Question

A hydrogen atom is in a ²Pr_1/2 state with total angular momentum up along the t-axis. In all parts of this problem show your computations and reasoning carefully.

(a) With what probability will the electron be found with spin down?

(b) Compute the probability per unit solid angle P(\(\theta, \phi\)) that the electron will be found at spherical angles \(\theta, \phi\) (independent of radial distance and spin).

(c) An experimenter applies a weak magnetic field along the positive z-axis. What is the effective magnetic moment of the atom in this field?

(d) Starting from the original state, the experimenter slowly raises the magnetic field until it overpowers the fine structure. What are the orbital and spin quantum numbers of the final state?

[Assume the Hamiltonian is linear in the magnetic field.]

(e) What is the effective magnetic moment of this final state?

Answer 1

(a) For the state ²P_1/2, l = 1, s = 1/2, J = 1/2, J_z = 1/2. Transforming the coupling representation into the uncoupling representation, we have

(b) As

(c) In the weak magnetic field, J and J_z are Good quantum numbers and the state remains unchanged. The effective magnetic moment is

(d) In a strong magnetic field, the interaction of the magnetic moment with the field is much stronger than the coupling interaction of spin and orbit, so that the latter can be neglected. Here 1 and s are good quantum numbers. The Hamiltonian related to the magnetic field is

When the magnetic field is increased slowly from zero, the state remains at the lowest energy. From the expression of W, we see that when the magnetic field becomes strong, only if 1, = -ti, s, = -A/2 can the state remain at the lowest energy. Thus the quantum numbers of the final state are l = 1, I_z = -1, s = l/2, s_z = -1/2.

(e) the effective magnetic moment of the final state is