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Confusion about How Dirac discovered Dirac's Equationby petergreat
Tags: dirac's equation 
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#1
Nov1810, 04:32 PM

P: 270

When I learned about Dirac's Equation, textbooks usually say that the earlier KleinGordon equation isn't linear in time derivative, contrary to what we expect from the timedependent Schrodinger equation, therefore Dirac had to come up with a version that's linear. However, I think this doesn't really make sense. KleinGordon equation is perfectly acceptable if electrons were bosons. The only justification for Dirac's equation is the fermionic nature of electrons.
The "linear time derivative" argument just seems to be some irrelevant outdated intuition from the structure of nonrelativistic QM, and its only benefit is that we preserve the form of Schrodinger's equation and we can still talk about "Hamiltonian" and "energy levels" in a rather similar manner, e.g. in atomic physics, without going into quantum field theory. In contrast, for the KleinGordon equation you must treat it as a quantum field to recover the concept of Hamiltonian (now a field Hamiltonian) and a timedependent Schrodinger equation which by definition is linear in time derivative. In short, the "linear time derivative" property just makes semiclassical treatment easy, and doesn't really have physical content. Does what I say make sense? Or am I confused about something? 


#2
Nov1810, 04:45 PM

P: 270

I'll summary my point again. Dirac's equation, with a suitable choice of representation for the gamma matrices, happens to look like a timedependent Schrodinger equation, while the KleinGordon equation lacks this property. However in view of QFT, this fact is not meaningful, because both of the are just classical field equations.



#3
Nov2010, 11:43 AM

P: 36

A system that evolves according to a DE that is second order in time needs state information at two previous instants(or state plus derivative) implying the state needs some sort of 'memory' to evolve in time. A 2nd order system also generates those awkward advanced solutions. At least that's the way I understand it. I quote from section 27 of "Princples of Quantum Mechanics": Dirac postulates: "...the state at one time determines the state at another time.." . 


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