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28 lines
1.1 KiB
Plaintext
28 lines
1.1 KiB
Plaintext
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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.
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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.
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The probability current is essentially a flux of probability. The wave function spreads with time across space. We have
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𝗷 = ħ/2mι (Ψ* ∇Ψ - Ψ ∇Ψ*).
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recognize p̂ = -ιħ∇.
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𝗷 = 1/2m (Ψ* p̂ Ψ - Ψ p̂ Ψ*).
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this is in the position basis
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∂/∂t ρ + ∇⋅𝗷 = 0 is the continuity equation, a statement that the density of a system can only lose quantity equal to its divergence.
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the probability density is Ψ*Ψ.
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in one dimension,
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∂/∂t Ψ*Ψ + d/dx j = 0.
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(∂/∂t Ψ*) Ψ + (Ψ* ∂/∂t Ψ) + d/dx j = 0.
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