初等量子力学数学手册1

Mathematical Handbook for Elementary Quantum Mechanics

Gaowei Qiu

Department of Physics, Zhejiang University; Hangzhou, Zhejiang, 310027, P. R. China

Contents

1 Pure Mathematics 1

1.1 Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3.1 Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.3 Feynman-Hellmann Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1

1 Pure Mathematics

Here are some fundamental maths

1

that are often used in quantum mechanics. Certainly, you might have

met them in high school, anyway they are gathered and listed below.

1.1 Trigonometry

1. Basic trigonometric formulas:

cos α cos β =

sin α cos β =

2

, (1.7)

cos α sin β =

2

. (1.8)

Reverse identities of prosthaphaeresis:

cos α + cos β = 2 cos



cos



α − β

2

, (1.11)

cos α − cos β = −2 sin



α + β

2





4. Half-angle formula:

2



1 − cos α

2

, (1.14)

cos

2



1 + cos α

2

, (1.15)

2



1 − cos α

sin α

1 − cos α

1

5. Duplication formula:

cos(2α) = cos

2

2

Generalization

3

:

sin(nα) =



k=1,3,5,···

(−1)



5

5

cos(nα) =



k=0,2,4,···

(−1)



6. Powers of trigonometric functions:

sin

2

cos

2

2