TimeTime--Independent Independent Perturbation TheoryPerturbation Theory

Lab Session 2: Lab Session 2: Sample Calculations in Quantum Sample Calculations in Quantum

MechanicsMechanics

ObjectivesObjectives

•• To review the fundamentals of timeTo review the fundamentals of time--independent perturbation theory in independent perturbation theory in quantum mechanics.quantum mechanics.

•• To get acquainted with some of To get acquainted with some of manipulation capabilities.manipulation capabilities.

Review of FundamentalsReview of FundamentalsThe SchrThe Schröödinger Equationdinger Equation

A main problem in quantum mechanics is solving A main problem in quantum mechanics is solving SchrSchröödingerdinger’’s equation, which in 1D reads:s equation, which in 1D reads:

Or in general:Or in general:

22

2( ) ˆ( ) ( ) ( )

2n

n n nd x

V x x E xm dx

ψ ψ ψ− + =

ˆn n nH EΨ = Ψ

Review of FundamentalsReview of FundamentalsEigenvalue ProblemsEigenvalue Problems

Eigenfunction (Eigenvector); State vector

Eigenvalue (Energy)Hamiltonian Operator

ˆn n nH EΨ = Ψ

Quantum number

Review of FundamentalsReview of FundamentalsThe Main ProblemThe Main Problem

Only in very rare cases is SchrOnly in very rare cases is Schröödingerdinger’’s s equation exactly solvable.equation exactly solvable.Researchers resort to simplifications Researchers resort to simplifications and/or approximations.and/or approximations.

Review of FundamentalsReview of FundamentalsApproximate Solution MethodsApproximate Solution Methods

Perturbation TheoryPerturbation Theory: Building a solution for : Building a solution for a difficult problem based on that of a simpler a difficult problem based on that of a simpler one.one.Variational PrincipleVariational Principle: Obtaining the lowest : Obtaining the lowest eigenvalue and corresponding eigenfunction eigenvalue and corresponding eigenfunction without solving the Schrwithout solving the Schröödinger equation.dinger equation.WKB ApproximationWKB Approximation: A method to solve : A method to solve SchrSchröödingerdinger’’s equation assuming that the s equation assuming that the potential function varies slowly.potential function varies slowly.

TimeTime--Independent Perturbation Independent Perturbation Theory: Main IdeaTheory: Main Idea

Given an eigenvalue problem which you can solve:Given an eigenvalue problem which you can solve:

Find the solution for another problem:Find the solution for another problem:

Such that:

0 0 0 0ˆn n nH EΨ = Ψ

ˆn n nH EΨ = Ψ

Such that:0ˆ ˆ ˆH H Hλ ′= +

TimeTime--Independent Perturbation Theory: Independent Perturbation Theory: Terminology and AssumptionsTerminology and Assumptions

is the exact Hamiltonian, is the is the exact Hamiltonian, is the ““unperturbedunperturbed”” Hamiltonian, is the Hamiltonian, is the ““perturbationperturbation””, and is a small number , and is a small number between 0 and 1.between 0 and 1.In timeIn time--independent perturbation theory, all independent perturbation theory, all terms making up the Hamiltonian are timeterms making up the Hamiltonian are time--independent.independent.

H 0HH ′

λ

TimeTime--Independent Perturbation Independent Perturbation Theory: Concept of DegeneracyTheory: Concept of Degeneracy

If two or more distinct states have the same If two or more distinct states have the same energy, they are called energy, they are called degeneratedegenerate. For n. For n≠≠m:m:

Two sets of different rules are used for Two sets of different rules are used for degenerate and nondegenerate and non--degenerate states.degenerate states.

, ,n n n m m m n mH E H E E EΨ = Ψ Ψ = Ψ =

, ,n n n m m m n mH E H E E EΨ = Ψ Ψ = Ψ ≠

degenerate

non-degenerate

NonNon--Degenerate TimeDegenerate Time--Independent Independent Perturbation Theory: General FormalismPerturbation Theory: General Formalism

First, we write the exact state vectors , and the First, we write the exact state vectors , and the exact energies in terms of the unperturbed state exact energies in terms of the unperturbed state vectors and energies, and using :vectors and energies, and using :

Where and are the kth corrections to the Where and are the kth corrections to the energy, and state vector, respectively.energy, and state vector, respectively.NonNon--degenerate theory is valid only when:degenerate theory is valid only when:

nΨnE

λ0 1 2 2

0 1 2 2

...

