Hamiltonian Dirac, time-independent

The Dirac equation is invariant to Lorentz transformations [8], a necessary requirement of a relativistic equation. In the limit of large quantum numbers the Dirac equation reduces to the Klein-Gordon equation [9,10]. The time-independent form of Dirac s Hamiltonian is given by... 

A proper definition of (quasi-)particle-creation and (quasi-)particle-annihilation operators an and a is provided by diagonalization of the (time-independent) unperturbed part Ho = //ext+Z/e-e of the total Hamiltonian. After the iteration is performed (e.g. on the Dirac-Fock level) the latter may be cast into the form... 

In this section we shall discuss the interactions that arise upon the introduction of electromagnetic fields in the relativistic electronic Hamiltonian, and we shall also consider the form of electromagnetic interactions in the non-relativistic limit. To simplify matters, we shall first limit our attention to one-electron systems. Consider the time-independent Dirac equation for a free particle... 

The functions in Fig. 7.5 approximate the Dirac delta function. Draw graphs of the corresponding functions that approximate the Heaviside step function with successively increasing accuracy. Quantum mechanics postulates that the present state of an undisturbed system determines its future state. Consider the special case of a system with a time-independent Hamiltonian H. Suppose it is known that at time to the state function is fo). Derive Eq. (7.101) by substituting the expansion (7.66) with g = i/i into the time-dependent Schrodinger equation (7.97) multiply the result by i/i, integrate over all space, and solve for c . 

We now have the necessary tools to discuss the transformation of the Dirac Hamiltonian. We consider the time-independent equation with a static scalar potential, (4.74)... 

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