Question

Why schrodinger equation is not invariant relativistically

Answer 1

Not sure why no one has brought this up, but there’s a fairly straightforward answer to this question…

Let’s just look at the Schrodinger equation for a free particle we have

iℏ∂∂tψ(x,t)=−ℏ22m∇2ψ(x,t)

where the Hamiltonian is just the Newtonian kinetic energy:

H=Etotal=p22m=12mv2

Clearly this is not the relativistic dispersion relation derived from the Lorentz invariant free-particle 4-momentum, E2=p2c2+m2c4, from which we could substitute E=iℏ∂∂t and p=−iℏ∇ to get

−ℏ2∂2∂t2ψ=(−ℏ2∇2c2+m2c4)ψ

or, after rearranging terms to the more familiar relativistic wave equation:

∇2ψ+1c2∂2∂t2ψ=m2c2ℏ2ψ

This equation, the Klein-Gordon equation, can be shown to be Lorentz invariant as it should stand to reason having been derived from the Lorentz invariant 4-momentum. Of course this equation will not suffice as a relativistic quantum mechanical wave function as it does not conserve probability, but it does expose the reasoning behind why the Schrodinger equation is not Lorentz invariant.

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