Off—Diagonal Hypervirial Theorem and Its Applications

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Abstract
A hypervirial operator, by definition, is a time‐independent linear operator with an arbitrary functional structure expressed in terms of the dynamical variables of the system under consideration. In the energy representation, the diagonal matrix elements of the hypervirial operator are constant in time; this is known as the hypervirial theorem. The off‐diagonal matrix elements of the hypervirial operator in the energy representation oscillate in time with frequencies related to the energy differences between their corresponding stationary states of the system. This is known as the off‐diagonal hypervirial theorem. The latter theorem is as powerful as the former for dealing with many quantum‐mechanical problems. Several aspects of the possible applications of the theorem to the problem of construction and improvement of wavefunctions are considered. As a special example, the restricted fourth alternative expression for dipole transition matrix elements is derived for pair functions not belonging to states with zero orbital angular momentum. A scheme is also outlined for improving wavefunctions of the excited states. The statistical implications of the sum rule, which is expressed in terms of an appropriate hypervirial operator, are discussed in connection with uncertainty principles.