6.2 Foundations of Quantum Physics.
E are given by
(6.2.2)
position operator: xˆψ=xψ,
momentum operator: pˆψ=−ih ̄∇ψ,energy operator: Eˆψ=i ̄h
∂ ψ
∂t.
Postulate 6.4. For a quantum systemψand a physical Hermitian operatorLˆ,ψcan be
expanded as
(6.2.3) ψ=∑αkψk+
∫
αλψλdλ,whereψkandψλare the eigenfunctions ofL corresponding to discrete and continuous eigen-ˆ
values respectively. In (6.2.3) for the coefficientsαkandαλ, their modular square|αk|^2 and
|αλ|^2 represent the probability of the systemψin the statesψkandψλ. In addition, the
following integral, denoted by
(6.2.4) 〈ψ|Lˆ|ψ〉=
∫
ψ†(Lˆψ)dx,represents the average value of physical quantityL of systemˆ ψ.
Postulate 6.5.For a quantum system with observable physical quantities l 1 ,···,lN, if they
satisfy a relation
R(l 1 ,···,lN) = 0 ,
then the quantum systemψ(see (6.2.1)) satisfies the equation
R(Lˆ 1 ,···,LˆN)ψ= 0 ,whereLˆkare the Hermitian operators corresponding to lk( 1 ≤k≤N), provided that R(Lˆ 1 ,···,LˆN)
is a Hermitian.
Two remarks are in order. First, in Subsection2.2.5, Postulates6.3and6.5are introduced
as Basic Postulates2.22and2.23.
Second, in addition to the three basic Hermitian operators given by (6.2.2), the other
Hermitian operators often used in quantum physics are as follows:
(6.2.5)
angular momentum: Lˆ=xˆ×pˆ=−ih ̄~r×∇,
spin operator: Sˆ=sh ̄~σ,
scalar momentum: hp ̄ 0 =ih ̄(~σ·∇) (massless fermion),
scalar momentum: pˆ 1 =−ih ̄(~α·∇) (massive fermion),
Hamiltonian energy : Hˆ=Kˆ+Vˆ+Mˆ,wheresis the spin,~σand~αare as in (2.2.47) and (2.2.48), andKˆ,Vˆ,Mˆare the kinetic energy,
potential energy, mass operators.