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

by OC3523515

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**preview**shows page 1. to view the full**4 pages of the document.**Chem 132B Quiz 1 Key

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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:

Ïˆ(x) = Aeikx +Beâˆ’ikx

a) Give an expression for the energy of the electron as a function of k.

Ek=k2Â¯h2

2m

b) A measurement reveals that the electron was moving to the right (toward

increasing xvalues) with a speed of v= 5.00 Ã—105m/s. Determine the

(numerical) value for kand the actual form of the electronâ€™s wavefunction.

Particle moves to the right, so wavefunction is Ïˆ(x) = Aeikx, with mo-

mentum Â¯h

i

d

dx Aeikx =kÂ¯hAeikx, which means that hpi=kÂ¯h

Comparing momentum:

mv = 9.109 Ã—10âˆ’31 kg Â·5.00 Ã—105m/s = 4.55 Ã—10âˆ’25 kg Â·m/s = kÂ¯h

so k= 4.32 Ã—109kgÂ·m

JÂ·s2. Note that kgÂ·m

JÂ·s2= mâˆ’1

c) Assume the electron has the wavefunction found in b), but with a new

kinetic energy of E= 2.00 Ã—10âˆ’19 J. At a certain moment, a barrier is

inserted along x, where V= 3.00 Ã—10âˆ’19 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 electronâ€™s wavefunction in this region.

Since V > E, we can write

k2=âˆ’2m(Vâˆ’E)/Â¯h2= (iÎº)2with Îº=p2m(Vâˆ’E)/Â¯h= 4.05Ã—109mâˆ’1.

So k=iÂ·4.05 Ã—109mâˆ’1

Wavefunction is Ïˆ(x) = Aeâˆ’Îºx =Aeâˆ’4.05Ã—109mâˆ’1Â·x

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