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HigherOrderPT

# HigherOrderPT - Higher Order Perturbation Theory Michael...

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Higher Order Perturbation Theory Michael Fowler 03/07/06 The Interaction Representation Recall that in the first part of this course sequence, we discussed the Schrödinger and Heisenberg representations of quantum mechanics here . In the Schrödinger representation, the operators are time-independent (except for explicitly time-dependent potentials) the kets representing the quantum states develop in time. In the Heisenberg representation, the kets stay the same, the time dependence is in the operators. These differing representations describe the same physics— matrix elements of operators between kets must be the same in both. The most natural to use depends on the problem at hand. In the classical limit, for example, the Heisenberg operators have the time dependence of the corresponding classical operators. In fact, for perturbation theory problems with a time-dependent potential, an intermediate representation, the interaction representation , is very convenient. Using a subscript S to denote the Schrödinger representation, ( ) ( ) ( ) ( ) ( ) 0 , S S S S S S d i t H t H V t dt ψ ψ ψ = = + = t we define the interaction representation by the unitary transformation ( ) ( ) 0 / S iH t I S t e t ψ ψ = = so the interaction representation kets and the Schrödinger representation kets coincide at t = 0, and if the interaction were zero, the interaction representation kets would be constant in time, like those in the Heisenberg representation. For nonzero ( ) V t , then, the time development of the interaction representation kets is entirely due to ( ) V t , and is easily found by differentiating both sides of the equation: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 / 0 / / 0 0 / / ( ) ( ) , S S S S S iH t I I S iH t iH t I S S iH t iH t S I I I d d i t H t e i t dt dt H t e H V t e t e V t e t V t t ψ ψ ψ ψ ψ ψ ψ = − + = − + + = = = = = = = = = I where we have introduced the interaction representation operator V I ( t ), defined by ( ) ( ) 0 0 / / .

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