A sigle wave
Superposition of waves
© The scientific sentence. 2010

Meaning of the wave function
1.Intensity of the electromagnetic wave
Let's recall that the power of a wave is the rate at which its energy
is propagated that is P = U/Δt Where U is the total
energy of the wave passing through Δx during the time Δt. The quantity that
characterizes the flow of energy is called intensity, that is the power of the
propagating wave per unit area of a surface ΔA.
This intensity is then written as I = P/ΔA(in watts per square meters: W/m^{2}).
We have: I = U /Δt/ΔA
For electromagntic waves, the total (electric and magnetic) energy per volume unit, that
is the energy density, is expressed as: u_{em} = ε_{o} E^{2};
where E is the electric field of the electromagnetic wave, that can be expressed as:
E = E_{o} sin (ωt  kx).
The related intensity is S = U /Δt/ΔA = u_{em} ΔV/Δt/ΔA,
where ΔV is the volume unit around Δx during the time Δt. Using ΔV = ΔA Δx,
we can write S = u_{em} Δx/Δt. The intensity S is the magnitude of the Pointing
vector of the the wave. Further, we can write S = u_{em} c; where c = Δx/Δt is the
speed of the electromagnetic wave, that is the speed of light.
But the Pointing Vector is defined as
S = E x B/μ_{o}. According to E = c B, we have: S = E^{2}/c μ_{o}.
As ε_{o}μ_{o} c^{2} = 1, then: S = c ε_{o}E^{2}.
( We have also the average S = (c ε_{o}/2)E_{o}^{2}, because
over the period T, the average of sin^{2} (ωt  kx) = 1/2 )
We have also:
I = U /Δt/ΔA = u_{em} ΔV /Δt/ΔA = u_{em} Δx /Δt
= c u_{em}. Where c = ΔV /ΔA = Δx , and c = Δx /Δt. Hence:
I = c ε_{o} E^{2}. We find the expreesion of the Pointing vector.
2. Maining of the wave function
We remark then, the intensity of the wave is proportional to the square of the
wave function. That is, if we write Ψ = E = E_{o} sin (ωt  kx), we can
simply write I = Constant Ψ^{2}. We can then express the total intensity as
I_{tot} = Σ I = Constant Σ Ψ^{2}.
The probability to find a particle light in the elementary volume dV, at
the time "t" is P(t) = I/I_{tot} = Ψ^{2}/Σ Ψ^{2} =
Ψ^{2} dV/∫ Ψ^{2}dV.
This integrale (from  ∞ to + ∞) is normalized to 1. We have then:
P(t) = Ψ ^{2} dV. As the wave function Ψ is often complex, we write instead:
P(t) = Ψ^{2} dV
P(t) = Ψ^{2} dV is the probabilty to find the particlewave at the
time t in the volume dV
Ψ^{2} is called the probability density of the wave Ψ.
This is what Max Born (1928) has set.
©: The scientificsentence.net. 2007.


