# Redshift Academy

Wolfram Alpha:

Last modified: January 27, 2018

The Schrodinger, Heisenberg and Dirac Pictures ---------------------------------------------- In the Schrodinger picture the operators do not change with time but the states do. In the Heisenberg picture the operators change with time but the states do not. <O> = <ψ(t)|O(0)|ψ(t)> ... Schrodinger. <O(t)> = <ψ(0)|O(t)|ψ(0)> ... Heisenberg. Using the completeness relationship (see Essential Mathematics of QM) we can expand the wavefunction, ψ, as: |ψ> = Σαj(t)|ej> |dψ/dt> = Σdαj/dt|ej> Now using the TDSE we can also write: |dψ/dt> = Σαj(t)(-iH/h)|ej> Therefore, by comparison: dαj/dt = -iαjH/h The solution to this equation is: αj(t) = αj(0)exp(-iHt/h) We cab use this information to compute the expectation value as of O(t) follows: <O(t)> = <ψ(t)|O|ψ(t)> = <Σαk*exp(iEkt/h)ek|0|Σαjexp(-iEjt/h)ej> = ΣΣαk*exp(iEkt/hjexp(-iEjt/h)<ek|0|ej> = ΣΣαkjexp(i(Ek - Ej)t/h)<ek|0|ej> = ΣΣαkj[exp(i(Ek - Ej)t/h)Okj] So the term in [] is the time evolution of the operator which can be written as: O(t) = exp(iEkt/h)Okjexp(-iEjt/h) Heisenberg Equation ------------------- Using the product rule. dO(t)/dt = (i/h)Ekexp(iEkt/h)Oexp(-iEjt/h) + exp(iEkt/h)(∂O/∂t)exp(iEjt/h) + (i/h)exp(iEkt/h)O(-Ej)exp(-iEjt/h) = (i/h)exp(iEkt/h)(EkO - OEj)exp(-iEjt/h) + exp(iEkt/h)(∂O/∂t)exp(-iEjt/h) = (i/h)(EkO(t) - O(t)Ej) + exp(iEkt/h)(∂O/∂t)exp(-iEjt/h) If we let Ej = Ek = E we get: dO(t)/dt = (i/h)[E,O(t)] + exp(iEt/h)(∂O/∂t)exp(-iEt/h) This is the Heisenberg equation. Dirac (Interaction) Picture --------------------------- In the Dirac picture, both the wavefunctions and the operators carry time-dependence. The interaction picture allows for operators to act on the state vector at different times and forms the basis for quantum field theory and many other newer methods. <O> = <ψ(t)|O(t)|ψ(t)> ... Dirac. HS = H0,S + H1,S (1 not I) = Free + Interaction A state vector in the interaction picture is defined as: ψI(t) = exp(iH0,St/h)|ψS(t)> or, ψS(t) = exp(-iH0,St/h)|ψI(t)> An operator in the interaction picture is defined as: OI(t) = exp(iH0,St/h)OSexp(-iH0St/h) For the free Hamiltonian, the interaction picture and Schrodinger pictures coincide: H0,I(t) = exp(iH0,St/h)H0,Sexp(-iH0,St/h) = H0,S(t) For the interacting Hamiltonian, the interaction picture is: H1,I = exp(iH0,St/h)H1,Sexp(-iH0,St/h) Time Evolution of States ------------------------ The Schrodinger equation for states in the interaction picture can be derived starting from the Schrodinger picture as follows: ih∂|ψS(t)>/∂t = HSS> Rewrite in the Dirac picture: ih∂exp(-iH0,St/h)|ψI(t)>/∂t = (H0,S + H1,S)exp(-iH0,St/h)|ψI(t)> Calculate, ih(-iH0,S/h)exp(-iH0,St/h)|ψI(t)> + ihexp(-iH0,St/h)∂|ψI>/∂t = H0,Sexp(-iH0,St/h)|ψI(t)> + H1,Sexp(-iH0,St/h)|ψI(t)> Simplify, H0,Sexp(-iH0,St/h)|ψI(t)> + ihexp(-iH0,St/h)∂|ψI>/∂t = H0,Sexp(-iH0,St/h)|ψI(t)> + H1,Sexp(-iH0,St/h)|ψI(t)> ihexp(-iH0,St/h)∂|ψI>/∂t = H1,Sexp(-iH0,St/h)|ψI(t)> ih∂|ψI>/∂t = exp(iH0,St/h)H1,Sexp(-iH0,St/h)|ψI(t)> Finally, ih∂|ψI>/∂t = H1,II(t)> This the SCHWINGER-TOMONAGA equation. Time Evolution of Operators --------------------------- If the operator OS itself does not have an explicit time dependence (is not a function of time) then the corresponding time evolution for OI(t) is given by: ∂OI(t)/∂t = -i/h[H1,I,H0,S] This is proven in the notes on "Time Evolution and Symmetry Operations".