What is a probability amplitude?

[ideogram]

I have no idea what it means for a probability to be an amplitude. Can someone give me some kind of intuitive explanation of this concept?

 

[tiny-tim]

Welcome to PF!

Hi ideogram ! Welcome to PF!

I have no idea what it means for a probability to be an amplitude. …

A probability is not an amplitude.

An amplitude is a complex number whose absolute value squared is a probability or probability density …

see http://en.wikipedia.org/wiki/Probability_amplitude" for details.

 

[ideogram]

Hi ideogram ! Welcome to PF!


A probability is not an amplitude.

An amplitude is a complex number whose absolute value squared is a probability or probability density …

see http://en.wikipedia.org/wiki/Probability_amplitude" for details.

Thank you! Yes, I read the wikipedia article but could not understand it. What is an amplitude then? Why does squaring it yield a probability? Why is it added or multiplied like a probability in calculating the actual probability of combined events? In particular, I do not understand this sentence: "a purely real formulation has too few dimensions to describe the system's state when superposition is taken into account."

It seems like there is a strong analogy between amplitudes as complex numbers and ordinary probability as one-dimensional values. The fact that they are added and multiplied like probabilities makes me wonder if they are in some sense probabilities "generalized" to two dimensions. Am I making any sense here?

 

[alxm]

What is an amplitude then? Why does squaring it yield a probability?

A wave is defined by its amplitude, frequency and phase. It's usually convenient to represent waves mathematically by using complex numbers, where the amplitude
is represented by the modulus (aka absolute value) and the phase is represented by the argument (aka phase) in the http://en.wikipedia.org/wiki/Complex_number#Polar_form" representation of complex numbers.

In quantum mechanics, the wave function is complex-valued, and the square of the absolute value yields a probability
(e.g. for the wave function in position space, the probability of the described particle(s) being at a certain location).
As for why, this is more or less a basic postulate of QM. Although there are plenty of 'interpretations', debated endlessly around here, theorizing why this is the case.

Why is it added or multiplied like a probability in calculating the actual probability of combined events?

It sounds like you're asking why probability acts like a probability?

In particular, I do not understand this sentence: "a purely real formulation has too few dimensions to describe the system's state when superposition is taken into account."

They're saying that amplitudes alone are not enough to fully describe a system. Since you have quantum-mechanical superposition (things being "in several states as once"),
it requires knowledge of the phase part as well to describe the behavior. This is analogous to classical interference between waves.

It's not really true that you can't have a "purely real" formulation of quantum mechanics (or classical wave mechanics, for that matter).
It's just cumbersome and mathematically unelegant.

It seems like there is a strong analogy between amplitudes as complex numbers and ordinary probability as one-dimensional values.

No, no, the amplitudes are real numbers. All (directly) observable properties are real numbers. The amplitude/modulus of a complex number is a real number.

 

[ideogram]

A wave is defined by its amplitude, frequency and phase. It's usually convenient to represent waves mathematically by using complex numbers, where the amplitude
is represented by the modulus (aka absolute value) and the phase is represented by the argument (aka phase) in the http://en.wikipedia.org/wiki/Complex_number#Polar_form" representation of complex numbers.

In quantum mechanics, the wave function is complex-valued, and the square of the absolute value yields a probability
(e.g. for the wave function in position space, the probability of the described particle(s) being at a certain location).
As for why, this is more or less a basic postulate of QM. Although there are plenty of 'interpretations', debated endlessly around here, theorizing why this is the case.



It sounds like you're asking why probability acts like a probability?



They're saying that amplitudes alone are not enough to fully describe a system. Since you have quantum-mechanical superposition (things being "in several states as once"),
it requires knowledge of the phase part as well to describe the behavior. This is analogous to classical interference between waves.

It's not really true that you can't have a "purely real" formulation of quantum mechanics (or classical wave mechanics, for that matter).
It's just cumbersome and mathematically unelegant.



No, no, the amplitudes are real numbers. All (directly) observable properties are real numbers. The amplitude/modulus of a complex number is a real number.

Ah I think I have my terminology wrong.

I'm watching the Douglas Robb Memorial Lectures and as I understand it calculation of quantum probabilities uses vector addition and multiplication where calculating classical probabilities would use real addition and multiplication. This seemed very mysterious to me and made me wonder what is being vector added and multiplied in quantum probability that is analogous to a classical probability being real added and multiplied. Apparently no one really knows?

 

[LostConjugate]

Ah I think I have my terminology wrong.

I'm watching the Douglas Robb Memorial Lectures and as I understand it calculation of quantum probabilities uses vector addition and multiplication where calculating classical probabilities would use real addition and multiplication. This seemed very mysterious to me and made me wonder what is being vector added and multiplied in quantum probability that is analogous to a classical probability being real added and multiplied. Apparently no one really knows?

This is just a convenient way of working with functions.

The wave function is being converted to a vector where each argument is a now a vector index and each value of the function at that argument is now the component along that index. The component of the new wave function vector along these new vector indexes is the amplitude you speak of.

If you looked at a one dimensional wave function graphed, the displacement of the function's value from $$y_0$$ at position $$x$$ would be your amplitude. Where $$y = f(x)$$