FOUNDATIONS OF QUANTUM MECHANICS

20 CHAPTER II. THE FORMALISM
With (II. 16), in an orthonormal basis we have
⎛ ⎞
c 1
c 2
|ψ⟩ = ⎜ ⎟
⎝ . ⎠
c n
(II. 17)
and hence ⟨ψ| = (c 1 ∗ , c 2 ∗ , . . . , c ∗ n), therefore
⎛
c 1 c
∗ 1 . . . c 1 c ∗ ⎞
n
⎜
|ψ⟩ ⟨ψ| = ⎝
.
. ..
⎟
⎠ , (II. 18)
c n c
∗ 1 c n cn
∗
from which it is evident that for the vectors of the orthonormal basis {|α i ⟩} it holds that
N∑
|α i ⟩ ⟨α i | = 11, (II. 19)
i=1
with 11 the identity mapping on H,
11 |ψ⟩ = |ψ⟩ ∀ |ψ⟩ ∈ H. (II. 20)
Using (II. 14) and (II. 16), we see that an orthonormal basis is indeed characterized by the relation
|ψ⟩ =
N∑
⟨α i | ψ⟩ |α i ⟩ =
i=1
N∑
|α i ⟩ ⟨α i | ψ⟩. (II. 21)
i=1
The definition of a finite - dimensional Hilbert space is now completed; it is a finite - dimensional
complex Hilbert space with an inner product which is related to the norm by means of (II. 11). A real
finite - dimensional Hilbert space is obtained by replacing C everywhere by R, i.e., the set of scalars is
in R and the inner product is always real. In section II. 6 we will see that for the infinite - dimensional
case the definition must be extended with two requirements, ‘separability’ and ‘completeness’, which
we can prove in the finite - dimensional case.
II. 2
OPERATORS
An operator A on a Hilbert space H is a linear mapping of H onto itself,
A : H → H, |ψ⟩ ↦→ A |ψ⟩ with A ( a |ψ⟩ + b |ϕ⟩ ) = a A |ψ⟩ + b A |ϕ⟩. (II. 22)
From (II. 16) we saw that in a given orthonormal basis |α 1 ⟩, . . . , |α N ⟩ the vectors |ψ⟩ ∈ H are
unambiguously represented by rows of N complex numbers c i = ⟨α i | ψ⟩. This corresponds to the