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Time-Independent Perturbation Theory 1. Source: D. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 2004) R. Scherrer, Quantum Mechanics An Accessible Introduction (Pearson Int’l Ed., 2006)
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Time-Independent Perturbation Theory 1 Source: D. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 2004) R. Scherrer, Quantum Mechanics An Accessible Introduction (Pearson Int’l Ed., 2006) R. Eisberg & R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (Wiley, 1974)
Perturbation Theory Perturbation Theory: A systematic procedure for obtaining approximate solutions to the unperturbed problem, by building on the known exact solutions to the unperturbed case.
Time-Independent Perturbation Theory Schroedinger Equation for 1-D Infinite Square Well Obtain a complete set of orthonormal eigenfunctions If potential is perturbed slightly Find the new eigenfunctions and eigenvalues of H.
Derivation of Corrections New Hamiltonian: H’ = perturbation H0= unperturbed quantity Write Ψn and En as power series of λ: Insert into Ist order correction to the nth value 2nd order correction to the nth value
Derivation of Corrections After insertion: Collecting like powers of λ,
First Order Correction to Energy Taking the inner product of: This means: Multiplying by and integrating. Replace But H0 is hermitean, so and Therefore: First order correction to energy: Expectation value of perturbation, in the unperturbed state.
First Order Correction to Wavefunction Rewrite Known function Becomes inhomogeneous DE Therefore: satisfies
First Order Correction to Wavefunction Equals Zero If l = n, m = n !st order energy correction First order correction to wavefunction If n = m, degenerate perturbation theory need to be used.
V(x) -d/3 d/3 Unperturbed State Perturbed State Example: V(x) Unperturbed Wave function of Infinitely Deep Square Well
Perturbed Energy Levels are obtained from: Energy is increased by 0.61 times the amount of additional potential energy at
To find the perturbed wave function: and Unperturbed levels are degenerate. Perturbation remove degeneracy.
Example Suppose we put a delta-function bump in the centre of the infinite square well. where α is a constant. Find the second-order correction to the energies for the above potential.
