THe Born Rule, in English, states that the probability density of finding a particle at a given point is proportional to the magnitude of the particle's wavefunction at that point, squared.

A classical wave has an amplitude A. In a mechanical medium, the displacement of particles follows a path along the wave at that amplitude. A quantum particle does not directly follow the path of the wave function, but rather the wave function, or the "probability amplitude", provides a spread that, when squared, gives the probability density of finding the particle at a given point.


The probability current is essentially a flux of probability. The wave function spreads with time across space. We have


    𝗷 = ħ/2mι (Ψ* ∇Ψ - Ψ ∇Ψ*).

    recognize p̂ = -ιħ∇.

    𝗷 = 1/2m (Ψ* p̂ Ψ - Ψ p̂ Ψ*).

    this is in the position basis


∂/∂t ρ + ∇⋅𝗷 = 0 is the continuity equation, a statement that the density of a system can only lose quantity equal to its divergence.

the probability density is Ψ*Ψ.

in one dimension,
    
    ∂/∂t Ψ*Ψ + d/dx j = 0.

(∂/∂t Ψ*) Ψ + (Ψ* ∂/∂t Ψ) + d/dx j = 0.

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