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# 5.P: Exercises

1. Demonstrate that the operators defined in Equations (427)-(429) are Hermitian, and satisfy the commutation relations (417).

2. Prove the Baker-Hausdorff lemma, (447).

3. Find the Pauli representations of the normalized eigenstates of and for a spin- particle.

4. 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 .

5. An electron is in the spin-state

in the Pauli representation. Determine the constant by normalizing . If a measurement of is made, what values will be obtained, and with what probabilities? What is the expectation value of ? Repeat the above calculations for and .

6. Consider a spin- system represented by the normalized spinor

in the Pauli representation, where and are real. What is the probability that a measurement of yields ?

7. An electron is at rest in an oscillating magnetic field

where and are real positive constants.
1. Find the Hamiltonian of the system.
2. 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.
3. Find the probability that a measurement of yields the result as a function of time.
4. What is the minimum value of required to force a complete flip in ?