Application to the Motion of a Single Particle

We can convert the Hamiltonian function for a single particle into the Hamiltonian operator by substituting the expressions for p, p and p into equation (7.5), This gives . 

Because the potential energy is a function of the coordinates alone and does not involve the momenta, it remains unchanged in this process. After multiplying out the terms, the equation becomes  

A comparison of this equation with equation (7.3) shows that this is indeed the Hamiltonian operator for a single particle, previously obtained from the Schrodinger equation. 

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