Quantization of the Nonrelativistic Hamiltonian

Using the correspondence principle, (4.1), we get the time-dependent Schrodinger equation  

This equation is obviously not Lorentz invariant— it has x, y, and z appearing quadrat-ically but t appearing linearly, which violates the relativistic principle of equivalence of spatial and temporal variables. Since we know that the nonrelativistic classical Hamiltonian is not Lorentz invariant, it is no surprise that neither is the nonrelativistic 

In most quantum chemical applications the electrons are not free, but are subject to electrostatic potentials in the form of Coulomb interactions with nuclei and other electrons. These are accounted for in the Hamiltonian by adding a term involving the scalar potential and the charge of the electron (—e)  

However, if we are concerned about Lorentz invariance, we should at this point remember that the scalar potential is only one component of a four-vector A = (A, i t /c). If the scalar potential modifies %, or equivalently E, then we would expect the vector potential to modify the momentum, which accounts for the remaining components of the four-vector. 

From classical mechanics (e.g., Goldstein 1950), we can show that the presence of a vector potential requires that the Hamiltonian function must be constructed using the kinetic momentum (or mechanical momentum), which is the momentum that is given in nonrelativistic theory by m. We must express this momentum in terms of the canonical momentum of Lagrangian mechanics, because it is the canonical momentum to which the quantization rule p —ihV applies. Here (and hereafter) we will use p for the canonical momentum and n for the kinetic momentum. The relation between the two is 

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We start this chapter by reexamining the quantization of the nonrelativistic Hamiltonian and draw out some features that will be useful in the quantization of the relativistic Hamiltonian. We then turn to the Dirac equation and sketch its derivation. We discuss some properties of the equation and its solutions, and show how going to the nonrelativistic limit reduces it to a Schrodinger-type equation containing spin. 

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