...

n n n n

n n n nE E E E

λ λ

λ λ

Ψ = Ψ + Ψ + Ψ +

= + + +knE k

nΨ

0 0 0 0k n n kH E E′Ψ Ψ −

NonNon--Degenerate TimeDegenerate Time--Independent Independent Perturbation Theory: General FormalismPerturbation Theory: General Formalism

Substituting the above definitions in Substituting the above definitions in SchrSchröödingerdinger’’s equation, and comparing the s equation, and comparing the coefficients of the powers of on either side, coefficients of the powers of on either side, we get for the first two energy corrections:we get for the first two energy corrections:

λ

1 0 0

20 02

0 0

n n n

m nn

n mm n

E H

HE

E E≠

′= Ψ Ψ

′Ψ Ψ=

−∑

NonNon--Degenerate TimeDegenerate Time--Independent Independent Perturbation Theory: General FormalismPerturbation Theory: General Formalism

As for state vectors corrections:As for state vectors corrections:

( )( )

( )

0 01 0

0 0

0 0 0 02 0

0 0 0 0

0 0 0 00

20 0

m nn m

n mm n

k nn k

n k nk n n

n n k nk

k n n k

HE E

H HE E E E

H H

E E

≠

≠ ≠

≠

′Ψ ΨΨ = Ψ

−′ ′Ψ Ψ Ψ Ψ

Ψ = Ψ− −′ ′Ψ Ψ Ψ Ψ

− Ψ−

∑

∑∑

∑

Degenerate TimeDegenerate Time--Independent Independent Perturbation Theory: General FormalismPerturbation Theory: General Formalism

Sometimes perturbation Sometimes perturbation ““breaksbreaks”” or or ““liftslifts””degeneracy; two states having the same energy degeneracy; two states having the same energy have different values of energy after have different values of energy after perturbation.perturbation.To get first order corrections for a degenerate To get first order corrections for a degenerate systems, we diagonalize the Hamiltonian matrix.systems, we diagonalize the Hamiltonian matrix.Explanation of the diagonalization process:Explanation of the diagonalization process:

( )0

0

H E H E

H E I

ψ ψ ψ ψ

ψ

′ ′ ′ ′= ⇒ − =

′ ′∴ − =

Degenerate TimeDegenerate Time--Independent Perturbation Independent Perturbation Theory: General FormalismTheory: General Formalism

To diagonalize the Hamiltonian in case of kTo diagonalize the Hamiltonian in case of k--fold degeneracy, we need to solve the fold degeneracy, we need to solve the following secular equation:following secular equation:

11 11 12 12 1 1

1 1

0 0 0* 0, , , ,

0 0 0* 0, , , ,

0

ˆ ˆ

n n k n k

k n k kk n kk

ij n j n i n i n j

ij n j n i n i n j

H E S H E S H E S

H E S H E S

H H H d

S d

ψ ψ τ

ψ ψ τ

′ ′ ′ ′ ′ ′− − −

=

′ ′ ′ ′− −

′ ′ ′≡ Ψ Ψ =

≡ Ψ Ψ =

∫∫

Degenerate TimeDegenerate Time--Independent Perturbation Independent Perturbation Theory: Example; twoTheory: Example; two--fold degeneracyfold degeneracy

Suppose that we want to solve a timeSuppose that we want to solve a time--independent independent perturbation in which two of the unperturbed perturbation in which two of the unperturbed bases are degenerate:bases are degenerate:

The perturbation Hamiltonian matrix reads:The perturbation Hamiltonian matrix reads:

