ABSTRACT
Up to this point in our study of quantum mechanics, we have ignored relativistic effects and taken spin as an extra observed property of particles with no specific model. In formulating an appropriate theory for the extension of quantum mechanics to include relativistic effects, it is logical to begin with the Hamiltonian based on our knowledge of relativity, where we have
Hˆ = c[(pˆ− eA)2 +m2c2]1/2 + eφ , (11.1)
where the first portion portion comes from W 2 = p2c2 + m2c4 using the canonical momentum pˆ = γmv + eA when there is a magnetic field present represented by the vector potential A, and then we have added the scalar electrostatic potential. This form for the Hamiltonian is very formidable, since it includes a square root with an operator inside (pˆ → (~/i)∇). By rearranging and squaring, Equation (11.1) can be put into a more symmetric form that avoids this difficulty, so that
(Hˆ − eφ)2 − (pˆ− eA)2c2 = m2c4 . (11.2)
One possible interpretation of this equation is that we should make the usual substitutions
pˆ→ ~ i ∇ and Hˆ → −~
i ∂
∂t ,
which leads to[( −~ i ∂
∂t − eφ
)2 − ( ~ i ∇− eA
)2 c2
] ψ = m2c4ψ . (11.3)
This equation is called the Klein-Gordon equation, and represents one possible way to generalize the Schro¨dinger equation to include relativistic effects.
