1. 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)

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

3. 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:

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

5. 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:

6. 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.

7. Stern-Gerlach Experiment

8. 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

9. Problem 1 9

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

11. Addition of Momenta

12. Addition of Momenta

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

14. PROBLEM 2

15. PROBLEM 3