Applications of quantum mechanical hypervirial theorems

Thesis



Two applications of the quantum mechanical hypervirial theorem are discussed. This states that for any system having eigenstates A and B of Hamiltonian H, an operator W has the matrix element relation

(A,(HW andminus; WH) B) = (E_A andminus; E_B) (A, W B)

providing W is such that H is Hermitean between A and W B.

The first application uses the operator W = e^itx to derive a further relationship between certain matrix elements. For one dimensional systems in which the potential V(x) involves only positive powers of x, this relationship becomes a differential equation in

f(t) = (A, e^itx B)

This differential equation has been solved analytically for the harmonic oscillator and in series for the quartic oscillator which were taken as example systems.

The second application is the off-diagonal hypervirial method, which seeks to require a wave function A to satisfy a number of hypervirial relations. In particular the wave function A is forced to obey as many conditions as it has variable parameters. To accomplish this a computational algorithm was developed which allowed one of these parameters to be non-linear. The results for oscillator test systems are not encouraging. The systems tested were the quartic and sixth power oscillators with a trial basis of harmonic oscillator eigenfunctions. In general the results were not as good as those obtainable by the variational method.

Authors: Nash, J; Nash, John C.

• Publication date: 1972
• DOI: 10.5287/ora-pv6ea0g69
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English