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6
votes
2answers
351 views
What does the Canonical Commutation Relation (CCR) tell me about the overlap between Position and Momentum bases?
I'm curious whether I can find the overlap $\langle q | p \rangle$ knowing only the following:
$|q\rangle$ is an eigenvector of an operator $Q$ with eigenvalue $q$.
$|p\rangle$ is an eigenvector of ...
4
votes
3answers
1k views
What is the relation between position and momentum wavefunctions in quantum physics?
I have read in a couple of places that $\psi(p)$ and $\psi(q)$ are Fourier transforms of one another (e.g. Penrose). But isn't a Fourier transform simply a decomposition of a function into a sum or ...
16
votes
6answers
1k views
Fourier transformation in nature/natural physics?
I just came from a class on Fourier Transformations as applied to signal processing and sound. It all seems pretty abstract to me, so I was wondering if there were any physical systems that would ...
0
votes
0answers
90 views
Prove that the position operator is $\hat{x} = i\hbar \frac{d}{{dp}}$ in the momentum representation [closed]
Proof that: $x = i\hbar \frac{d}{{dp}}$
I did this, could you tell me if I am false or true
$\begin{array}{l}
x{e^{\frac{{ipx}}{\hbar }}} = - i\hbar \frac{{d{e^{\frac{{ipx}}{\hbar }}}}}{{dp}} = ...
5
votes
1answer
182 views
Fourier Transform on a Riemannian Manifold
The question is quite simple: What would be the definition of Fourier Transform (and it's inverse) on a Riemannian Manifold?
I've found that a similar question has been asked at Mathematics.SE but ...
2
votes
0answers
94 views
Number theoretical function applied in physics? [closed]
I have a series of number theoretic phenomena (mathematics) that I can describe exactly by the superpositions or linear combination of the below function (I know it is an inverse Fourier type). Does ...
4
votes
4answers
519 views
Uncertainty Principle for a Totally Localized Particle
If a particle is totally localized at $x=0$, its wave function $\Psi(x,t)$ should be a Dirac delta function $\delta(x)$. Accordingly, its Fourier transform $\Phi(p,t)$ would be a constant for all $p$, ...
4
votes
3answers
404 views
Very simple example of the way the Fourier transform is used in quantum mechanics?
According to a book I'm reading, the Fourier transform is widely used in quantum mechanics (QM). That came as a huge surprise to me. (Unfortunately, the book doesn't go on to give any simple examples ...
3
votes
1answer
390 views
Physical Significance of Fourier Transform and Uncertainty Relationships
What is the physical significance of a fourier transform?
I am interested in knowing exactly how it works when crossing over from momentum space to co ordinate space and also how we arrive at the ...
2
votes
1answer
227 views
Is there a relation between quantum theory and Fourier analysis?
These days I was studying the quantum theory.I found that some theories about that is similar to Fourier Transform theory.For instance, it says "A finite-time light's frequency can't be a certain ...
5
votes
4answers
984 views
Optics of the eye - do we see Fourier transforms?
I've recently been learning about Fourier optics, specifically, that a thin lens can produce the Fourier transform of an object on a screen located in the focal plane.
With this in mind, does the ...
3
votes
4answers
653 views
Intuitive explanation of why momentum is the Fourier transform variable of position?
Does anyone have a (semi-)intuitive explanation of why momentum is the Fourier transform variable of position?
(By semi-intuitive I mean, I already have intuition on Fourier transform between ...
