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CHEM 132B Study Guide - Quiz Guide: Momentum Operator, Linear Combination, Symmetric FunctionExam


Department
Chemistry
Course Code
CHEM 132B
Professor
Eric Potma
Study Guide
Quiz

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Chem 132B Quiz 1 Key
Name (Last, First):
Student ID number:
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Problem 1
Encircle all that are true.
a) A set of eigenfunctions must be normalized otherwise the eigenfunctions
are not orthogonal. False
b) Operators that correspond to real observables are Hermitian. True
c) Operators that correspond to real observables must commute with the
Hamiltonian. False
d) Any wavefunction can be expressed as a linear combination of eigenfunc-
tions of a given operator that form a complete set. True
Problem 2
An electron is moving along the x-axis against a background where V(x) = 0.
The general solution of its wavefunction is:
ψ(x) = Aeikx +Beikx
a) Give an expression for the energy of the electron as a function of k.
Ek=k2¯h2
2m
b) A measurement reveals that the electron was moving to the right (toward
increasing xvalues) with a speed of v= 5.00 ×105m/s. Determine the
(numerical) value for kand the actual form of the electron’s wavefunction.
Particle moves to the right, so wavefunction is ψ(x) = Aeikx, with mo-
mentum ¯h
i
d
dx Aeikx =k¯hAeikx, which means that hpi=k¯h
Comparing momentum:
mv = 9.109 ×1031 kg ·5.00 ×105m/s = 4.55 ×1025 kg ·m/s = k¯h
so k= 4.32 ×109kg·m
J·s2. Note that kg·m
J·s2= m1
c) Assume the electron has the wavefunction found in b), but with a new
kinetic energy of E= 2.00 ×1019 J. At a certain moment, a barrier is
inserted along x, where V= 3.00 ×1019 J between x= 0 and x=L,
while V= 0 anywhere else. Determine the (numerical) value for kin the
barrier and the actual form of the electron’s wavefunction in this region.
Since V > E, we can write
k2=2m(VE)/¯h2= ()2with κ=p2m(VE)/¯h= 4.05×109m1.
So k=i·4.05 ×109m1
Wavefunction is ψ(x) = Aeκx =Ae4.05×109m1·x
1
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