Download Chemistry 354 - Homework Set IV

Chemistry 354 - Homework Set IV
(Due 3/19/13)
Name:
1. (10 points) An electron is confined to a one-dimensional box with dimensions Lx = 2
bohr and Ly = 1.5 bohr. Find all energy levels below 2500 eV, and provide a table with
energies, quantum numbers and the level degeneracy associated with each energy.
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2. (10 points, requires some google research) Express the following quantities in both SI and
atomic units: the energy content of a “King Size” (108 g) Butterfinger bar; the enthalpy of
combustion of a single molecule of sucrose; the distance from sideline to sideline on a football
field; the time required for an electron to make one circuit of the nucleus in the Bohr atom; the
mass of a hydrogen atom; the mass of Haystacks Calhoun (erstwhile professional wrestler);
the distance between the n = 1 and n = 2 energy levels of an electron in a one-dimensional
box of length 1 nm; and finally, the speed of light. Make a table of these numbers, and note
that the “atomic-scale” quantities are all “reasonable” numbers in atomic units, while those
for “regular life” quantities are “unreasonable”, and that the opposite behavior is seen for
the numbers in SI units. Finally, come up with names for the atomic units of mass, time
and speed. The “best” answers will be recognized.
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3. (10 points) Show, by direct application of the Hamiltonian, that the state function
"
πx
2πy
Ψ = N sin
sin
Lx
Ly
!
2πx
πy
− π sin
sin
Lx
Ly
!#
is an eigenfunction of the Hamiltonian only under one condition. Find this condition, the
associated eigenvalue, and also solve for the value of the normalization constant N .
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4. (10 points, independent thinking rewarded) Make up, and solve, a problem dealing with
the quantum mechanical harmonic oscillator. Laura will be on the lookout – and will give
a grade of zero – for problems taken directly from internet sources or textbooks, so rely on
your own creativity and initiative for this one.
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5. (10 points, extra credit) Using the solution to the expectation value for the energy of the
one-dimensional particle in the box state function
Ψ = N x sin(
πx
).
Lx
obtained in the last problem set and the content of the recent handout on an alternative
solution to the same problem, prove the identity:
∞
X
n4
π 4 + 6π 2
=
2
4
12288
n=1 (4n − 1)
If you can show this using just series summation techniques, that would certainly represent
an acceptable (and impressive) answer.
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