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2 Quantum Mechanics – I
2.68 (a) Let f = e
i A ; then f
†
=
e
i A
† = e
−i A
Therefore f
† f = e
−i A e
i A
= 1
Thus e
i A is unitary.
(b) (a) Momentum (b) Parity
2.69 (a) exp(iσ x θ ) = 1 + iσ x θ + (iσ x θ)
2
/2! + (iσ x θ )
3
/3! + · · ·
= (1 − θ
2
/2! + θ
4
/4! . . .) + iσ x (θ − θ
3
/3! + θ
5
/5!. . . .)
= cos θ + iσ x sin θ
(where we have used the identity σ
2
x = 1)
(b)
ϕ
∗ (dψ/dx)dx = ϕ
∗
ψ −
(dϕ
∗
/dx)ψdx
But ϕ
∗
ψ = 0
Hence
ϕ
∗
dψ
dx
dx =
−
dϕ
∗
dx
ψdx =
−(dϕ/dx)
†
ψdx
Therefore
d
dx
† = −d/dx
2.70 (a) [x, P x ]ψ = x P x ψ − P x xψ
= x
−i
∂
∂ x
ψ + i
∂
∂ x
(xψ)
= −ix
∂ψ
∂ x
+ ix
∂ψ
∂ x
+ iψ
= iψ
∴ [x, P x ] = i
(b) [x
2
, P x ]ψ = x
2 (−i∂ ψ/∂x + i
∂
∂ x
(x
2
ψ)
= −ix
2
∂ ψ/∂x + i x
2
∂ψ/∂x + i(2x)ψ
= 2ixψ
Therefore [x
2
, p x ] = 2ix
2.71 By definition a transformation A is said to be linear if for any constant (possibly complex) λ
A(λX ) = λ A X
And if for any two vectors x and y
A(x + y) = Ax + Ay
If H is a hermitian operator
(x, H λy) = (H x, λy) = λ(H x, y) = λ(x, H y) = (x, λH y)
Or H λy = λH y
for any y. Furthermore
(z, H (x + y)) = (H z, x + y) = (H z, x) + (H z, y)
= (z, H x) + (z, H y) = (z, H x + H y)
∴ H (x + y) = H x + H y
2.72 Consider the equation
∂
∂ x
(ψ
∗
ψ) = ψ
∗ ∂ψ
∂ x
+ ψ
dψ
∗
dx
(1)