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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......
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- Spring '04
- Chemistry, H0, ∂t, heisenberg picture