Fall 2006 15 Qm

Two electrons each of mass m are placed in a one-dimensional box of width L placed in an external magnetic field B in the z direction. The interaction Hamiltonian of the electrons is

(1)
\begin{align} H_{int} = A \bold{S_1} \cdot \bold{S_2} - (\bold{\mu_1} +\bold{\mu_2}) \cdot \bold{B} \end{align}

where the magnetic moment is $\bold{\mu_{1,2}} = - \mu_0 \bold{S_{1,2}}$, where $\mu_0$ is a constant.

(a) Find the possible energies of the system, and the quantum numbers (i.e. spatial and spin quantum numbers) and multiplicities of the allowed states.

(b) Write the energy as a multiple of $\hbar^2 / (2 m L^2)$ in terms of the dimensionless parameters $a = m L^2 A$ and $b= \mu_0 B m L^2 / \hbar$.

(c) Find the ground state quantum numbers as a function of a and b. Give your answer by marking the quantum numbers in the a - b plane.

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