# Probability and probability amplitude

• phyky
In summary, the conversation discussed the concepts of probability amplitudes and probabilities in quantum mechanics. The speaker questioned whether <A|A> is equal to a^2+b^2 and if there is an interference term in <A|A>. They also mentioned the difference between transition amplitudes and probability amplitudes, and asked about the interference between orthogonal states. It was clarified that for a normalized wave function, a^2+b^2=1, but <A|A> is not equal to a^2+b^2 and may also include a*b and ab*. It was suggested that using <A|A>=|A>|^2 may help in understanding the concept.
phyky
|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?!

|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?

## 1. What is probability and how is it different from probability amplitude?

Probability is a measure of the likelihood that an event will occur. It is represented by a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. Probability amplitude, on the other hand, is a complex number used in quantum mechanics to describe the probability of different outcomes of a quantum system. It is related to probability by taking the square of its absolute value.

## 2. How is probability amplitude calculated?

Probability amplitude is calculated using mathematical equations, such as the Schrödinger equation, which takes into account the initial state of a system and how it evolves over time. It also involves the use of wave functions, which represent the probability amplitude of a particle being at a certain position in space.

## 3. What is the difference between classical and quantum probability?

In classical probability, events are described by classical mechanics, which follows the laws of classical physics. The outcomes of events are deterministic and can be predicted with certainty. In quantum probability, events are described by quantum mechanics, which follows the laws of quantum physics. The outcomes of events are probabilistic and can only be predicted with a certain degree of probability.

## 4. How does the concept of superposition relate to probability amplitude?

In quantum mechanics, superposition is the state of a system where it exists in multiple states at the same time. This is represented by a linear combination of the different possible states, each with its own probability amplitude. The probability of observing a particular outcome is determined by the square of its probability amplitude.

## 5. Can probability amplitude be negative?

Yes, probability amplitude can be negative. This is because it is a complex number, which means it has both a real and imaginary component. The negative sign in the imaginary component is what allows for interference effects in quantum systems, where the probability of an event can be canceled out or enhanced by the probability amplitudes of other events.

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