ENERGY CORRECTION FIRST ORDER WAVE FUNCTION CORRECTION APPLICATIONS AND SIGNIFICANCE REFERENCES

3. INTRODUCTION The postulates of quantum mechanics have been successfully used for deriving exact solutions to S chrodinger equation for problems like A particle in 1 Dimensional box Harmonic oscillator Rigid rotator Hydrogen atom However for a multielectron system, the SWE cannot be solved exactly due to inter-electronic repulsion terms. For example, He atom consists of one nucleus and 2 electrons. The H amiltonian for the system is, The last term in H amiltonian expression is the interelectronic repulsion term which causes difficulty in solving SWE. The SWE is solved by method of seperation of variables .  

4. However, the inter-electronic repulsion term cannot be solved because the variables cannot be seperated and the SWE cannot be solved . Approximate methods have helped to generate solutions for such and even more complex real quantum systems. Approximate methods have been developed for solving Schrodinger equation to find wave function and energy of the complex system under consideration. Two widely used approximate methods are, Perturbation theory Variation method

5. VARIATION METHOD The variation method is based on the variation theorem or variation principle. In this method a suitable wave function called trail function is chosen and is assumed to be the solution of Schrodinger equation. The variation theorem states that the expectation value of the energy calculated using trial function ψ will always be greater than the true value or eigen value E  

6. PERTURBATION THEORY Perturbation theory is an approximate method that describes a complex quantum system in terms of a simpler system for which the exact solution is known . Perturbation theory has been categorized into , Time independent perturbation theory, proposed by Erwin Schrodinger, where the perturbation H amiltonian is static. Time dependent perturbation theory, proposed by Paul Dirac, which studies the effect of time dependent perturbation on a time independent H amiltonian H .

7. PERTURBATION THEOREM Consider a function f(x) [which is not an exact wave function like ψ , ψ 1 , . . . . . ψ n ] but satisfies the same boundary condition as ψ , ψ 1 , . . . . ψ n then function f(x) can be expressed as a linear combination of all these functions ψ i ….(1) The values of coefficients a n determines the value of arbitrary function f(x) To find the value of a n , multiply both sides of equation (1) by ψ * m and integrate with in the boundary condition, ….(2) ….(3) Case 1: If n=0 Case 2: If n=1 Case 3: If n=m  

8. This means that m and n states are orthogonal and therefore for all n ≠ m and hence we could conclude from case 3, …..(4) TIME INDEPENDENT NON-DEGENERATE PERTURBATION Let us consider a system with non-degenerate ground state i.e., with discrete energy levels. Let us consider a quantum system for which the Schrodinger equation cannot be solved , ….(5) Where ψ is the wave function of the system and E is the total energy of the system and .…(6) Where the perturbation parameter and the term in equation (6) is the first order perturbation term . In general, Hamiltonian , wave function and energy for any complex system can be expressed using Taylor series of expansion as,  

9. …(7) Where is exactly solvable and the first term in each of the expressions in equation (7) represents the unperturbed term i.e ., , and . While the remaining terms in each of the expressions of equation (7) represents the correction terms which indicate perturbation effect . The basic assumption in perturbation theory is that the perturbation terms are smaller than unperturbed term . If the perturbation terms are not very large, then the solution derived using perturbation method should be close to the solution of unperturbed system .  

10. FIRST ORDER PERTURBATION THEORY To derive a solution for a complex system using first order perturbation, the H amiltonian, energy and wave function for the system can be written as, ….(8) Assuming and E as exact solutions when perturbation terms are not present, the Schrondinger equation becomes ….(9) Substituting equation (8) in eq (5) …(10) Rearranging equation (10) ….(11)  

11. The first term on each side of eq (11) cancels because of eq (9) Further, the last terms on each side can be neglected as they represent the product of 2 small perturbation terms and hence eq (11) reduces to ….(12) Where ψ 1 and E 1 are to be determined . The terms in eq (12) are all first order in while last terms on each side in eq (11) so neglected are of second order which represent second order corrections . Rearranging eq (12) gives, ….(13) This eq (13) is used to calculate first order corrections to energy and wave function.  

12. FIRST ORDER ENERGY CORRECTION Considering perturbation theorem, the first order correction to the wave function can be written as a linear combination of orthonormal set of functions ….(14) Substituting eq (14) in eq (13) gives, ….(15) Multiplying both sides of eq (15) by ψ * and integrating ….(16) Rearranging the expression gives ….(17) Solving LHS of eq (17) Case:1 n=0 ,  

13. ….( 18) Case:2 n=1 ….( 19) ( is an Hermitian operator and eigen functions ψ of Hermitian operators are orthogonal ) Equating LHS to RHS of eq (17) ….(20) Equation (20) is the first order correction to energy term Total energy of the system becomes, ….(21)  

14. FIRST ORDER WAVE FUNCTION CORRECTION Using equation (15) Multiplying both sides of eq (15) by and integrating ….(22) Solving LHS of equation(22) Case:1 n=m ….(23) Case:2 n=0 Case:3 n=1  

15. Equating eq (23) with RHS of eq (22) ….(24) …(25) Rearranging eq (25) gives, ….(26) Substituting the value of coefficient a m from eq (26) into eq (14) gives, …(27) Equation (27) is first order correction to wave function T otal wave function of the system becomes, ….(28)  

16. APPLICATIONS OF PERTURBATION METHOD Perturbation method has been successfully used for a particle in 1 Dimensional box with slanted bottom where the first order correction to the perturbed system shifts the energy level by a factor of in comparison to unperturbed system. Perturbation method is applied in solving multi-electron system like He atom. It even helps in solving anharmonic oscillator and non rigid rotator. SIGNIFICANCE OF PERTURBATION METHOD The experimental value for the He atom energy is -2.9033au , comparing this with first order perturbation energy -2.7500au gives a result about 5% error , variation method yeilds -2.8477au giving about 2% error . This result indicates that variation method gives good results for complex quantum system.  

17. REFERENCES ‘Physical chemistry- A molecular approach’- Donald A McQuarrie , John D. Simon, viva book edition. ‘Introduction to Quantum Mechanics’- David.J.Griffiths , second edition, Pearson publication. ‘Quantum Chemistry’- Ira.N.Levine . 6 th edition, PHI Learning Private Limited, New Delhi. ‘Quantum Chemistry’- R.K.Prasad , 4 th revised edition, New age international publication. Quantum Chemistry-Fundamentals to applications’- Tamas Veszpremi and Miklos Feher , Springer international edition. Web source ; http://epgp.inflibnet.ac.in

18. THANK YOU