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zheng89120
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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.
Can someone explain what a propagator is and does?
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In summary, the conversation discusses the use of the propagator in quantum mechanics, which is a mathematical tool used to calculate the probability (amplitude) of a quantum state evolving from one time to another. Sakurai's book is recommended for further understanding of the concept.
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juanrga
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zheng89120 said:
A propagator propagates
, for instance a quantum state from time t1 to time t2.- #3
kith
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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'>".
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'>".
Related to Can someone explain what a propagator is and does?
1. What is a propagator?
A propagator is a mathematical tool used in quantum mechanics to describe the time evolution of a quantum state. It tells us how a quantum state evolves over time, taking into account the effects of external forces.
2. How does a propagator work?
A propagator works by considering the initial state of a quantum system and then calculating how it evolves over time. This is done by integrating the system's Hamiltonian, which describes the energy of the system, over a specific period of time.
3. What does a propagator tell us?
A propagator tells us the probability of a quantum system transitioning from one state to another over a specific period of time. It also takes into account the quantum nature of the system, which means that the probabilities are not always deterministic.
4. Why is a propagator important?
A propagator is important because it allows us to make predictions about the behavior of quantum systems. By using the propagator, we can calculate the probability of a system transitioning from one state to another, which is crucial in understanding and studying quantum mechanics.
5. Are there different types of propagators?
Yes, there are different types of propagators depending on the specific system being studied. Some common examples include the position propagator, momentum propagator, and time-evolution propagator. Each type of propagator is used to describe a different aspect of the quantum system's behavior.
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