# 5.P: Exercises

- Demonstrate that the operators defined in Equations (427)-(429) are Hermitian, and satisfy the commutation relations (417).
- Prove the Baker-Hausdorff lemma, (447).
- Find the Pauli representations of the normalized eigenstates of and for a spin- particle.
- Suppose that a spin- particle has a spin vector that lies in the - plane, making an angle with the -axis. Demonstrate that a measurement of yields with probability , and with probability .
- An electron is in the spin-state
- Consider a spin- system represented by the normalized spinor
- An electron is at rest in an oscillating magnetic field
- Find the Hamiltonian of the system.
- If the electron starts in the spin-up state with respect to the -axis, determine the spinor that represents the state of the system in the Pauli representation at all subsequent times.
- Find the probability that a measurement of yields the result as a function of time.
- What is the minimum value of required to force a complete flip in ?

### Contributors

- Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)