Complete Solution Manual for Introduction to Environmental Engineering and Science, 3rd Edition by Gilbert M. Masters.

All 12 Chapters Covered




SOLUTIONS

,2




Contents

1 The Wave Function 4

2 The Time-Independent Schrödinger Equation 16

3 Formalism 78

4 Quantum Mechanics in Three Dimensions 109

5 Identical Particles 168

6 Sỵmmetries and Conservation Laws 197

7 Time-Independent Perturbation Theorỵ 235

8 The Variational Principle 301

9 The WKB Approximation 333

10 Scattering 354

11 Quantum Dỵnamics 372

12 Afterword 420

A Linear Algebra 427

,4 CHAPTER 1. THE WAVE FUNCTION




Chapter 1

The Wave Function

Problem 1.1
(a)

hji 2 = 21 2 = 441.

2 1 X 2 1 2 2 2 2 2 2


hj i = j N (j) = (14 ) + (15 ) + 3(16 ) + 2(22 ) + 2(24 ) + 5(25 )
N 14
1 6434
459.571.
= (196 + 225 + 768 + 968 + 1152 + 3125) = =
14 14

j ∆j = j − hji
14 14 − 21 = −7
15 15 − 21 = −6
(b) 16 16 − 21 = −5
22 22 − 21 = 1
24 24 − 21 = 3
25 25 − 21 = 4


1 X 1
σ2 = (∆j)2 N (j) = (−7)2 + (−6)2 + (−5)2 · 3 + (1)2 · 2 + (3)2 · 2 + (4)2 · 5
N 14
1 260
= (49 + 36 + 75 + 2 + 18 + 80) = = 18.571.
14 14


√
σ = 18.571 = 4.309.

(c)

hj 2 i − hji 2 = 459.571 − 441 = 18.571. [Agrees with (b).]

, CHAPTER 1. THE WAVE FUNCTION 5


Problem 1.2
(a)
2
Z h 2 1 1 2 5/2 h h2

hx i = x √ dx = √ x = .
5 5
0 2 hx 2 h 0

2
2 2 2
h2 h 4 2
σ = hx i − hxi = − = h ⇒ σ=
5 3 45


(b)
Z x+ 1 1 √ x+ 1 √ √
P =1− √ dx = 1 − √ (2 x) =1− √ x+ − x− .
x− 2 hx 2 h x− h




x+ ≡ hxi + σ = 0.3333h + 0.2981h = 0.6315h; x− ≡ hxi − σ = 0.3333h − 0.2981h = 0.0352h.


√ √
P = 1 − 0.6315 + 0.0352 = 0.393.




Problem 1.3
(a)
Z ∞ 2
1= Ae−λ(x−a) dx. Let u ≡ x − a, du = dx, u : −∞ → ∞.
−∞
Z ∞ r
π
1=A e−λu 2du =A ⇒
−∞ λ

(b)


Z ∞ Z ∞
xe−λ(x−a) dx = A
2
hxi = A (u + a)e−λu du
2


−∞ −∞
Z ∞ Z ∞
r
2 2 π
=A ue−λu du + a e−λu du = A 0 + a = a.
−∞ −∞ λ


Z ∞
x2e−λ(x−a) dx
2
2
hx i = A
− ∞ Z Z
Z
∞ ∞
2 2 ∞
u2e−λu du + 2a ue−λu du + a2 e−λu du
2
=A

−∞ −∞ −∞
r r
1 π π
=A + 0 + a2 =
2λ λ λ 2λ