Physics 249 Midterm Exam 2
Nov 16th 2012
1) List of the quantum numbers of all the electrons in ground state Sodium (Z=11) in the energy order they
are generally filled. For each n and l combination state how many radial probability distribution minimum
and maximum there are.
(15 points)
2) Which atoms through Z=11 (you can refer to them by Z) have the highest ionization energies,
lowest ionization energies, and highest electron affinity. Explain why they have these properties in each of
the three cases
(15 points)
3) Many of the standard atomic energy levels are split into doublet energy levels. Which type of orbital
does not split and why?
(15 points)
4) Using the quantum version of the argument that energy of a rotational orbit is
𝐿2
𝐸=
2𝐼
Compare the energy of the rotational transitions between the lowest rotational energy states in an H2
molecule and the H atom using this treatment. (Treat the H atom as a rotating system of proton and electron
using the same equation for rotational energy above.) Express your answer in eV. Comment on why this
treatment is valid for the H2 molecule and not the H atom.
Proton mass: 1.6726x10-27 kg or 938.27 MeV/c2
Electron mass: 9.1094x10-31 kg or 0.51100 MeV/c2
Bohr radius: 0.52918x10-10 m
H2 separation distance: 0.074nm
ℏ = 1.0546x10-34 J*s, 6.5821x10-16 eV*s
c = 3.00x108 m/s
ℎ𝑐 = 1240𝑒𝑉𝑛𝑚
ℏc = 197.3eVnm
(25 points)
(SEE NEXT PAGE)
5) Calculate the rate of transmission for E>0 particles through a one dimensional delta function barrier
potential at x=0: 𝑉(𝑥) = 𝜆𝛿(𝑥) where 𝜆>0. The delta function is infinite at x=0 and 0 everywhere else.
This case is representative of the tunneling between a thin barrier between conductors since the potential is
very high for an electron to be outside the wire.
Qualitatively you can think of the delta "function" as the limit of a process, which starts with a square
barrier, and reduces the thickness of the barrier, but increases the height of the barrier so that the area under
the barrier function remains the same and the barrier is finally infinitesimally thin in space and infinitely
high in space. Since, product of width times height of the potential barrier is finite the wave function does
not have to be zero at x=0.
(30 points, 5 points for each part)
a) Write down the Schrödinger equation for this system.
b) Write down a proposed wave equation solution.
Hint: Since, product of width times height of the delta function potential barrier is finite, as in the finite
potential barrier problem we solved, the wave function does not have to be zero at x=0
c) Write down an equation relating the wave functions at the region boundary.
d) Write down an equation relating the first derivatives of the wave function at the region boundary.
Hint: You can evaluate the form of the first derivative accounting for the impact of the delta function by
integrating the Schrodinger equation in an infinitesimally small region, 𝑥 = −𝜀 to 𝑥 = −𝜀 around x=0 and
considering the limit as that region shrinks to zero.
Hint: By definition of the delta function: ∫ 𝑓(𝑥) 𝛿(𝑥)𝑑𝑥 = 𝑓(0) over any range including x=0.
e) Solve the system of equations to understand the normalization constants of the transmitted wave
compared to the incoming wave (the overall normalization constant does not need to be understood).
f) Write down the transmission rate.