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The Derivation of the Source and Current Densities of a Quantum Field |
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Let ψ be the wave function for a non-relativistic quantum field. It satisfies the time-dependent Schrödinger equation
Let ψ* denote the complex conjugate of ψ. The complex conjugate of the above equation is
Multiply the first of the above equations by iψ* to obtain
Multiply the second of the above equations by −iψ to obtain
When the two equations created by the two multiplication operations are added together the result is
The Laplacian operation is equal to the divergence of the gradient of its scalar argument; i.e., ∇·(∇(φ))]. Thus
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
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.
A free particle has a wave function of the form
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
The probability density current J is then
Source:
Gordon Kane, Modern Elementary Particle Physics, Addison-Wesley Publishing Co., New York, 1987, p 22.
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