Instructors Solutions Manual An Introduction to Groups and their Matrices for Science Students

INSTRUCTOR SOLUTION MANUAL

Solutions Manual to accompany

AN INTRODUCTION TO GROUPS AND THEIR MATRICES FOR SCIENCE STUDENTS KOLENKOW

Copyright c 2021 by Robert Kolenkow

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Acknowledgment The style code used for typesetting this manual was written by the experienced and skillful Ali Woollatt for use by Cambridge University Press.

CONTENTS

1

FUNDAMENTAL CONCEPTS

1

2

MATRIX REPRESENTATIONS OF DISCRETE GROUPS

17

3

MOLECULAR VIBRATIONS

40

4

CRYSTALLINE SOLIDS

53

5

BOHR’S QUANTUM THEORY AND MATRIX MECHANICS

65

6

WAVE MECHANICS, MEASUREMENT, AND ENTANGLEMENT

74

7

ROTATION

88

8

QUANTUM ANGULAR MOMENTUM

97

9

THE STRUCTURE OF ATOMS

112

10

PARTICLE PHYSICS

127

1.1 Apply Tx = −x.

Tx = −x Te x = e(−x) = e−x Here is another approach. Apply the operator to e x expressed in series form. ! x2 x3 x + + ...

Te = T 1 + x + 2! 3! ! (−x)2 (−x)3 + + ... = 1−x+ 2!

3! ! x2 x3 = 1−x+ − + ...

2! 3! = e−x The operator has changed the sign of the odd terms.

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FUNDAMENTAL CONCEPTS

1.2 Apply Tx = −x.

T cos (x) = cos (−x) = cos (x) This result can also be understood by considering the series expansion of cos (x). x2 x4 x6 + − + ...

2! 4! 6!

T cos (x) = cos (−x) cos (x) = 1 −

(−x)2 (−x)4 (−x)6 + − + ...

2!

4!

6! x2 x4 x6 =1− + − + ...

2! 4! 6! = cos (x) =1−

The series expansion of cos (x) contains only even terms, so Tx = −x changes nothing.

1.3 According to the group axioms (AB)(AB)−1 = E Multiply by A−1 from the left.

A−1 AB(AB)−1 = A−1 E B(AB)−1 = A−1 Multiply by B−1 from the left.

B−1 B(AB)−1 = B−1 A−1 (AB)−1 = B−1 A−1