   # Evaluate 〈 L z 〉 for 3 p x , Compare it to the answer in Example 11.25, and explain the difference in the answers. ### Physical Chemistry

2nd Edition
Ball + 3 others
ISBN: 9781133958437

#### Solutions

Chapter
Section ### Physical Chemistry

2nd Edition
Ball + 3 others
ISBN: 9781133958437
Chapter 11, Problem 11.87E
Textbook Problem
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## Evaluate 〈 L z 〉 for 3 p x , Compare it to the answer in Example 11.25, and explain the difference in the answers.

Interpretation Introduction

Interpretation:

The Lz for 3px is to be stated and the answer is to be compared with Example 11.25.

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, Ψmrepresents 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.

### Explanation of Solution

The wavefunction 3px can be written as follows.

3px=12(Ψ3p+1+Ψ3p1)Ψ3px=181(2πa3)1/2(6rar2a2)er/2asinθcosϕ

The value of Lz can be calculated as follows.

002π0πΨ3px*LzΨ3pxdτ=i812πa3002π0π(6rar2a2)2e2r/3asin3θcos

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