Can someone explain what a propagator is and does?

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A propagator is a mathematical tool used in quantum mechanics to describe the evolution of quantum states over time, specifically from an initial time t1 to a final time t2. It is defined within the framework of Hilbert space, allowing for expressions such as and <ψ'(t2)|ψ(t1)>. Shankar's "Simple Problems in One Dimension" and Sakurai's texts provide foundational insights into the concept and its applications in non-relativistic quantum mechanics.

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  • Understanding of quantum mechanics principles
  • Familiarity with Hilbert space formalism
  • Knowledge of unitary evolution in quantum systems
  • Basic grasp of quantum state notation (e.g., |x>, |ψ(t)>)
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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.
 
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zheng89120 said:
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 :biggrin:, for instance a quantum state from time t1 to time t2.
 
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'>".
 

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