QuantumPhysics.dvi

3 Mathematical Formalism of Quantum Physics

We begin by reviewing the mathematical formalism of quantum physics, including Hilbert

spaces of finite and infinite dimensions, and of linear operators acting on Hilbert spaces.

Those in hand, we shall give the postulates of quantum physics in thenext section.

3.1 Hilbert spaces

A Hilbert spaceHis a complex vector space, endowed with a Hermitian positive definite

inner product, denoted by (·,·). For finite-dimensionalH, this definition will be complete,

while for infinite-dimensionalH, some additional convergence properties will have to be

supplied. We shall use the Dirac notation, and denote the vectors inHbykets|u〉,|v〉etc.

• The fact thatHis a complex vector space requires that

(α+β)|u〉 = α|u〉+β|u〉

α(|u〉+|v〉) = α|u〉+α|v〉

(αβ)|u〉 = α(β|u〉) (3.1)

for all|u〉,|v〉∈Hand for allα,β∈C.

• A Hermitian inner product (·,·) is a map fromH×H→C, such that

(|v〉,|u〉) = (|u〉,|v〉)∗

(|u〉,α|v〉+β|w〉) = α(|u〉,|v〉) +β(|u〉,|w〉)

(α|u〉+β|v〉,|w〉) = α∗(|u〉,|w〉) +β∗(|v〉,|w〉) (3.2)

The inner product is linear in the second entry, butanti-linearin the first entry. The inner

product notation (·,·) is primarily a notation of mathematicians. In physics instead, we use

the notation invented by Dirac,

〈v|u〉≡(|v〉,|u〉)

bra ket

(3.3)

This notation actually also has mathematical significance, as it leads naturally to interpret a

bra〈v|as alinear form onH. The space of all (continuous) linear forms onHis by definition

the Hilbert space dual toH, and is denotedH+.

• Positive definiteness of the inner product means that

〈u|u〉= (|u〉,|u〉)≡||u||^2 ≥ 0 for all|u〉∈H

〈u|u〉= 0 ⇒ |u〉= 0 (3.4)

Here,|u〉= 0 stands for the unit element 0 under the addition of the vector spaceH.