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alphaone
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could somebody please explain to me why position and momentum space are related to one another by a Fourier transform, meaning why do I get momenta when I do a Fourier transform of an expression in position space?
alphaone said:could somebody please explain to me why position and momentum space are related to one another by a Fourier transform, meaning why do I get momenta when I do a Fourier transform of an expression in position space?
alphaone said:could somebody please explain to me why position and momentum space are related to one another by a Fourier transform, meaning why do I get momenta when I do a Fourier transform of an expression in position space?
alphaone said:Thanks for all your replies. I am sorry for having phrased my question badly. I know the mathematics of how to transform from position space to momentum space and how to get the Fourier factor. However what I really shoud have asked is: Is there a physically intuitve reason why we would expect to get to momentum space when we do a Fourier transform? By a physically intuitive reason I mean something along the same lines as calculating commutation relations of poincare generators with the Pauli-Lubanski vector: We could either plug in definitions and do the mathematics(which is a similar method to the earlier replies) or we could say that the pauli lubanski vector is a vector and so we know how it will transform under poincare transformations(which is kind of physically intuitive).
alphaone said:I am sorry, but I did not really understand your last reply. I am used to usual Fourier analysis if that is what you mean, however I have never heard of a physical interpretation of such a transform and considering that it maps momentum space to position space and vice versa I was wondering whether such an interpretation exists. So if you know about any please let me know.
Are you asking why a wave packet can be described by some equationalphaone said:could somebody please explain to me why position and momentum space are related to one another by a Fourier transform, meaning why do I get momenta when I do a Fourier transform of an expression in position space?
alphaone said:could somebody please explain to me why position and momentum space are related to one another by a Fourier transform?
alphaone said:could somebody please explain to me why position and momentum space are related to one another by a Fourier transform, meaning why do I get momenta when I do a Fourier transform of an expression in position space?
samalkhaiat said:Now, why is it that coordinate and wave number spaces related by the above Fourier transformation? well, the one and only answer is; because eq(F) represents the most general SUPERPOSITION of PLANE WAVES.
regards
sam
Position space refers to the physical space in which a particle or object exists, while momentum space refers to the mathematical space in which the momentum of a particle or object is described. In position space, the position of a particle is described by its coordinates (x, y, z), while in momentum space, the momentum of a particle is described by its momentum vector (px, py, pz).
Question 2: How are position space and momentum space related?Position space and momentum space are related through the mathematical concept of Fourier transform. The Fourier transform converts a function from its representation in position space to its representation in momentum space and vice versa. This allows us to study the properties of a particle or object in both position space and momentum space.
Question 3: What is the uncertainty principle in position space and momentum space?The uncertainty principle states that it is impossible to know both the position and momentum of a particle with absolute certainty. In position space, this means that the more precisely we know the position of a particle, the less precisely we know its momentum, and vice versa in momentum space. This is a fundamental principle in quantum mechanics and is related to the wave-particle duality of matter.
Question 4: How do we visualize position space and momentum space?Position space and momentum space are abstract mathematical concepts, so we cannot visualize them in the same way we visualize physical space. However, we can use mathematical tools such as graphs and diagrams to represent the properties of particles or objects in position space and momentum space. For example, a wavefunction graph is often used to represent the position and momentum of a particle simultaneously.
Question 5: Why is it important to study position space and momentum space?Studying position space and momentum space allows us to understand the behavior and properties of particles and objects at the quantum level. It is important in fields such as quantum mechanics, particle physics, and materials science. By understanding the relationship between position and momentum, we can make predictions and calculations about the behavior and interactions of particles, which has important implications for technology and our understanding of the universe.