Operators and Quantum Mechanics

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

$$\hat{H} = - \frac{\hbar^2}{2} \sum_i \frac{1}{m_i} \nabla_i^2 + V$$ (31)

where $i$ is an index over all the particles of the system. We have already encountered the single-particle Hamiltonian in equation (25). The average value of an observable A represented by an operator $\hat{A}$ for a quantum molecular state $\psi({\bf r})$ is given by the ``expectation value'' formula

$$<A> = \int \psi^{*}({\bf r}) \hat{A} \psi({\bf r}) d{\bf r}$$ (32)