Wave-Function, Fourier Transform, and Speed

In summary, the conversation discusses the concept of Fourier decomposition in quantum mechanics and its relation to wave-functions and basis states. The speaker asks for clarification on whether wave-functions undergo Fourier decomposition and if so, whether it is done with respect to position, time, or both. They also inquire about the connection between momentum and position wave-functions as Fourier transforms of each other and their relation to the de Broglie relations. Lastly, the topic of basis states as wave equations and their potential relationship to decomposition is mentioned, as well as the possible physical implications of the speed of a basis state wave-equation.
  • #1
danmay
48
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Hi, I'm pretty much an amateur in quantum mechanics. If anyone could clarify the following, that would be greatly appreciated!

When you write a wave-function (phi or "amplitude" for example) in terms of basis states (either position or momentum), does it undergo a Fourier decomposition? If so, do you actually perform it with respect to position, time, or both?

Does this process have anything to do with how momentum and position wave-functions are Fourier transforms of each other? Does this also have anything to do with the de Broglie relations (which one, frequency-energy or wavelength-momentum, or both as related through the constant c)?

Finally, regarding the basis states, are they also wave equations? If so, do their wave-numbers and frequencies have any relation to the wave-function undergoing decomposition, or can they be arbitrarily chosen? In either case, does the speed of a basis state wave-equation have any physical implications?
 
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  • #2
Anyone knows? ...
 

FAQ: Wave-Function, Fourier Transform, and Speed

What is a wave-function and why is it important in physics?

A wave-function is a mathematical description of a quantum system that contains all the information about its physical properties. It is important in physics because it allows us to predict the behavior of particles at the quantum level.

What is the Fourier transform and how is it related to wave-functions?

The Fourier transform is a mathematical operation that allows us to break down a function into its individual frequency components. It is related to wave-functions in that it can be used to represent a wave-function as a combination of different frequency components.

How does the speed of a particle relate to its wave-function?

The speed of a particle is related to its wave-function through the uncertainty principle in quantum mechanics. The more precisely we know the position of a particle, the less precisely we can know its speed, and vice versa.

Can the Fourier transform be used to calculate the speed of a particle?

No, the Fourier transform cannot be used to directly calculate the speed of a particle. It is a mathematical tool that helps us understand the frequency components of a wave-function, but it cannot provide information about the speed of a particle.

How does the speed of a particle change when its wave-function collapses?

When a particle's wave-function collapses, its position becomes more certain and its speed becomes less certain. This means that the speed of the particle will also change, becoming more precise but with a larger uncertainty in its value.

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