Physical Chemistry
Physical Chemistry
2nd Edition
ISBN: 9781285969770
Author: Ball
Publisher: Cengage
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Chapter 11, Problem 11.90E
Interpretation Introduction

Interpretation:

The forms of the five real 3d wavefunctions are to be stated.

Concept introduction:

The wavefunction for 3-D rotational motion involves spherical polar coordinates to explain the motion in three dimensions. It is given by the expression as follows.

Ψm,l=12πeimϕθl,ml

Where, Ψm represents the wavefunction dependent on quantum number m and l. The values of m and l start from 0. Also, they can take up both negative and positive integral values. According to Euler’s identity, eimϕ can be written as cosϕ+isinϕ.

Expert Solution & Answer
Check Mark

Answer to Problem 11.90E

The forms of the five real 3d wavefunctions are given below.

Ψ32(1)=181(Z3πa3)1/2Z2r2a2eZr/3asinθcosθsinϕΨ321=181(Z3πa3)1/2Z2r2a2eZr/3asinθcosθcosϕΨ32(2)=1162(Z3πa3)1/2Z2r2a2eZr/3asin2θsin2ϕΨ322=1162(Z3πa3)1/2Z2r2a2eZr/3asin2θcos2ϕ

Ψ320=1486(6Z3πa3)1/2Z2r2a2eZr/3a(3cos2θ1)

Explanation of Solution

From Table 11.4, the equations used to determine five 3d wavefunctions are given below.

Ψ320=1486(6Z3πa3)1/2Z2r2a2eZr/3a(3cos2θ1)Ψ32(1)=181(Z3πa3)1/2Z2r2a2eZr/3asinθcosθeiϕΨ321=181(Z3πa3)1/2Z2r2a2eZr/3asinθcosθeiϕΨ32(2)=1162(Z3πa3)1/2Z2r2a2eZr/3asin2θe2iϕ

Ψ322=1162(Z3πa3)1/2Z2r2a2eZr/3asin2θe2iϕ

Use Euler’s identity in the above wavefunctions as given below.

Ψ32(1)=181(Z3πa3)1/2Z2r2a2eZr/3asinθcosθ(cosϕisinϕ)Ψ321=181(Z3πa3)1/2Z2r2a2eZr/3asinθcosθ(cosϕ+isinϕ)Ψ32(2)=1162(Z3πa3)1/2Z2r2a2eZr/3asin2θ(cos2ϕisin2ϕ)Ψ322=1162(Z3πa3)1/2Z2r2a2eZr/3asin2θ(cos2ϕ+isin2ϕ)

The imaginary part of wavefunctions with ml and the real part of wavefunctions with ml are taken to get the real forms of wavefunctions. Thus, the forms of the five real 3d wavefunctions can be written as follows.

Ψ32(1)=181(Z3πa3)1/2Z2r2a2eZr/3asinθcosθsinϕΨ321=181(Z3πa3)1/2Z2r2a2eZr/3asinθcosθcosϕΨ32(2)=1162(Z3πa3)1/2Z2r2a2eZr/3asin2θsin2ϕΨ322=1162(Z3πa3)1/2Z2r2a2eZr/3asin2θcos2ϕ

Ψ320=1486(6Z3πa3)1/2Z2r2a2eZr/3a(3cos2θ1)

Conclusion

The forms of the five real 3d wavefunctions are given below.

Ψ32(1)=181(Z3πa3)1/2Z2r2a2eZr/3asinθcosθsinϕΨ321=181(Z3πa3)1/2Z2r2a2eZr/3asinθcosθcosϕΨ32(2)=1162(Z3πa3)1/2Z2r2a2eZr/3asin2θsin2ϕΨ322=1162(Z3πa3)1/2Z2r2a2eZr/3asin2θcos2ϕ

Ψ320=1486(6Z3πa3)1/2Z2r2a2eZr/3a(3cos2θ1)

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