AN ATOMIC POPULATION AS THE EXPECTATION VALUE OF A QUANTUM OBSERVABLE

Dirac defines an observable to be a real dynamical variable with a complete set of eigenstates. It is shown that the density operator rho = SIGMA(i) delta(r(i) - r), is a quantum-mechanical observable whose expectation value is the particle density and that the integral form of this operator, the number operator N, is also a quantum-mechanical observable whose expectation value is the average number of particles. The principle of stationary action defines the expectation value of the equation of motion for every observable. Using this principle it is demonstrated that an atomic population is the expectation value of the observable N when rho is the electron density operator. An atom and its population are defined in terms of experimentally measurable expectation values of the observables rho and N.