LECTURE 15

Example: Second Order Perturbation 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.

Example: Continue

Degenerate Perturbation Theory If two or more Eigen-States share the same energy: and ∞

Degenerate Perturbation Theory Let be the “good” unperturbed state in generic form with a and b adjustable. Matrix Element of H’ Perturbed energies of degenerate eigenstates:

Potential is zero in the box and infinite outside. PARTICLE IN A BOX Potential is zero in the box and infinite outside. x y z a c b when x =0, x = a y = 0, y = b z = 0, z = c …………..(1) …………..(2) (3) (4) Substitute Equation (1) into Equation (2) (5)

Insert (3), (4) and (5) into (1) and divide by XYZ …………..(6) Separate Equation (6) by making each component a constant Using Boundary conditions: At

Degeneracy – Number of states that have the same energy. Lowest Energy: No quantum number of a particle in a square box is zero because if so Degeneracy – Number of states that have the same energy. Example: Second level above ground level. , and are degenerates if a = b = c, Degeneracy = 3

Example: Degeneracy Consider the 3-D infinite cubical well: This was solved in Lecture 8 The stationary states are: The allowed energies are: Ground State The first excited state is triply degenerate:

Perturbation on Degenerate Energy Levels: Example Ist order correction of ground state:

Perturbation on Degenerate Energy Levels: Example Construct matrix W : The z integral is zero. It can be shown that: Therefore:

Perturbation on Degenerate Energy Levels: Example The eigenvalues are: The first order in l: The “good” unperturbed states are linear combinations of the form:

Perturbation on Degenerate Energy Levels: Example The coefficients (a, b and g) form the eigenvectors of the matrix W: The “good” states are:

Fine Structure of Hydrogen: Relativistic Correction Relativistic Momentum and Kinetic Energy: Expanding T in the powers of small number of (p/mc) since p << mc in the non-relativistic limit. The lowest order relativistic correction to Hamiltonian:

Fine Structure of Hydrogen: Relativistic Correction First order correction: Schroedinger Equation in unperturbed state: Substitute: and Relativistic Correction

Spin-Orbit Coupling Electron in orbit around a nucleus. Electrons point of view: proton is circling around it. A magnetic field B is set up, in electron frame. The Hamiltonian m :Dipole moment of e. Magnetic field of proton: Hydrogen atom from electron’s perspective. B is in the same direction as L Use to eliminate mo in favor of eo.

Spin-Orbit Coupling Magnetic dipole moment of the electron. Magnetic dipole moment of ring of charge Moment inertia, mr2 X angular velocity (2p/T) Gyromagnetic Ratio: Classical Value Actual Value A ring of charge rotating about its axis. Kinematic Correction: Thomas Precession:

Spin –Orbit Coupling Total Angular Momentum: As a result of Spin-Orbit coupling, H’so no longer commutes with L and S. H’so commute with L2, S2 and J. Eigenvalues of L.S In terms of En: Fine-Structure=Realativistic Correction + Spin-Orbit Coupling Combining with Bohr Formula: Pg. 149Eq. 4.70 Griffith.

Spin-Orbit Coupling Energy Levels of Hydrogen, including Fine Structure