# Can someone explain what a propagator is and does?

## Main Question or Discussion Point

So, Shankar in his famous book started to use the propagator a bunch of times in his Simple Problems in One Dimension chapter, and I have been confused to what it is and does.

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So, Shankar in his famous book started to use the propagator a bunch of times in his Simple Problems in One Dimension chapter, and I have been confused to what it is and does.
A propagator propagates , for instance a quantum state from time t1 to time t2.

kith
I recommend Sakurai's book for some nice descriptive views on the non-relativistic propagator.

Starting from the abstract Hilbert space formalism, it may be helpful to write down a few equivalent expressions for the propagator: <x'|U(t2-t1)|x> = <x'|U+(t2)U(t1)|x> = <ψ'(t2)|ψ(t1)>

So you take a state |x>, let it evolve unitarily from initial time t1 to final time t2 and ask "what's the probability (amplitude) to find it in state |x'>".