FOUNDATIONS OF QUANTUM MECHANICS

II. 1. FINITE - DIMENSIONAL HILBERT SPACES 19
An important inequality is the Cauchy - Schwarz inequality
|⟨ϕ | ψ⟩| 2 ⟨ϕ | ϕ⟩ ⟨ψ | ψ⟩. (II. 12)
EXERCISE 2. Prove (a) the Cauchy - Schwarz inequality (II. 12),
(b) the definition of the norm satisfies the standard requirements for a norm.
The n vectors |α 1 ⟩, . . . , |α n ⟩ are called (linearly) independent if it follows from
n∑
c i |α i ⟩ = 0 (II. 13)
i=1
that all coefficients c i are equal to zero, otherwise the vectors are called dependent.
EXERCISE 3. Prove that mutually orthogonal vectors are linearly independent.
A set of vectors |α 1 ⟩, . . . , |α N ⟩ in H is complete 1 if every vector |ψ⟩ ∈ H can be written as a
linear combination of this set,
|ψ⟩ =
N∑
c i |α i ⟩. (II. 14)
i=1
A complete, independent set of vectors is called a basis. A basis is called orthonormal if
⟨α i | α j ⟩ = δ ij , (II. 15)
where δ ij is the Kronecker delta. It can be proved that every basis of a space H contains the same
number of elements, this number is, by definition, the dimension of H, and is written dim H. The
dimension of a Hilbert space is infinite if every finite set of linearly independent vectors is incomplete.
If |α 1 ⟩, . . . , |α N ⟩ is an orthonormal basis, with N = dim H, then it follows from (II. 15) that the
coefficients in (II. 14) are given by
c i = ⟨α i | ψ⟩, (II. 16)
and the vectors |ψ⟩ can thus be represented in such a basis by columns of N complex numbers.
Therefore, an N - dimensional Hilbert space can also be written as C N .
1 The use of the term ‘complete’ for a system of vectors should not be confused with the same phrase as used within the
context of the foundations of quantum mechanics, that is, as a property of a physical theory.