126
2 Quantum Mechanics – I
This gives a = 1, b = 0 for λ = 1 and a = 0, b = 1 for λ = 3
Hence the eigen states of A are
1
0
and
0
1
(c) As A = A
† , A is Hermitian and hence an observable.
2.76 (a) A general rule for commutators is
[A
2
, B] = A[A, B] + [A, B]A
Here H = P
2
/2μ
[H, X ] = (1/2μ)[P
2
, x] = (1/2μ)(P[P, x] + [P, x]P)
= (1/2μ)2 p/i = p/iμ
Therefore [x, H ] = i p/μ
(b) [[x, H ], x] =
iP x
μ
, x
=
i
μ
[P x , x] =
i
μ
(−i) =
2
μ
2.77 [ A
2
, B] = A A B − B A A = A A B − A B A + A B A − B A A
= A[ A, B] + [A, B]A
2.78 (σ. A)(σ.B) = (σ x A x + σ y A y + σ z A z )(σ x B x + σ y B y + σ z B z )
= A x B x σ
2
x + A y B y σ
2
y + A z B z σ
2
z + σ x σ y A x B y + σ x σ z A x B z
+ σ y σ x A y B x + σ y σ z A y B z + σ z σ x A z B x + σ z σ y A z B y
= A.B + iσ z (A x B y − A y B x ) + iσ x (A y B z − A z B y )
+ iσ y ( A z B x − B z A x )
= A.B + i[σ.( A × B)] z + i[σ.( A × B)] x + i[σ.( A × B)] y
= A.B + i σ.( A × B)
where we have used the identities in simplifying:
σ
2
x = σ
2
y = σ
2
z = 1
and σ y σ x = −σ x σ y etc.
2.79 (a) σ yμ = σ μy † as can be seen from the matrix elements of σ y . Therefore σ y is
Hermitian. It is the matrix of a Hermitian operator whose eigen values are
real.
(b) The eigen values λ are found by setting
σ y 11 −λ σy 11
σ y 21 σ y 22 −λ
=
−λ −i
i −λ
= 0
λ
2
− 1 = 0, λ = ±1