0 0 0 0 0 0 0 0 0 0ˆ ˆ, , 0m m n n m nH E H Eψ ψ ψ ψ ψ ψ= = =

11 12

21 22

H HH

H H

′ ′⎛ ⎞⎟⎜ ⎟⎜′ = ⎟⎜ ⎟⎜ ′ ′ ⎟⎜⎝ ⎠

Degenerate TimeDegenerate Time--Independent Perturbation Independent Perturbation Theory: Example; twoTheory: Example; two--fold degeneracyfold degeneracy

The secular equation for this matrix is:The secular equation for this matrix is:

The energy correction is:The energy correction is:

11 12

21 220

H E H

H H E

′ ′ ′−=

′ ′ ′−

( )2 211 22 11 22 12

1 42

E H H H H H±⎡ ⎤′ ′ ′ ′ ′ ′= + ± − +⎢ ⎥⎣ ⎦

Degenerate TimeDegenerate Time--Independent Perturbation Independent Perturbation Theory: Example; twoTheory: Example; two--fold degeneracyfold degeneracy

Taking , we the solution:Taking , we the solution:2 1a ≡

( )2 211 22 11 22 12

121

4

2

H H H H Ha

H±

⎡ ⎤′ ′ ′ ′ ′− ± − +⎢ ⎥⎣ ⎦= ′

Vectors and Matrices in Vectors and Matrices in

Vectors in Vectors in take the form of sets:take the form of sets:

Operations on vectors, summation, products:Operations on vectors, summation, products:A = {1,2,3};B = {4,5,6};

×

A + B A - B

a * A A.B

A B A * B

Vectors and Matrices in Vectors and Matrices in

Matrices in Matrices in take the form of take the form of

Some operations on matrices:Some operations on matrices:

m = {{1,2,3},{4,5,6}};

n = {{4,6,d},{a,g,s}};

m + n m - n

a * m m.n

m.A

Some MatrixSome Matrix--Specific Specific

MarixForm[]MarixForm[] displays a matrix in standard displays a matrix in standard mathematical form.mathematical form.IdentityMatrix[]IdentityMatrix[] generates an identity matrix.generates an identity matrix.DiagonalMatrix[]DiagonalMatrix[] generates a diagonal matrix.generates a diagonal matrix.Inverse[]Inverse[] finds the inverse of a matrix.finds the inverse of a matrix.Transpose[]Transpose[] finds the transpose of a matrix.finds the transpose of a matrix.Det[]Det[] finds the determinant of a matrix.finds the determinant of a matrix.

The Function The Function Part[]Part[]

This is a very important and useful This is a very important and useful

It extracts parts of an expressionIt extracts parts of an expressionExamples:Examples:

m = {{1,2,3},{4,5,6}};

Part[m,1] Part[m,1,2]

m[[1]] m[[1,2]]

The Function The Function Solve[]Solve[]

Solve[]Solve[] is used to solve a system of is used to solve a system of simultaneous algebraic equations.simultaneous algebraic equations.Examples:Examples:

2

2

Solve[ax + bx + c == 0,x]

Solve[{ax + by == 1,cx + d y == 5},{x,y}]

Solving Eigensystems with Solving Eigensystems with

has builthas built--in capabilities to in capabilities to eigensystems.eigensystems.Given a squareGiven a square--matrix, matrix, can can equation (diagonalize it), and find the equation (diagonalize it), and find the eigenvectors for it.eigenvectors for it.Relevant functions include:Relevant functions include:

Eigensystem[]Eigensystem[]Eigenvalues[]Eigenvalues[]Eigenvectors[]Eigenvectors[]

Solving Eigensystems with Solving Eigensystems with : Example: Example

Solve the eigensystem that resulted in the Solve the eigensystem that resulted in the twotwo--fold degeneracy handled previously.fold degeneracy handled previously.Solution:Solution:

Eigensystem[{{H11,H12},{H21,H22}}]Eigensystem[{{H11,H12},{H21,H22}}]

•• This slide show was prepared by This slide show was prepared by Usama alUsama al--BinniBinni in March 2005 as in March 2005 as part of the material used in teaching the Software Package in Phpart of the material used in teaching the Software Package in Physics ysics Course at the department of Physics at the University of Jordan.Course at the department of Physics at the University of Jordan.

BINNBINNÆÆUSUS