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Explanation of a failed approach to relativize Schrodinger equation

  1. Oct 7, 2012 #1
    I'm reading the Wikipedia page for the Dirac equation



    I am not sure how one gets a new [itex]\rho[/itex] and [itex]J^\mu[/itex]. How does one do to derive these two? And can anyone show me why the expression for density not positive definite?
     
  2. jcsd
  3. Oct 7, 2012 #2

    mfb

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    I think those formulas are guessed, and it can be shown that they satisfy the relevant equations afterwards. As you can see from the nonrelativistic probability flow, it is nothing completely new...

    Consider a function ##\psi## where ##\rho>0## and look at ##\psi^*##.
    ##\rho(\psi)=-\rho(\psi^*)##
     
  4. Oct 7, 2012 #3

    Bill_K

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    Take ψ to be a plane wave, ψ(x,t) = ei(k·x - ωt), which will be a solution provided ω2 = k2 + m2. For this solution, ρ = ħω/m, obviously positive/negative whenever ω is positve/negative. For the general solution which is a superposition of plane waves, ρ is an integral over the positive frequency solutions minus an integral over the negative frequency ones.

    According to the continuity equation, Q = ∫ρ d3x is a conserved quantity. Although the 'derivation' of this usually consists of simply writing it down, its existence is no accident. For a complex scalar field, Q represents the total charge. 'Charge' can mean either the ordinary electric charge or some other charge such as strangeness.
     
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