Can someone explain what a propagator is and does?

[zheng89120]

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.

 

[juanrga]

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'>".