Grifitths Introduction to Quantum Mechanics 3rd
edition Problem 5.6
terU3760
September 2025
1
Moses on Mount Sinai — Moses ascended the mountain alone. No crowd
followed him, and no human witness confirmed his ascent.
Yet God met him on the mountain.
Whether Moses truly climbed that height was not something for people to
verify — God Himself knew.
The mountain mattered not because it was seen, but because God bore
witness to Moses being there.
“I went there, and God knew.”
“The journey was not unseen — God bore witness.”
“Not every path needs a human witness; God remembers where I have been.”
“Whether I went or refused, the journey was already known to God.”
2
(a)
By Equation (5.23),
⟨(x1 − x2 )2 ⟩d = ⟨x2 ⟩a + ⟨x2 ⟩b − 2⟨x⟩a ⟨x⟩b
= ⟨ψn∗ |x2 |ψn ⟩ + ⟨ψl∗ |x2 |ψl ⟩ − 2⟨ψn∗ |x|ψn ⟩⟨ψl∗ |x|ψl ⟩
1
1
a a
1
1
= a2
− 2 2 + a2
− 2 2 −2· ·
3 2n π
3 2l π
2 2
2
1
1
a
1
+ 2 a2
=
− 2
6
2π
n2
l
(b)
By Equation (5.25),
2
⟨(x1 − x2 ) ⟩d = ⟨x2 ⟩a + ⟨x2 ⟩b − 2⟨x⟩a ⟨x⟩b − 2|⟨x⟩ab |2
1
1
1
1
2
= a2 − 2
+
a2 − 2 ⟨ψn |x|ψℓ ⟩
6
π
2n2
2ℓ2
=
1
1 2
a − 2
6
π
1
1
+ 2
2n2
2ℓ
a2 − 2
2
a (−1)n−l − 1 (−1)n+l − 1
−
π2
(n − ℓ)2
(n + ℓ)2
1
1
1 a2 − 1
+
a2 ,
6
π 2 2n2
2ℓ2
2
1 2
1
1
1
1
2
1
2
+ 2 a − 4
−
a2 ,
a − 2
6
π
2n2
2ℓ
π
(n + ℓ)2
(n − ℓ)2
(n ̸= ℓ)
n + ℓ is even
n + ℓ is odd
(c)
By Equation (5.25),
2
⟨(x1 − x2 ) ⟩d = ⟨x2 ⟩a + ⟨x2 ⟩b − 2⟨x⟩a ⟨x⟩b − 2|⟨x⟩ab |2
1
1
1 2
1
2
= a − 2
+ 2 a2 − 2 ⟨ψn |x|ψℓ ⟩
6
π
2n2
2ℓ
1
1
= a2 − 2
6
π
1
1
+ 2
2
2n
2ℓ
2
a +2
2
a (−1)n−l − 1 (−1)n+l − 1
−
π2
(n − ℓ)2
(n + ℓ)2
1
1
1
1 2
2
6 a − π 2 2n2 + 2ℓ2 a ,
2
1
1
1
1
2
1
1 2
2
+
−
a
+
a2 ,
a − 2
6
π
2n2
2ℓ2
π4
(n + ℓ)2
(n − ℓ)2
3
n + ℓ is even
n + ℓ is odd
(n ̸= ℓ)
0
