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entropy1
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Apart from the fact that it is, what is the physical significance of the fact that you can get the momentum distribution of a particle by taking the Fourier transform of its position distribution?
Why is that?jtbell said:Momentum is directly related to a wave property
Do you mean an energy potential?jtbell said:for a certain potential function V(x)
It comes from solving Schrodinger's equation. The momentum operator is ##\hat{p}=i\hbar\frac{\partial}{\partial{x}}## and that leads to the momentum appearing in the solutions as jtbell described. So your question may come down to asking why the momentum operator is defined the way it is.entropy1 said:Why is that?
I understand it can be found by differentiating the position expectation value with respect to time: [itex]\frac{\mathrm{d} \langle X \rangle}{\mathrm{d} t}[/itex], correct?Nugatory said:So your question may come down to asking why the momentum operator is defined the way it is.
entropy1 said:Apart from the fact that it is, what is the physical significance of the fact that you can get the momentum distribution of a particle by taking the Fourier transform of its position distribution?
entropy1 said:I understand it can be found by differentiating the position expectation value with respect to time: [itex]\frac{\mathrm{d} \langle X \rangle}{\mathrm{d} t}[/itex], correct?
I ment the momentum operator. (#5) [itex]\frac{\mathrm{d} \langle X \rangle}{\mathrm{d} t}[/itex] leads to the speed EV <V> which is the momentum EV <P> short for a mass constant m. <P> is [itex]\langle \psi | \hat{p} | \psi \rangle[/itex], and you can read [itex]\hat{p}[/itex] off the equation to be ##\hat{p}=i\hbar\frac{\partial}{\partial{x}}##. (And you need the schroedinger equation)SergioPL said:[itex]\frac{\mathrm{d} \langle X \rangle}{\mathrm{d} t}[/itex] would give the momentum expected value but not the momentum itself,
How does that lead to the momentum being the Fourier transform of position?jtbell said:1. Momentum is directly related to a wave property, the wavenumber ##k##, via ##p = \hbar k##.
2. Wavefunctions have the property of superposition. If ##\psi_1 = e^{i k_1 x}## and ##\psi_2 = e^{i k_2 x}## are solutions of the Schrödinger equation for a certain potential function V(x), then ## A \psi_1 + B \psi_2## is also a solution.
How does one come to this result?vanhees71 said:$$\langle x|p \rangle=\frac{1}{\sqrt{2 \pi}} \exp(\mathrm{i} p x),$$
entropy1 said:I ment the momentum operator. (#5) [itex]\frac{\mathrm{d} \langle X \rangle}{\mathrm{d} t}[/itex] leads to the speed EV <V> which is the momentum EV <P> short for a mass constant m. <P> is [itex]\langle \psi | \hat{p} | \psi \rangle[/itex], and you can read [itex]\hat{p}[/itex] off the equation to be ##\hat{p}=i\hbar\frac{\partial}{\partial{x}}##. (And you need the schroedinger equation)
You use the commutator relationsentropy1 said:How does one come to this result?
Momentum is a measure of an object's motion, and it is defined as the product of an object's mass and its velocity. In other words, it is the quantity of motion an object has.
The Fourier transform is a mathematical tool used to convert a function of time or space into a function of frequency. It is commonly used to analyze signals and waves in various fields such as physics, engineering, and mathematics.
This is because momentum and position are conjugate variables, meaning that they are related through the Heisenberg uncertainty principle in quantum mechanics. The Fourier transform is used to describe the position and momentum of a particle in a quantum system, where the position and momentum operators are related through the Fourier transform.
The Fourier transform is used in quantum mechanics to describe the position and momentum of a particle in a quantum system. It allows us to determine the probability of finding a particle at a certain position or with a certain momentum. It is also used to analyze wave functions and quantum states.
Understanding momentum and the Fourier transform is crucial in many fields of science, such as physics, engineering, and mathematics. It allows us to analyze and describe the behavior of particles and waves, and it is a fundamental concept in quantum mechanics. It also has many practical applications, such as in signal processing and medical imaging.