Why Does Transition Amplitude Not Equate to Time Evolution in Quantum Mechanics?

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Discussion Overview

The discussion revolves around the relationship between transition amplitudes and time evolution in quantum mechanics. Participants explore the implications of these concepts in the context of state evolution, measurement, and perturbations, with a focus on energy eigenstates and superpositions.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Technical explanation

Main Points Raised

  • One participant questions why the probability amplitude for transitioning from one state to another is represented as the inner product of the "another state" and the propagator acting on the initial state, suggesting confusion over its application to general states versus eigenstates.
  • Another participant argues that measurements can lead to new states that differ from the initial state, providing an example involving spin states to illustrate that a non-zero probability can exist for transitioning to a different state.
  • A subsequent reply challenges the idea that an energy eigenstate can evolve into a superposition of other eigenstates, emphasizing that this should not occur regardless of the time elapsed.
  • One participant introduces the concept of transition amplitudes in the presence of a perturbation Hamiltonian, suggesting that external influences can facilitate transitions between states.
  • Another participant provides a simplified explanation, stating that if a system is in a specific state, it can have a certain probability to transition to another state, implying a shared component between the states.
  • It is noted that transition amplitudes do not equate to time evolution, as they represent the probability of finding the system in a new state upon measurement, which involves physical interaction with the system.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between transition amplitudes and time evolution, with some asserting that measurements can lead to state changes while others maintain that energy eigenstates cannot evolve into superpositions. The discussion remains unresolved with multiple competing perspectives.

Contextual Notes

Participants reference specific scenarios involving energy eigenstates and measurement outcomes, indicating a reliance on definitions and assumptions that may not be universally accepted. The role of perturbations in state transitions is also highlighted but not fully explored.

HomogenousCow
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I cannot see why the probaability ampltiude for an initial state to turn into another state is the inner product between the "another state" and the propogator acting on the initial state, since this is just equivalent to the inner product between the "another state" and the evolved initial state at the time of the evaluation.
For example, if I try it with the initial state as an energy eigenstate |1> and want to know the probability of it turning into a state |1>+|2> ( normalization constants implied), I would get a non zero result, but then that must be wrong because my initial state had zero probability of being in |2>. Thus my initial state should never be able to turn into |1>+|2>.
I can only see this working with eigenstates but not general ones, what am I missing?
 
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When you do the measurement, of course you can get to a new state that is different from the one you started with. When you start with a spin in +z direction and ask for the amplitude to measure a spin in +x-direction, the probability will be non-zero, eben though +x can be wirtten as a linear combination of +z and -z.
 
but then how does it reconcile my example, an energy eigen state cannot evolve into a superposition of other eigenstates no matter how much time has passed
 
As far as I knwo, there is a transition amplitude when there is a perturbation hamiltonian H1 so that <2|H1|1> <> 0. There are other facts as the spontanous emision that can make a state to fall without need of H1 into another of less energy. Anyway the main idea is that an external perturbation can cause a change from a state to another.
 
not talking about that here, bump need exlplanation
 
there is a simple explanation.if it is in state|1> then (with normalization factor of 1/√2),it has 50% probability to turn into the other one where |1> shares of course 50% part.
 
The transition amplitude is not the time evolution. It gives you the probability of finding the system in the new state when actually doing a measurement. Since this will force the system to a new state, the measurement is a physical interaction with the system.
 

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