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# Exercises

1. Demonstrate that the operators defined in Equations (5.11)-(5.13) are Hermitian, and satisfy the commutation relations (5.1).

2. Prove the Baker-Campbell-Hausdorff lemma, (5.31).

3. Let the , for , be the three spin- Pauli matrices. Demonstrate that , and, hence, that     and .

4. Find the Pauli representations of the normalized eigenstates of (a) and (b) for a spin- particle.

5. 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 .

6. An electron is in the spin-state

in the Pauli representation. (a) Determine the constant by normalizing . (b) If a measurement of is made, what values will be obtained, and with what probabilities? What is the expectation value of ? Repeat the previous calculations for (c) and (d) . [61]

7. 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 ? [53]

8. 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 ?

9. In the Schrödinger/Pauli representation, the generalization of Schrödinger's time-dependent wave equation for an electron moving in electromagnetic fields is written

where the vector potential, the magnetic field-strength, the scalar potential, and the spinor-wavefunction. The term involving the Pauli matrices comes from the electron's intrinsic magnetic moment. (See Section 5.5.) Demonstrate that this equation can also be written

The previous expression is known as the Pauli equation [80].

10. Let , , and be the Pauli matrices for a spin- particle.
1. Show that

where

2. Show that

where

3. Show that

where

4. Hence, deduce that the spinor matrices for rotations through an angle about the three Cartesian axes are

where and .

5. 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 , 0 , and with probabilities , , and , respectively.

Next: Addition of Angular Momentum Up: Spin Angular Momentum Previous: Spin Greater Than One-Half
Richard Fitzpatrick 2016-01-22