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/m²).
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 E²; 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 = E²/c μ_o.
As ε_oμ_o c² = 1, then: S = c ε_oE². ( We have also the average S = (c ε_o/2)E_o², because over the period T, the average of sin² (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 E². 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 Ψ². We can then express the total intensity as I_tot = Σ I = Constant Σ Ψ².

The probability to find a particle light in the elementary volume dV, at the time "t" is P(t) = I/I_tot = Ψ²/Σ Ψ² = Ψ² dV/∫ Ψ²dV.
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

This is what Max Born (1928) has set.