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Time-Independent Perturbation Theory Prepared by: James Salveo L. Olarve Graduate Student January 27, 2010

Introduction The presentation is about how to solve the approximate new energy levels and wave functions to the perturbed problems by building on the known exact solutions to the unperturbed case. The intended reader of this presentation were physics students. The author already assumed that the reader knows Dirac braket notation. This presentation was made to facilitate learning in quantum mechanics.

Nondegenerate Perturbation Theory

The Perturb Hamiltonian 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)

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

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.

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: Eq. 1 Eq. 2

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 First Order Correction Eq. 1

First Order Correction So, We find first order correction for energy 1D continuous spectrum

First Order Correction Solving for the 1st order change in the wave function Since form a complete set then The eigenvalue equation for unperturbed state m Taking inner product with

First Order Correction Cases: I. So, Now, Cases: II. So, Therefore the wave function correction to first order is:

Second Order Correction Taking the inner product with yields Now, But, Second order correction for energy Eq. 2

Finally, The Eigenenergy The Wave Function

Degenerate Perturbation Theory

Twofold Degeneracy When the unperturbed states are degenerate then two or more distinct states share the same energy. As a consequence of that the ordinary perturbation theory fails. Suppose that: Note that any linear combination of these states, is still an eigenstate of , with the same eigenvalue Typically the perturbation will break the degeneracy We can’t even calculate the first-order energy because we don’t know what unperturbed states to use.

Twofold Degeneracy The “good” unperturbed states in the general form with Plugging these and collecting like powers of Taking the inner product with

Note: Twofold Degeneracy Then, Let: where:

Twofold Degeneracy Similarly, the inner product with yields Multiplying at the right hand side from

Twofold Degeneracy a) Suppose Then, here,

Twofold Degeneracy b) Suppose From: Which is consistent with

The states and were already the “correct” linear combinations. The answers for are precisely what we would have obtained using nondegenerate perturbation theory. IMPLICATION: In matrix form: Evidently the are nothing but the eigenvalues of the -matrix. And the “good” linear combinations of the unperturbed states are the eigenvectors of W.

Reference: Retrieved from http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/Time_Ind_PT.htm , January 19 2010, Michael Fowler. Introduction to Quantum Mechanics. David J. Griffiths. 1994

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

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