Why is state transition probability symmetric?

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SUMMARY

The discussion centers on the symmetry of state transition probabilities in finite dimensional quantum systems, specifically using the Trace inner product. When transitioning from state S1 to eigenstate S2, the probability is calculated as p = Trace(S1*S2). Conversely, transitioning from state S2 to S1 yields the same probability p, highlighting the inherent symmetry in quantum mechanics. This symmetry is rooted in the time-symmetry principle of nonrelativistic quantum mechanics, where the probabilities are proportional and sum to 1 due to the trace condition of the states.

PREREQUISITES
  • Understanding of finite dimensional quantum mechanics
  • Familiarity with the concept of eigenstates and eigenvalues
  • Knowledge of the Trace operation in linear algebra
  • Basic principles of time-symmetry in quantum mechanics
NEXT STEPS
  • Explore the mathematical foundations of the Trace inner product in quantum mechanics
  • Study the implications of time-symmetry in nonrelativistic quantum systems
  • Investigate the role of eigenstates and eigenvalues in quantum state transitions
  • Examine experimental setups that demonstrate state transitions in quantum systems
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Quantum physicists, researchers in quantum mechanics, and students studying the principles of state transitions and time-symmetry in quantum systems will benefit from this discussion.

normvcr
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Restricting to finite dimensional QP, suppose a system is in a state S1, an experiment is done, and state S2 is one of the eigenstates (assume all eigenvalues are distinct). The probability that the system transitions from S1 to S2 is p = Trace( S1*S2), using state operator notation. On the other hand, if a system is in state S2, a different experiment is done, and state S1 is one of the eigenstates,, the probability that the system transitions from S2 to S1 is, again, p, due to the symmetry of the Trace inner product. Is there a physical rationale why these two state transition probabilties are the same?
 
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It is not directly the probability, it is just proportional to it.

This is the time-symmetry of (nonrelativistic) quantum mechanics.
 
Time symmetry is an interesting perspective. The difficulty I have with this, though, is that the two directions of state transition require two different experiments.
BTW, it is directly probability, as the p's add up to 1, owing to the states having trace 1.
 

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