# 3 - p. 21 MIT Department of Chemistry 5.74, Spring 2004:...

This preview shows pages 1–3. Sign up to view the full content.

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: p. 21 MIT Department of Chemistry 5.74, Spring 2004: Introductory Quantum Mechanics II Instructor: Prof. Andrei Tokmakoff SCHRÖDINGER AND HEISENBERG REPRESENTATIONS The mathematical formulation of the dynamics of a quantum system is not unique. Ultimately we are interested in observables (probability amplitudes)—we can’t measure a wavefunction. An alternative to propagating the wavefunction in time starts by recognizing that a unitary transformation doesn’t change an inner product. ϕ j ϕ = ϕ j U † U ϕ i i For an observable: A ϕ i = ( ϕ j U † ) A U ϕ ) = ϕ U † AU ϕ ( ϕ j i j i Two approaches to transformation: 1) Transform the eigenvectors: ϕ i → U ϕ i . Leave operators unchanged. 2) Transform the operators: A → U † AU . Leave eigenvectors unchanged. (1) Schrödinger Picture : Everything we have done so far. Operators are stationary. Eigenvectors evolve under U t , t ( ) . (2) Heisenberg Picture : Use unitary property of U to transform operators so they evolve in time. The wavefunction is stationary. This is a physically appealing picture, because particles move – there is a time-dependence to position and momentum. Schrödinger Picture We have talked about the time-development of ψ , which is governed by ∂ i = ψ = H ψ in differential form, or alternatively ∂ t ψ ( ) t = U t , t 0 ) ψ ( t 0 ) in an integral form. ( p. 22 ∂ A Typically for operators: = 0 ∂ t What about observables? Expectation values: A(t) = ψ ( ) ψ ( ) or......
View Full Document

## This note was uploaded on 11/28/2011 for the course CHEM 5.74 taught by Professor Robertfield during the Spring '04 term at MIT.

### Page1 / 7

3 - p. 21 MIT Department of Chemistry 5.74, Spring 2004:...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online