November 18, 1997

PHY 441 - Summarizing Activity # 2

Note: This is an "exam-like" activity which can be turned in any time before 10 AM on November 24, 1997. 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 a Stern-Gerlach type experiment where, in the original coordinate system, a beam of electrons ($s=\frac{1}{2}$) is prepared in pure eigenstates of S_z -- $\vert m_s=+\frac{1}{2}\!\gt$. Determine the probability of finding eigenstates of $S_{z^{\prime}}$ ($\vert m_{s^{\prime}}=+\frac{1}{2}\!\gt$ and $\vert m_{s^{\prime}}=-\frac{1}{2}\!\gt$) in a new coordinate system described by the rotation:

$$\left( \begin{array} {c} x^{\prime} \\ y^{\prime} \\ z... ...begin{array} {c} x \\ y \\ z \\ \end{array} \right).$$ (1)

2.

Consider a another Stern-Gerlach type experiment where, in the original coordinate system, a beam of deuterons (s=1) is prepared in pure eigenstates of S_z -- $\vert m_s=+1\!\gt$. Determine the probability of finding eigenstates of $S_{z^{\prime}}$ ($\vert m_{s^{\prime}}=+1\!\gt$, $\vert m_{s^{\prime}}=0\!\gt$, and $\vert m_{s^{\prime}}=-1\!\gt$) in a new coordinate system described by the rotation (same as for problem #1 above):

$$\left( \begin{array} {c} x^{\prime} \\ y^{\prime} \\ z... ...begin{array} {c} x \\ y \\ z \\ \end{array} \right).$$ (2)

3.

Consider the stationary state perturbation problem ${\cal{H}} = {\cal{H}}_0 + V$, where ${\cal{H}}_0$ represents the ideal Hamiltonian of a hydrogen-like ion of nuclear charge Z, and V represents the perturbation caused by the finite size of the nucleus approximated as:

$$V(r) = \left\{ \begin{array} {ll} -\frac{3 Z e^2}{2A} + \f... ...r} & \;\;\; {\rm {for} } \;\;\; r \geq A. \end{array} \right.$$ (3)

where the nuclear radius is denoted by A. Find the first order correction to the energy of the lowest eigenstate.

4.

Consider an unperturbed system described by a spherically symmetric Hamiltonian having degenerate eigenstates of orbital and spin angular momentum (${\bf l}$) and (${\bf s}$) : $\vert\ell\; m\; s\; m_s\!\gt$ with $\ell$ = 1 and s = $\frac{1}{2}$.

(a)

Suppose that the system is perturbed by an interaction of the form

$$\cal{H}^{\prime} = A {\bf l} \cdot {\bf s} - \gamma {\bf s} \cdot {\bf B}$$

where A and $\gamma$ are scalar constants and ${\bf B}$ denotes a magnetic field (which can be assumed to lie along the z-axis). Set up the degenerate perturbation equations to find the first order energies of this system.

(b)

Obtain an algebraic solution for the first order energies as a function of the parameters. Sketch the energies as a function of $\vert{\bf B}\vert$, indicating the functional dependence of the first order energies in the limits $\vert{\bf B}\vert << A$ and $\vert{\bf B}\vert \gt\gt A$.