"Probabilty amplitude" meaning ?

[microsansfil]

Hello,

http://en.wikipedia.org/wiki/Probability_amplitude

Do you know this concept in the frame of probability theory ? If yes what is mathematical sense ?

Patrick

 

[HallsofIvy]

That is NOT a concept in mathematical probability. As the Wikipedia article linked to says, it is a concept used in Quantum Physics used to link wave properties with probability of finding a particle in a given location.

 

[microsansfil]

That is NOT a concept in mathematical probability. As the Wikipedia article linked to says, it is a concept used in Quantum Physics used to link wave properties with probability of finding a particle in a given location.

A probability amplitude, becomes a probability density after squaring. The mathematical concept of probability density is well known, but Physicists speak of probability amplitude as being a mathematical concept.

https://www.physicsforums.com/threa...or-nature-of-probability.774898/#post-4878368Patrick

 

[Stephen Tashi]

Using a complex number to represent a discrete probability distribution with two possible outcomes is an application of mathematics. It is a convention for coordinatizing.

We could make an analogy. Vectors and n-tuples of numbers are different mathematical concepts. Using n-tuples of numbers to represent vectors is so convenient that the two concepts merge in some people's minds. However, the abstract definition of a vector space can be stated without referring to n-tuples of numbers. The n-tuples are a particular way to coordinatize certain kinds of vectors. .

Explaining why it is useful to use the repesentation of probability amplitudes in QM instead of dealing with probability distributions as lists of real numbers would involve explaining the physical facts of QM. (My favorite explanation is http://henry.pha.jhu.edu/quantum.html ). It is tempting to think that mathematics ought to imitate Nature and thus it should found probability theory on probability amplitudes - similar to the way that some introductory textbooks teach the theory of vectors by defining them as n-tuples of real numbers. However, there are many applications of probability theory where using complex numbers to coordinatize a probability distribution is inconvenient..

 

[microsansfil]

Thank for your answer.

Using a complex number to represent a discrete probability distribution with two possible outcomes is an application of mathematics. It is a convention for coordinatizing.

I don't see here the concept of amplitude probability distribution.

Explaining why it is useful to use the repesentation of probability amplitudes in QM instead of dealing with probability distributions as lists of real numbers would involve explaining the physical facts of QM.

The physical meaning is clear :

http://skepticaleducator.org/wp-content/uploads/2014/08/2SlitExplanationCrackpot3.pdf
1.2 Probability amplitudes : http://www.physics.ox.ac.uk/qubit/tutes/The%20Physics%20of%20Quantum%20Mechanics,%20Binney%20and%20Skinner.pdf

Patrick

 

[Stephen Tashi]

I don't see here the concept of amplitude probability distribution.

Why would you? You wouldn't necessarily see information about to coordinatize a probability distribution in a link to the concept of a probability distribution. The sum of the values of a discrete probability distribution is 1. So when you represent a distribution with two values as a complex number you can stipulate the convention that the sums of the squares of the real and imaginary parts is 1. From a purely mathematical point of view, you could stipulate some other convention. For example, you might stipulate that of the real and imaginary parts are non-negative real numbers that (unsquared) sum to 1. The convenience of one convention or the other depends on how probability distributions behave in a particular application.

HallsofIvey is correct. The concept of a probability amplitude is not a concept used in the axiomatics of mathematical probability theory.

 

[microsansfil]

Why would you?

As a comparison with probability density may be made"Advantages of probability amplitude over probability density in quantum mechanics". Thus in the context of mathematics we hope to find a clear definition of the level of that of the probability density. in Quantum Physics, probability amplitude it is a concept used to link wave function with a probability distribution for the purpose of prediction.

Patrick

 

[Demystifier]

Any non-negative real number $p$ can be written as $p=\psi^*\psi$, where $\psi$ is a complex number and * denotes complex conjugation (of the symbol at the left from *). If $p$ is probability, then $\psi$ is called probability amplitude. This concept is useful in quantum mechanics, because the probability amplitude (unlike the probability itself) turns out to obey linear equations, which makes probability amplitude simpler to calculate than probability itself.

Even in classical (i.e. not quantum) physics probability can be expressed in terms of a complex probability amplitude, but there it doesn't obey a linear equation:
http://lanl.arxiv.org/abs/quant-ph/0505143