Probability and probability amplitude

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SUMMARY

The discussion centers on the concepts of probability and probability amplitude in quantum mechanics, specifically the relationship between the inner product and its interpretation as probability. The participants clarify that represents the probability amplitude, while ||^2 denotes the probability. They emphasize that interference terms arise in transition amplitudes, such as , and not in the self-inner product . The normalized wave function |A> satisfies the condition a^2 + b^2 = 1, but includes additional terms that account for interference when transitioning between states.

PREREQUISITES
  • Understanding of quantum mechanics terminology, including "probability amplitude" and "inner product."
  • Familiarity with the concept of normalized wave functions in quantum states.
  • Knowledge of transition amplitudes and their role in quantum state interactions.
  • Basic grasp of interference in quantum mechanics and its mathematical representation.
NEXT STEPS
  • Study the mathematical formulation of quantum states and their inner products.
  • Learn about transition amplitudes and how to calculate interference terms using ||^2.
  • Explore the implications of normalized wave functions and their significance in quantum mechanics.
  • Investigate the role of interference in quantum systems, particularly between orthogonal states.
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Students and professionals in quantum mechanics, physicists analyzing quantum states, and anyone interested in the mathematical foundations of probability in quantum theory.

phyky
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|A>=a|0>+b|1>
<A|A>=probability amp |<A|A>|^2=probability
If the state is same, so <A|A>= a^2+b^2=1. then i find the probability and no interference term in it!
so can i say <A|A>=|<A|A>|^2 and no square is needed? since if i square it become a^4+b^4=??
what wrong with it! and where should i place interference term?!
 
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|A> is the probability amplitude.
<A|A> is the probability.
Maybe you are confusing the situation with transition amplitudes where <A|B> is the amplitude of transition from A to B?
 
so if transition from A to B it need |<B|A>|^2 to find interference term? and <A|A> show only probability but nothing about interference? but its outer product |A><A| does show interference?
 
|A>=a|0>+b|1> so <A|A>=a^2<0|0>+b^2<1|1>+a*b<0|1>+b*a<1|0> where <i|j>=0 <i|i>=1. where the interference term?
 
You looking for interference between orthogonal states then. Why would you expect there to be some?
 
As for a normalized wave function |A>, a^2+b^2=1. However, <A|A>is not equal to a^2+b^2but a^2+b^2+a*b+ab* , generally speaking, the latter is not equal to 1. <A|A>=|A>|^2, would it help?
 

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