LECTURE 12 SPINS 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)

Topics Today Electron in Magnetic Field Angular Momentum Operators Addition of Momenta

Electron in Magnetic Field Spinning charge constitute a magnetic dipole. Magnetic dipole moment, m is proportional to spin angular momentum: When a dipole is placed in a magnetic field, t = Energy associated with a torque: Hamiltonian of a spinning charged particle:

Larmor Precession A particle of spin ½ at rest in a uniform a magnetic field: The general solution to the time-dependent Scroedinger Equation:

Larmor Precession At t = 0: a = cos (a /2) b = sin (a/2) Significance of a will be obvious later <S> is tilted at constant angle a to the z-axis, and precesses about the field at the Larmor frequency:

Stern-Gerlach Experiment In an inhomogeneous magnetic field: This force separates out particles with particular spin orientation. Consider a beam of relative neutral relatively heavy neutral atoms, traveling in the y-direction in an inhomogenous magnetic field: Because of Larmor precision about B0, Sx oscillates rapidly and averages to zero, the net force is in the z-direction.

Stern-Gerlach Experiment

Stern-Gerlach Experiment The beam is deflected up and down, in proportion to the z-component of the spin angular momentum. Beam splits into 2s + 1 separate streams. Atom with s = 1/2

Problem 1 9

Addition of Momenta Two spin-1/2 particles: example: electron and proton in ground state. Total Angular momentum: Extra state m = 0

Addition of Momenta

Addition of Momenta

Addition of Momenta If spin s1 is combined with spin s2 Special cases of this general form with s1 = ½, s2 = ½

PROBLEM 2

PROBLEM 3