Assumption of quantum theory of radiation

Fundamental assumptions of Quantum Mechanics

In a particular representation and applied to a system consisting of a single, structure-less particle the fundamental assumptions of Quantum Mechanics are:

• The quantum state of a particle is characterized by a wave function Ψ(r,t), which contains all the information about the system an observer can possibly obtain.
• The wave function Ψ(r,t) is interpreted as a probability amplitude of the particles presence.  |Ψ(r,t)|² is the probability density.  The probability that a particle is at time t in a volume element d³r situated at r is  dP(r,t) = C|Ψ(r,t)|²d³r.  For a single particle the total probability of finding it anywhere in space at time t is equal to 1.  (In non-relativistic Quantum Mechanics, material particles, unlike photons, are neither created nor destroyed.)
  ∫_{all space}P(r,t)d³r = 1,   C∫_{all space}|Ψ(r,t)|²d³r = 1,   ∫_{all space}|Ψ(r,t)|²d³r = finite.
  A proper wave function must be square-integrable.  The wave function must also be single valued and continuous for the probability amplitude interpretation to make sense.
• The principle of spectral decomposition applies to the measurement of an arbitrary physical quantity A.
  • The result of a measurement belongs to a set of eigenvalues {a}.
  • Each eigenvalue is associated with an eigenfunction Ψ_a(r).
    If Ψ(r,t₀) = Ψ_a(r) then a measurement of A at t = t₀ will yield the eigenvalue a.
  • Any Ψ(r,t₀) can be expanded in terms of eigenfunctions, Ψ(r,t₀) = ∑_a c_a Ψ_a(r).
    The probability that a measurement at t = t₀ will yield the eigenvalue a’ is P_a’ = |c_a’|²/(∑_a|c_a|²).
  • If a measurement of A yields a, then the wave function immediately after the measurement is Ψ_a(r).
• The Schroedinger equation describes the evolution of Ψ(r,t).
  iħ∂Ψ(r,t)/∂t = -(ħ²/(2m))∇²Ψ(r,t) + U(r,t)Ψ(r,t)
  is the Schroedinger equation for a particle of mass m whose potential energy is given by U(r,t).

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