QuantumPhysics.dvi

4 The Principles of Quantum Physics

The mathematical description of a physical quantum system is in terms of a separable Hilbert

spaceHand certain linear operators onH.

Principle 1 Everyphysical stateof a quantum system is represented by a vector inH.

Two vectors,|φ〉,|φ′〉 ∈ Hcorrespond to the same physical state if and only if|φ′〉=λ|φ〉

for some non-zeroλ∈C. Using this equivalence, a physical state is really described by a

rayinH, and one often chooses||φ||= 1.

Principle 2 Everyobservableof a physical system is represented by a self-adjoint operator

onH. A state|φi〉has a definite measured valueaifor an observableAprovided|φi〉is an

eigenvector ofA,

A|φi〉=ai|φi〉 (4.1)

In any quantum system, the outcomes of any experiment on the system are the possible

eigenvalues of various observables. States associated with different eigenvalues are orthogonal

in view of the fact thatAis self-adjoint.

Principle 3 Let|φ〉be an arbitrary state inH, and let{|ψi〉}denote a set of mutually or-

thogonal states, such as, for example, the eigenstates of an observable. Then, the probability

pfor measuring the state|φ〉in one of the states|ψi〉are given by

p

(

|φ〉→|ψi〉

)

=|〈ψi|φ〉|^2 (4.2)

for normalized states satisfying〈φ|φ〉= 1 and〈ψi|ψj〉=δi,j.

Principle 4 Time-evolution, also referred to as dynamics, of a quantum systemis generated

by a self-adjoint HamiltonianH, which is itself an observable associated with the total energy

of the system. In the Schr ̈odinger picture of a closed system, the states of the system evolve

in time, and the observables are time independent. The Schr ̈odinger equation gives the time

evolution of any state|φ(t)〉, according to

i ̄h

∂

∂t

|φ(t)〉=H|φ(t)〉 (4.3)

In the Heisenberg formulation, the states remain time independentbut the observables ac-

quire time-dependence, according to

i ̄h

d

dt

A(t) = [A(t),H] (4.4)

The Heisenberg and Schr ̈odinger formulations are equivalent to one another, as we shall

confirm shortly.