Questions tagged [fourier-transform]
A unitary linear operator which resolves a function on $\mathbb{R}^N$ into a linear superposition of "plane wave functions". Most often used in physics for calculating the response of a time shift invariant linear system as the sum of its response to time harmonic excitation or for transforming a quantum state in position co-ordinates into one in momentum co-ordinates and contrawise. There is also a discrete, fast Fourier transform for discretised data.
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Singing glass frequency - why is it audible?
I'm thinking about the typical singing glass demonstration, where you rub a finger around the edge of a wine glass and can hear a loud sound at the natural frequency of the glass. The fact that it is ...
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Units of wave functions in real and reciprocal space
I'm confused about the units of wave functions in reciprocal space and their Fourier transform in real space. On one hand, I believe the Fourier transform of a reciprocal space wave function in 2D is ...
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Analysis to periodic diffusion at long times (Sethna Q2.7)
This is problem 2.7 from Entropy, Order Parameters, and Complexity by James P. Sethna. Consider a one-dimensional diffusion equation $\frac{\partial \rho}{\partial t} = D\frac{\partial^{2}\rho}{\...
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Diagrammatic interpretation of the four-point correlation function in momentum space
In Peskin & Schroeder's book (chapter 7.2,p227), they claim that the exact 4-point function, expressed by $$ \left( \prod_{i=1}^{2} \int d^4 x_i \, e^{i p_i \cdot x_i} \right) \left( \prod_{i=1}^{...
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Why can we extract the value of $S$-matrix element by simply analyzing the pole of correlation function?
In Peskin & Schroeder's book (Chapter 7.2, p.223-226), they analyze the pole of the correlation function by discarding the exponential term. They first calculate the accurate value of the ...
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Computing position two-point function and Fourier transform of $p^4 \ln p$
I am computing a two-point correlator in 4D Euclidean space and I am struggling with one particular term. I have found that in momentum space my correlator goes as $$\langle \mathcal{O}(p)\mathcal{O}(...
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Incompleteness of positive-energy solutions to Klein-Gordon equation
In non-relativistic Quantum Mechanics, plane waves (didn't attempt to normalized below) or eigenstates of momentum operator form a complete set of basis of the state space $L^2(\mathbb{R}^3)$ (...
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A particle confined without forming a standing wave
Can a particle remain confined given that the wavefunction $ψ$ at the boundaries is 0, even if it doesn't form a standing wave? What exactly is confinement? I think it's a condition where $Δx$ (the ...
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How do we express the quantum state of a particle in momentum eigenfunctions basis given that eigenfunctions of momentum are not square-integrable?
As far as I know, for a particle without spin the quantum state of the particle is described by a "point" in the Hilbert space of the equivalence classes of $L^2$ square-integrable functions ...
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Advanced Green's function for the classical forced harmonic oscillator
In Ashok Das' Field theory: A path integral approach (second edition, World Scientific, 2006), page 47, for the forced harmonic oscillator with Newton's equation of motion $$ \ddot{x} + m\omega^2x = F(...
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Finding $Ψ(x, t)$ from $Ψ(x, 0)$ for different QM-systems
To obtain Ψ(x, t) for a free particle having $Ψ(x, 0) = Ae^{-αx^2}$ the following method is applied: Find $Φ(k) = \frac{1}{√{2π}}\int^{+\infty}_{-\infty}Ψ(x,0)e^{-ikx}dx$ Find $Ψ(x,t) = \frac{1}{√{...
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Energy in Fourier modes for pseudo-quadratic fields
Consider the continuum analog of the Hooke's potential energy, $1/2\,\alpha\, s^2(x)$, where $s$ is the local displacement field and $\alpha$ is the spring constant. The potential energy in Fourier ...
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Proving positivity of kinetic energy operator [closed]
As a preparation for an exam I have soon, I'm going through the suggested exercises in the lecture notes. One of the suggested exercises is to show that $$\left\langle\psi,-\Delta\psi\right\rangle \...
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Decomposition of a gauge-invariant $3$-dimensional spin network state over the Wigner matrix elements
Below follows the passage of Rovelli and Vidotto's Covariant Loop Quantum Gravity that I do not understand. To give the context, let me clarify that a state $\psi$ is a function in $L^2[\text{SU}(2)^L]...
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Converting momentum-space hamiltonian of Hopf insulator into real-space tight-binding hamiltonian
I want to convert the Momentum-Space Hamiltonian for a Hopf-Insulator given in Equation 8 of this paper by Moore, Ran, and Wen. The momentum-space representation is $$ \mathcal{H}(k_x, k_y,k_z) = \vec{...
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