3.3 Solutions
161
3.9 (ψ, Qψ) = (ψ, qψ) = q(ψ, ψ)
(Qψ, ψ) = (qψ, ψ) = q
∗ (ψ, ψ)
since Q is hermitian,
(ψ, Qψ) = (Qψ, ψ) and that q = q
∗
That is, the eigen values are real. The converse of this theorem is also true,
namely, an operator whose eigen values are real, is hermitian.
3.10 (a) The normalization condition requires
∞
−∞
|ψ|
2 dx =
a
−3a
|c|
2 dx = 1 = 4a|c|
2
Therefore c = 1/2
√
a
(b)
a
0 |ψ|
2 dx =
α
0 c
2 dx = 1/4
3.11 (a) The expectation values are
< x >=
∞
−∞
ψ
∗ x ψ dx =
a
−3a
x
dx
4a
= −a
< x
2
>=
∞
−∞
ψ
∗ x
2
ψ dx =
a
−3a
(1/4a) x
2 dx =
7
3
a
2
xσ
2
=< x
2
> − < x >
2
=
7
3
a
2
− (−a)
2
=
4
3
a
2
(b) Momentum probability density is |ϕ( p)|
2
ϕ( p) = (2π )
−1/2
∞
−∞
dx ψ (x)e
−i px/
= (2π )
−1/2
a
−3a
dxce
−i px/
=
ic
p
2π
1/2
⎡
⎣ e
−
i pa
− e
3i pa
⎤
⎦
=
−
ic
p
2π
1/2
e
i pa/
⎡
⎣ e
2i pa
− e
−2i pa
⎤
⎦
=
2c
p
2π
1
2 e
i pa
sin
2 pa
Therefore |ϕ( p)|
2
=
2πap 2 sin
2
2pa