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The Derivation of the Source and Current
Densities of a Quantum Field

Let ψ be the wave function for a non-relativistic quantum field. It satisfies the time-dependent Schrödinger equation

2mi(∂ψ/∂t) + ∇²ψ = 0

Let ψ* denote the complex conjugate of ψ. The complex conjugate of the above equation is

−2mi(∂ψ*/∂t) + ∇²ψ* = 0

Multiply the first of the above equations by iψ* to obtain

−2mψ*(∂ψ/∂t) + iψ*∇²ψ = 0

Multiply the second of the above equations by −iψ to obtain

−2mψ(∂ψ*/∂t) − iψ(∇²ψ* = 0

When the two equations created by the two multiplication operations are added together the result is

−2m[ψ*(∂ψ/∂t)+ψ(∂ψ*/∂t)] + i[ψ*∇²ψ−ψ∇²ψ*] = 0
which is equivalent to
−2m(∂(ψψ*)/∂t) + i[ψ*∇²ψ−ψ∇²ψ*] = 0

The Laplacian operation is equal to the divergence of the gradient of its scalar argument; i.e., ∇·(∇(φ))]. Thus

[ψ*∇²ψ+ψ(∇²ψ*] = (∇²ψ)ψ* − (∇²ψ*)ψ
∇·[(∇ψ)ψ* − (∇ψ*)∇]

If ρ is defined as ψψ*, which is also expressed as |ψ|², and ψ*∇ψ − ψ∇ψ* is defined as 2mJ (the red coloring denoting that J is a vector) then the previous equation may be expressed with some rearrangement as

(∂ρ/∂t) + ∇·J = 0

which is the equation, for example, of electrical charge and its flow. The variable ρ is the charge density and J is the density of current flow. In the context of quantum theory a normalized ρ would be the probability density and J the probability density flow.

The theorem for the above is that if a function ψ satisfies the time dependent Schrödinger equation then ψ may be used to generate a probability density function ρ and a probability density flow.

Illustration

A free particle has a wave function of the form

ψ(r, t) = Q·exp(i(p·r−ωt))

where r is the position vector of the particle and p is its momentum. Q is a parameter and ω is the frequency of oscillation.

The probability density ρ works out to be

ρ = Q²

The probability density current J is then

J = ρ(p/m)


Source:

Gordon Kane, Modern Elementary Particle Physics, Addison-Wesley Publishing Co., New York, 1987, p 22.

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