FOUNDATIONS OF QUANTUM MECHANICS

II. 2. OPERATORS 21
representation of an operator A as an N × N - matrix A in a basis {|α i ⟩}, and the coefficients of the
vector A|ψ⟩ in this basis are, using (II. 19),
with
⟨α i | A | ψ⟩ = ⟨α i | A 11 | ψ⟩ =
N∑
⟨α i | A | α j ⟩ ⟨α j | ψ⟩ =
j=1
N∑
A ij c j , (II. 23)
j=1
A ij := ⟨α i | A | α j ⟩. (II. 24)
Operators A and B can be added and multiplied,
(A + B) |ψ⟩ := A |ψ⟩ + B |ψ⟩ and (A B) |ψ⟩ := A ( B |ψ⟩ ) . (II. 25)
The adjoint A † of an operator A is defined by the following equation
⟨ψ | A † | ϕ⟩ = ⟨ϕ | A | ψ⟩ ∗ ∀ |ϕ⟩, |ψ⟩ ∈ H. (II. 26)
EXERCISE 4.
( ) A
† = A ∗
ij ji .
Show that for the matrix representation in an orthonormal basis it holds that
Every operator on a finite - dimensional vector space has a unique adjoint, and the following holds
(c A) † = c ∗ A † ,
(A + B) † = A † + B † ,
(A B) † = B † A † ,
(
A
† ) † = A. (II. 27)
An operator B is called an inverse of A if
A B = B A = 11. (II. 28)
In this case we write A −1 for B, because the inverse, if it exists, is unique. Not every operator has an
inverse, an example in the Hilbert space C 2 is
( ) 0 1
. (II. 29)
0 0
The trace of an operator A is defined as follows,
Tr A :=
N∑
⟨γ i | A | γ i ⟩, (II. 30)
i=1
where |γ 1 ⟩, . . . , |γ N ⟩ is an arbitrary orthonormal basis and N = dim H.