February 12, 1998

PHY 742 - Summarizing Activity # 1

Note: This is an "exam-like" activity which can be turned in any time before 9 AM on February 18, 1998. Please attach full list of resources (including computer work, if appropriate) used to complete these problems, under the guidelines of the honor code.

1.

Consider the following three-dimensional harmonic oscillator for a spin $\frac{1}{2}$ particle of mass m and a ``spring" constant $m \omega^2$ described by the Hamiltonian:

$${\cal{H}} = -\frac{\hbar^2}{2m} \nabla^2 + \frac{1}{2} m \omega^2 r^2.$$ (1)

(a)

Find the general form of the eigenvalues of this system and describe the lowest eigenvalues and eigenfunctions in more detail.

(b)

A constant magnetic field ${\bf{B}} = B_0 {\bf{\hat{z}}}$ is introduced into the system.

i.

Choose a convenient gauge and write down the new Hamiltonian for the system.

ii.

Solve for the form of the eigenstates of the system in the presence of this magnetic field, describing the lowest eigenvalues and eigenfunctions in more detail.

iii.

Determine the magnetization of the ground state of the system.

2.

Consider following three-dimensional two-electron model in which each electron (coordinates ${\bf{r_1}}$ and ${\bf{r_2}}$) is bound to a center with a harmonic oscillator potential having a force constant of $m \omega^2$ and the electrons experience the normal Coulomb repulsion. (See, for example, Phys. Rev. A ${\bf{48}}$, 3561 (1993).)

$${\cal{H}} = -\frac{\hbar^2}{2m} ( \nabla_1^2 + \nabla_1^2 )+... ...ega^2 (r_1^2 + r_2^2) + \frac{e^2}{\vert{\bf{r_1 - r_2}}\vert}$$ (2)

(a)

Using harmonic oscillator states for each electron (with quantum numbers $n_1 \ell_1 m_1 \sigma_1$ and $n_2 \ell_2 m_2 \sigma_2$), determine the ground state and first few excited states of this system in the Hartree-Fock approximation, labeling states according to the single oscillator and total angular momentum and spin eigenvalues (such as ¹S, ³S, ¹P, ³P, etc.). Write an expression for the energy of these states in terms of the one-electron energies for the oscillators and in terms of the ${\cal{F}}^L$ and ${\cal{G}}^L$ radial integrals (which you do not need to evaluate). You may wish to use the ``gaunt" and ``clebsch" computer programs to evaluate some of the angular integrals.

(b)

With the change of variables ${\bf{R}} \equiv \frac{1}{2} ({\bf{r_1 + r_2}})$ and ${\bf{r}} \equiv {\bf{r_1 - r_2}}$, show that ${\cal{H}}$ becomes separable. Consequently, the spatial part of the wavefunctions can be determined exactly in terms of quantum numbers of the form $n_R \ell_R m_R$ and $n_r \ell_r m_r$. (Actually n_R is related to harmonic oscillator eigenstates but in general n_r corresponds to numerically determined radial functions.)

(c)

From symmetry considerations show that the Pauli exclusion principle requires that the spin states associated with exact solutions of the Schrödinger equation are restricted to total spin S = 0 if $\ell_r$ is even and S = 1 if $\ell_r$ is odd.

(d)

Show that the total angular momentum in the new variables and the old variables are related according to ${\bf{L}} = {\bf{L_1}} + {\bf{L_2}} \equiv {\bf{L_R}} + {\bf{L_r}}$.

(e)

Using the above analysis, label the lowest energy exact eigenstates (with quantum numbers $n_R, \ell_R, n_r, \ell_r$ and total spin S) according to the atomic term notation (¹S, ³S, ¹P, ³P, etc.), including any possible degeneracies.