CHAPTER 1. THE WAVE FUNCTION 3 Chapter 1 The Wave Function Problem 1.1 (a) hji2 = 212 = 441. hj 2 i = = 1 X 2 1
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CHAPTER 1. THE WAVE FUNCTION
3
Chapter 1
The Wave Function Problem 1.1 (a) hji2 = 212 = 441. hj 2 i = =
1 X 2 1 2 j N (j) = (14 ) + (152 ) + 3(162 ) + 2(222 ) + 2(242 ) + 5(252 ) N 14 1 6434 (196 + 225 + 768 + 968 + 1152 + 3125) = = 459.571. 14 14 j 14 15 16 22 24 25
(b)
σ2 = =
σ=
∆j = j − hji 14 − 21 = −7 15 − 21 = −6 16 − 21 = −5 22 − 21 = 1 24 − 21 = 3 25 − 21 = 4
1 1 X (∆j)2 N (j) = (−7)2 + (−6)2 + (−5)2 · 3 + (1)2 · 2 + (3)2 · 2 + (4)2 · 5 N 14 260 1 (49 + 36 + 75 + 2 + 18 + 80) = = 18.571. 14 14
√
18.571 = 4.309.
(c) hj 2 i − hji2 = 459.571 − 441 = 18.571.
[Agrees with (b).]
c
2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
Full file at https://testbanku.eu/Solution-Manual-for-Introduction-to-Quantum-Mechanics-2nd-Edition-by-Griffiths
4
CHAPTER 1. THE WAVE FUNCTION
Problem 1.2 (a) h
Z
2
1 1 x √ dx = √ 2 hx 2 h 2
hx i = 0
σ 2 = hx2 i − hxi2 =
h2 − 5
h 2 5/2 h2 = x . 5 5 0
2 h 4 2 2h = h ⇒ σ = √ = 0.2981h. 3 45 3 5
(b) √ x+ 1 √ 1 1 √ √ dx = 1 − √ (2 x) = 1 − √ x+ − x− . 2 hx 2 h h x−
x+
Z P =1−
x−
x+ ≡ hxi + σ = 0.3333h + 0.2981h = 0.6315h; √
P =1−
0.6315 +
√
x− ≡ hxi − σ = 0.3333h − 0.2981h = 0.0352h.
0.0352 = 0.393.
Problem 1.3 (a) Z
∞
2
Ae−λ(x−a) dx.
1=
Let u ≡ x − a, du = dx, u : −∞ → ∞.
−∞
Z
∞
−λu2
1=A
e −∞
r
r π du = A λ
⇒ A=
λ . π
(b) Z
∞
2
xe−λ(x−a) dx = A
hxi = A
−∞ ∞
ue
−λu2
Z
Z
∞
e −∞
∞
r π = a. du = A 0 + a λ
2
x2 e−λ(x−a) dx
−∞ ∞
Z
2
u2 e−λu du + 2a
=A
Z
−∞
−λu2
du + a
−∞
hx2 i = A
2
(u + a)e−λu du
−∞
Z =A
∞
Z
1 =A 2λ
r
∞
2
ue−λu du + a2
−∞
π + 0 + a2 λ
σ 2 = hx2 i − hxi2 = a2 +
r
π λ
Z
∞
2
e−λu du
−∞
1 = a2 + . 2λ
1 1 − a2 = ; 2λ 2λ
1 σ=√ . 2λ
c
2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
Full file at https://testbanku.eu/Solution-Manual-for-Introduction-to-Quantum-Mechanics-2nd-Edition-by-Griffiths
CHAPTER 1. THE WAVE FUNCTION (c)
5
l(x) A
x
a
Problem 1.4 (a) |A|2 1= 2 a
Z
a
|A|2
2
Z
(
b 2
2
(b − x) dx = |A| 2 (b − a) a r b−a 3 2 a 2b = |A| + ⇒ A= = |A| . 3 3 3 b x dx +
0
1 a2
a b ) x3 1 (b − x)3 + − 3 0 (b − a)2 3 a
(b) ^ A
a
b
x
(c) At x = a. (d) a
Z
|A|2 |Ψ| dx = 2 a 2
P = 0
Z 0
a
a x dx = |A| = . 3 b 2
2a
P = 1 if b = a, X P = 1/2 if b = 2a. X
(e) Z a Z b 1 1 2 x|Ψ|2 dx = |A|2 2 x3 dx + x(b − x) dx a 0 (b − a)2 a ( 4 a b ) 2 3 1 x 1 x3 x4 2x + b = − 2b + b a2 4 0 (b − a)2 2 3 4 a 2 3 = a (b − a)2 + 2b4 − 8b4 /3 + b4 − 2a2 b2 + 8a3 b/3 − a4 2 4b(b − a) 4 b 2 3 1 3 2a + b 2 2 = −a b + a b = (b3 − 3a2 b + 2a3 ) = . 2 2 4b(b − a) 3 3 4(b − a) 4 Z
hxi =
c
2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
Full file at https://testbanku.eu/Solution-Manual-for-Introduction-to-Quantum-Mechanics-2nd-Edition-by-Griffiths
6
CHAPTER 1. THE WAVE FUNCTION
Problem 1.5 (a) Z
2
2
∞
Z
−2λx
|Ψ| dx = 2|A|
1=
e
2
dx = 2|A|
0
∞ e−2λx |A|2 = ; −2λ 0 λ
A=
√
λ.
(b) Z hxi =
2
2
Z
∞
xe−2λ|x| dx = 0.
x|Ψ| dx = |A|
[Odd integrand.]
−∞
2
2
Z
∞ 2 −2λx
hx i = 2|A|
x e 0
2 1 = dx = 2λ . (2λ)3 2λ2
(c) σ 2 = hx2 i − hxi2 =
1 ; 2λ2
√
1 σ=√ . 2λ
|Ψ(±σ)|2 = |A|2 e−2λσ = λe−2λ/
2λ
√
= λe−
2
= 0.2431λ.
|^| 2
h
.24h