Operators Matrices and Spin

We have already solved many problems in Quantum Mechanics using wavefunctions and differential operators. Since the eigenfunctions of Hermitian operators are orthogonal (and we normalize them) we can now use the standard linear algebra to solve quantum problems with vectors and matrices. To include the spin of electrons and nuclei in our discussion of atomic energy levels, we will need the matrix representation.

These topics are covered at very different levels in
**Gasiorowicz Chapter 14,**
**Griffiths Chapters 3, 4** and, more rigorously, in
**Cohen-Tannoudji et al. Chapters II, IV and IX.**

- The Matrix Representation of Operators and Wavefunctions
- The Angular Momentum Matrices
***** - Eigenvalue Problems with Matrices
- An
System in a Magnetic Field
***** - Splitting the Eigenstates with Stern-Gerlach
- Rotation operators for
***** - A Rotated Stern-Gerlach Apparatus
***** - Spin
- Other Two State Systems
*****

- Examples
- Harmonic Oscillator Hamiltonian Matrix
- Harmonic Oscillator Raising Operator
- Harmonic Oscillator Lowering Operator
- Eigenvectors of
- A 90 degree rotation about the z axis.
- Energy Eigenstates of an System in a B-field
- A series of Stern-Gerlachs
- Time Development of an System in a B-field: Version I
- Expectation of in General Spin State
- Eigenvectors of for Spin
- Eigenvectors of for Spin
- Eigenvectors of
- Time Development of a Spin State in a B field
- Nuclear Magnetic Resonance (NMR and MRI)

- Derivations and Computations
- The
Angular Momentum Operators
***** - Compute
Using Matrices
***** - Derive the Expression for Rotation Operator
***** - Compute the
Rotation Operator
***** - Compute the
Rotation Operator
***** - Derive Spin Operators
- Derive Spin
Rotation Matrices
***** - NMR Transition Rate in a Oscillating B Field

- The
Angular Momentum Operators
- Homework Problems
- Sample Test Problems

Jim Branson 2013-04-22