RayleighRitz

RayleighRitz - PHY4221 Quantum Mechanics I Fall 2004...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: PHY4221 Quantum Mechanics I Fall 2004 Rayleigh-Ritz linear variational method Given the eigenvalue problem of the Hamiltonian operator H : H | Φ i = E | Φ i , (1) the Variational Principle states that any normalized state vector | Ψ i for which the expectation value E ([Ψ]) ≡ h Ψ | H | Ψ i of the Hamiltonian operator H , i.e. the average energy, is stationary is an eigenvector of H , and the corresponding energy eigenvalue is the stationary value of E ([Ψ]). That is, if we could actually carry out a variation of | Ψ i , starting with some initial | Ψ Initial i and varying | Ψ i by small steps, namely by rotating | Ψ i by small amounts in the Hilbert space, such that we could end by finding the true minimum of the functional E ([Ψ]), we would attain the ground state eigenvector and eigenenergy exactly. Similarly, a local minimum would give us an excited state eigenvector and eigenenergy. Obviously, this is not feasible even with modern computers....
View Full Document

Page1 / 2

RayleighRitz - PHY4221 Quantum Mechanics I Fall 2004...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online