# CHEM 132B Study Guide - Quiz Guide: Spherical Harmonics, Position Operator, Electron Rest MassExam

by OC3523515

This

**preview**shows pages 1-2. to view the full**7 pages of the document.**Practice Final Key

Version A

Problem 1

Consider the orthonormal set |pi, which are eigenfunctions of the linear momen-

tum operator bp. State whether the following are true or false.

a) The states |piare eigenstates of the Hamiltonian b

H(x) with b

V(x) = 0.

True

b) The states |piare eigenstates of the position operator bx.False

c) hp′|bp2|pi= 0 if p′6=p.True

d) hp|[bx, bp]|pi= 0. False

Problem 2

Consider the spherical harmonic functions Yl,ml(θ, φ). State whether the follow-

ing are true or false.

a) The magnitude of the angular momentum of an orbital with Y3,2is 2√3¯h.

True

b) There are 2l+ 1 nodes in φfor each l.False

c) The function Y1,−1(θ, φ) is orthogonal to all other spherical harmonic

functions. True

d) The probability distribution of ﬁnding the electron within a given surface

area Aon the sphere (θ, φ) is P(r) = r2|Rn,l|2.False

Problem 3

Molecular bond vibrations can be described well as quantum harmonic oscilla-

tors. Assuming an atom A and an atom B, the eﬀective mass of the oscillator

is given by µ=mAmB/(mA+mB). Assume the spring constant of diﬀerent

molecular bonds to be roughly the same. State whether the following are true

or false.

a) The excitation energy of the v= 0 to v= 1 transition for the C-C bond

vibration is higher than for the C-H bond vibration. False

b) The excitation energy of the v= 1 to v= 2 transition for the O2bond

vibration is higher than for the N2bond vibration. False

c) The mean square extension of the O2bond vibration in the v= 0 state is

smaller than in the v= 10 state. True

d) The average extension of the N2bond vibration is larger in the v= 0 state

than in the v= 10 state. False

Only pages 1-2 are available for preview. Some parts have been intentionally blurred.

Problem 4

Consider the hydrogen atom. The atom is initially in the 2s1conﬁguration.

a) Give the term symbol(s) of the atom in this conﬁguration.

L= 0, S=1

2, so 2S+ 1 = 2

2S1/2

b) Consider the total angular momentum of the electron in this hydrogen

atom. What are the projections along zof its total angular momentum?

J=1

2, so projections are MJ=−1

2,1

2

c) The electron is excited to the n= 3 state. A measurement reveals that the

magnetic quantum number (ml) of the state is 0. Give a full expression

of the wavefunction Ψ(r, θ, φ) of the excited state.

The electron is the 3p state with n= 3, l= 1 and ml= 0:

Ψ(r, θ, φ) = 2

81√10π1

a7/2r2e−r/3acos θ

d) Give the term symbols of the excited state. Indicate which state lies low-

est in energy.

2P3/2and 2P1/2, with 2P1/2lowest in energy.

###### You're Reading a Preview

Unlock to view full version