(a) The initial wave function of the atom is
is the eigenstate of Lx = FL.
At t = 0 a strong magnetic field Be, is switched on. Then for t ≥ 0 the Hamiltonian of the system is
For a not too strong magnetic field B \(\sim\) 105 Gs, we can neglect the B2 term and take as the Hamiltonian
The expectation value of Lx is given by \(\langle\psi(r, t)|L_x|\psi(r, t)\rangle\). As \(L_x =(L_+ + L_-)/2\),
(b) The effects of electron spin can be neglected if thee additional energy due to the strong magnetic field is much greater than the coupling energy for spin-orbit interaction, i.e.,
Thus when the magnetic field B is greater than 106 Gs, the effects of electron spin can be neglected.
(c) If the magnetic field is very weak, the effects of electron spin must be taken into consideration. To calculate the time dependence of the expectation value of L,, follow the steps outlined below.
(i) The Hamiltonian is now
which is the Hamiltonian for anomalous Zeeman effect and we can use the coupling representation. When calculating the additional energy due to the \(\hat s_x\) term, we can regard \(\hat s_x\) as approximately diagonal in this representation.
(ii) Write down the time-dependent wave function which satisfies the initial condition Lx = +h and sx = 1/2 h. At time t = 0 the wave function is
\(\psi_0(r, s_x) = R_{21}(r) \Theta(\theta, \phi)\phi_s\)
where \(\theta\) and \(\phi_s\) are the eigenfunctions of Lx = h and sx = h/2 in the representations (l2, I,), (s2, sx) respectively. Explicitly,
where \(\phi_{jmj}\) is the eigenfunction of (j2, jz) for the energy level Enljmi. Therefore, the time-dependent wave function for the system is
(iii) Calculate the expectation value of Lx in the usual manner:
