FOUNDATIONS OF QUANTUM MECHANICS

18 CHAPTER II. THE FORMALISM
and that every vector has an additive inverse, i.e., for every |ϕ⟩ ∈ H there is a vector |ϕ ′ ⟩ ∈ H, also
provable unique, such that
|ϕ⟩ + |ϕ ′ ⟩ = 0 . (II. 5)
The scalar multiplication is distributive and associative,
(a + b) ( |ϕ⟩ + |ψ⟩ ) = a |ϕ⟩ + a |ψ⟩ + b |ϕ⟩ + b |ψ⟩, (II. 6)
a ( b |ϕ⟩ ) = (a b) |ϕ⟩, (II. 7)
and we demand that
1 |ψ⟩ = |ψ⟩. (II. 8)
Incidentally we also write
a |ψ⟩ ≡ |a ψ⟩ ≡ |ψ⟩ a. (II. 9)
EXERCISE 1. Prove (a) 0|ϕ⟩ = 0 ,
(b) the additive inverse of |ϕ⟩ equals −1|ϕ⟩.
An inner product on a vector space is a mapping H × H → C, where the image in C
of ( |ϕ⟩, |ψ⟩ ) ∈ H × H is written as ⟨ϕ | ψ⟩. The inner product has the following properties:
(i)
⟨ϕ | a ψ + b χ⟩ = a ⟨ϕ | ψ⟩ + b ⟨ϕ | χ⟩,
(ii) ⟨ϕ | ψ⟩ = ⟨ψ | ϕ⟩ ∗ ,
(iii) ⟨ϕ | ϕ⟩ 0, (II. 10)
(iv) ⟨ϕ | ϕ⟩ = 0 iff |ϕ⟩ = 0 .
The value
∥ψ∥ := √ ⟨ψ | ψ⟩ (II. 11)
is called the norm of |ψ⟩ and meets the usual requirements for a norm; its value is positive, except
for the zero vector which is assigned 0, it is homogeneous, in the sense that ∥aψ∥ = |a|∥ψ∥, and it
satisfies the triangle inequality ∥ψ + ϕ∥ ∥ψ∥ + ∥ϕ∥. A vector is called a unit vector if the norm
equals 1.