mirror of
https://asciireactor.com/otho/phy-520.git
synced 2024-11-22 15:35:05 +00:00
28 lines
1.1 KiB
Plaintext
28 lines
1.1 KiB
Plaintext
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.
|
||
|
||
... |