QuantumPhysics.dvi

3.1.4 Finite-dimensional Hilbert spaces

All Hilbert spaces of finite dimensionNare isomorphic to one another. Let{|n〉}n=1,···,Nbe

an orthonormal basis in anN-dimensional Hilbert spaceH, satisfying〈m|n〉=δmn. Every

vector|φ〉inHmay be represented by a 1×N column matrix Φ whose matrix elements are

the complex numbersφn=〈n|φ〉,

|φ〉 ↔ Φ =

    

φ 1

φ 2

·

φN

    

〈φ| ↔ Φ†= (φ∗ 1 φ∗ 2 ··· φ∗N) (3.15)

The inner product of two vectors is given by matrix contraction,

(|ψ〉,|φ〉) =〈ψ|φ〉= Ψ†Φ =

∑N

n=1

ψn∗φn (3.16)

There is another combination that will be very useful,

|ψ〉〈φ| ↔ ΨΦ†=

   

ψ 1 φ∗ 1 ψ 1 φ∗ 2 ··· ψ 1 φ∗N

ψ 2 φ∗ 1 ψ 2 φ∗ 2 ··· ψ 2 φ∗N

···

ψNφ∗ 1 ψNφ∗ 2 ··· ψNφ∗N

   

(3.17)

Notice that a basis vector|m〉corresponds to a column matrix Φ whose entries areφn=δmn.

3.1.5 Infinite-dimensional Hilbert spaces

A infinite-dimensional Hilbert space Hin quantum physics will be separable and have a

countable orthonormal basis{|n〉}n∈N. All separable Hilbert spaces are isomorphic to one

another, but they may arise in different ways. An arbitrary vector|φ〉 ∈ Hmay be repre-

sented by the expansion,

|φ〉=

∑ n

cn|n〉 ||φ||^2 =