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In quantum mechanics, physical observables (e.g., energy, momentum,
position, etc.) are represented mathematically by operators. For
instance, the operator corresponding to energy is the Hamiltonian
operator

(31) 
where i is an index over all the particles of the system. We have
already encountered the singleparticle Hamiltonian in equation
(25). The average value of an observable A represented
by an operator
for a quantum molecular state
is given by the ``expectation value'' formula

(32) 