Background and Formalism

Quantum mechanics represents one of the cornerstones of modem physics. Though there were a variety of different clues (such as the ultraviolet catastrophe associated with blackbody radiation, the low-temperature specific heats of solids, the photoelectric effect and the existence of discrete spectral lines) which each pointed towards quantum mechanics in its own way, we will focus on one of these threads, the so-called wave-particle duality, since this duality can at least point us in the direction of the Schrodinger equation. 

When written in time-independent form for a single particle interacting with a potential V (r), the Schrodinger equation takes the form 

The relation between a given classical Hamiltonian and its quantum counterpart depends upon the transcription between classical dynamical variables and certain operators that act upon the space of wave functions. The basic idea is to take the classical Hamiltonian, replace the relevant dynamical variables by their operator analogs, and then to solve the resulting differential equation. 

This equation is a restatement of eqn (3.1) and we have introduced the notation that a above a quantity signals that quantity as an operator. Note that we are considering the time-independent setting and for now have made reference only to a single particle. To make the classical-quantum transcription more complete, the recipe is to replace classical dynamical variables by their operator analogs according to 

The Schrodinger equation corresponding to a given classical Hamiltonian is then obtained by replacing all of the dynamical variables in the original Hamiltonian with their operator analogs. 

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