Hamiltonian operator

The hamiltonian is the quantum operator corresponding to the measurement of the total energy of a particle.

Position-momentum system

For a typical position-momentum single-particle system with time-independent potential, the hamiltonian is given by

where and are the position and momentum operators respectively and is the potential. Thus

Verification of hermiticity

Let where is some infinite space. Then

Since is real-valued, applying ^GE1 twice, and using the fact that both wavefunctions and all derivatives are zero at the boundary

as required.

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