Zee-Mfl~ t. byye (!Tl~ -t-o +~ eIV: ~ .~ ~
II ') c
I(J+ I ) -
- i I
f. (1-(- \).+ 5/1 -Jfi'
1). Ignoring all other interactions besides the attractive Coulomb interaction
between the electrons and nucleus:
(a). Find expressions for the lowest three energies of a two-electron atom with
an arbitrary value for Z, w
1). This is essentially Griffiths 9.20, which is a long(ish) and somewhat difficult(ish)
problem, but it is one of the most important applications of quantum mechanics, so
here we go
A spin-1/2 particle with a gyromagneti
1). This is a slight restatement of Griffiths problem 6.21 (pg. 279): Consider the
eight n=2 states of hydrogen, |2jmj>. Find the energy of each state resulting
from a weak-field Zeeman splitting, and construct a diagram
1). This is Griffiths 11.1 (pg. 397):
Coulomb scattering is one of the most important scattering problems. Youll solve
this problem classically here. It turns out that the quantum result is the same. An
1). This is Griffiths 11.8 (p. 412): Show, by direct substitution, that the integral
form of the Schrdinger equation
(r ) 0 (r )
ik r r '
r r ' V (r ') (r ') dr '
is consistent with the standard differential f
Physics 487: Week 10 Discussion
(1) Because of spontaneous emission, there is a finite lifetime associated with an electron
in an eigenstate of any system, such as eigenstates associated with the Hydrogen atom.
What is the expected frequency spread of an
1). In this problem, you will get some practice constructing many-particle
wavefunctions for real systems. Consider the two 2p electrons in the Carbon atom.
(a). What are the possible values of Lto
In class, we derived an expression for Rabi oscillations of a quantum system,
which describes the time-dependent probability that the system will make a
transition between an initial state |a> and a final s
(1). We found in class that, at very low energies (ka < 1), only the = 0 partial wave
is important. In this problem, well consider what happens when there is also a small
amount of the = 1 partia
1). This is similar to Griffiths, problem 5.35 (p. 246): The Fermi energy discussed
in class has a dramatic astrophysical effect: A white dwarf is the remnant of a
normal star. Once the star has used up its
The P r3+ ion has 2 f -electrons outside a series of filled shells (.5s2 5p6 4f 2 ).
1: Maximize spin. There are two electrons in the outer shell, with spins s1 = s2 = 21 . The
total spin can take on values S = s1 + s2 , ., |s1
1). Suppose you have three noninteracting particles in thermal equilibrium in a 1-D
harmonic oscillator potential, with Etot = (9/2).
(a). In the table below, write the possible occupation number
1). In this problem, you will consider the very different effects of the Boltzmann
and Bose-Einstein distributions on the occupancy of the ground and first-excited
states of an atom. Consider an a
Physics 487: Week 11 Discussion
(1) We consider the scattering of a plane wave, eikz , off of a spherical shell potential, centered
about the origin and defined by
V (r) = (r a).
Because the incident wave is along the z-axis, there is no preferred directi
( Ne ) 3
where V = 43 R3 . So,
3 2 3
( ) 3 ( Ne ) 3
10me R 4
If R decreases by a factor , then E increases by a factor 2 .
Pressure can be calculated from its relationship
1). Consider the following linear molecule, which consists of 3 atoms, each of which has one
single atomic orbital with s symmetry available to participate in bonding:
If the electron is the atomic orbit
1). Perturbation theory refresher (this is Griffiths, problem 6.3 (p. 255): Two
identical bosons are placed in an infinite square well potential. They interact with
one another via the potential: V(x1,x2) = -aVo(x1 x2),
1). In class, we showed that the density of electronic states (dN/dE) in a threedimensional box is dN/dE ~ E1/2. Determine the expression for the density of
electronic states in a two-dimensional box (Note: this is import
1). Consider a system of three noninteracting particles confined to move in a onedimensional infinite square well potential of length L:
V(x) = 0 for 0 < x < L and V(x) = elsewhere.
Determine the second excited state ener