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ideogram
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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?
ideogram said:I have no idea what it means for a probability to be an amplitude. …
tiny-tim said: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.
ideogram said: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.
alxm said: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.
ideogram said: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?
A probability amplitude is a complex number that represents the probability of a quantum system to transition from one state to another.
A probability amplitude is calculated by taking the square root of the probability of a given event. This is known as the absolute value or modulus of the amplitude.
A probability amplitude and a probability are related by the Born rule, which states that the square of the absolute value of the amplitude represents the probability of a quantum system to transition from one state to another.
Probability amplitudes play a crucial role in quantum mechanics as they are used to calculate the probabilities of different outcomes in a quantum system. They also help to explain the wave-like behavior of particles at the quantum level.
Yes, probability amplitudes can be negative. In quantum mechanics, both positive and negative values are used to represent the amplitude of a system. However, when calculating probabilities, the negative value is squared and becomes positive, ensuring that the final probability value is always positive.