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 calcalating the response of a time shift invariant linear system as the sum of its response to time harmonic excitation or for transforming a ...

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7
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3answers
252 views

Why is the Fourier transform more useful than the Hartley transform in physics?

The Hartley transform is defined as $$ H(\omega) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty f(t) \, \mbox{cas}(\omega t) \mathrm{d}t, $$ with $\mbox{cas}(\omega t) = \cos(\omega t) + \sin(\omega ...
3
votes
3answers
132 views

How can $F_0\cos\omega t$ change to $F_0e^{i\omega t}$ in driven oscillator equation?

I have one thing that confuses me on deriving the solution for the Linear Forced Oscillator. Suppose we have the equation as $$ma + rv + kx = F_0 \cos \omega t$$ What confuses me is when the driving ...
0
votes
1answer
93 views

Derivation of group velocity using Fourier transform

The aim is to determine the group velocity of a wave packet with the general form $$\Psi\left(x,t\right)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \phi\left(x\right)e^{i\left(kx-\omega ...
0
votes
0answers
51 views

How can we fix the constant of the energy eigenstates of a quantum free particle such that they satisfy the orthonormality condition?

For a quantum free particle, the momentum and energy eigenstates are compatible. The constants of the momentum eigenstates are fixed by their orthonormality. Similarly, how can we fix the constant for ...
0
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2answers
49 views

$Ae^{\mathrm{i}\omega t}$ assumption for oscillating systems (formal & intuitive)

When we obtain a system of ODE's for $n$ masses connected with springs (or otherwise obtained by small amplitudes approximation), the next steps are usually assuming a solution in form $Ae^{i\omega ...
4
votes
2answers
110 views

Shifting momentum by a constant in the Schrodinger Equation

My book states that if we perturb a given Hamiltonian for the Schrödinger Equation $$ H = \frac{p^2}{2m} +V(x) $$ to $$ H' = \frac{p^2}{2m} + V(x) + \frac{\lambda p}{m} $$ then we can rewrite ...
0
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0answers
45 views

Doubt in Path integral equation

In Pokorski's "Gauge Field Theories" book, page 108 we find equation (2.87) ...
4
votes
2answers
198 views

Derivation of canonical position-momentum commutator relation

We know that the position-momentum commutator is fundamental in quantum mechanics, but would it be possible to derive it starting from a different set of first principles, more specifically starting ...
0
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2answers
67 views

How does one get the first few terms of the S-matrix expansion?

According to a set of notes I'm reading $$\langle p_f | S | p_i \rangle = \delta(p_f-p_i) + 2 \pi \delta(E_f-E_i) \bigg[\langle p_f | V | p_i \rangle + \cdots\bigg] \tag{1.29}$$ I don't understand ...
0
votes
1answer
59 views

Using the fourier series to analyze the motion of a finite string [closed]

Q: Find the Fourier series for the motion of a string of length L if (a) $y(x,0) = Ax(L-x); \frac{\partial y}{\partial t}_{t=0}=0.$ (b) $y(x,0) = 0; \frac{\partial y}{\partial ...
2
votes
1answer
213 views

Analogy to Fourier transform in spherical coordinates with boundary at a certain radius

Suppose, we have a wavefuction $\phi(\vec{x})$ which is restricted in a sphere, with the spherical boundary condtion $$\phi(\vec{x}=R)=\phi_0.$$ How can I do the 'Fourier transformation' as the case ...
0
votes
0answers
28 views

Units of Fourier Transform [duplicate]

I am a bit confused about the units of continuous time Fourier transform. Let's say that $x(t)$ is an input signal and has units of volts. Taking the Fourier transform of this yields $X(f)$. I would ...
0
votes
2answers
30 views

Can you use Fourier transformations (or other) to read multiple superimposed barcodes?

If you printed bar codes on tracing paper/acetate etc. and then positioned several in front of one another, could you extract the individual codes from the aggregate overlaid image? I feel intuitively ...
0
votes
1answer
80 views

How Quantum Fourier Transform equal to Hadamard Transform on 4-by-4 matrix?

I just don't understand why $QFT_4$ become the same as Hadamard Transform $H_4$ The Hadamard matrix is as follwoing, $$ H_2 = \frac12 \begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & -1 & ...
0
votes
0answers
73 views

How to Fourier transform creation/annihilation operators?

Zee's QFT in a Nutshell pages 65-66. For a complex scalar QFT $$ \varphi(\vec{x},t) = \int\frac{d^Dk}{\sqrt{(2\pi)^D2\omega_k}}\left[a(\vec{k})\mathrm{e}^{-i(\omega_kt-\vec{k}\cdot\vec{x})} + ...
0
votes
1answer
42 views

locator equation of motion

I strugle with folowing problem. I do start with the locator equation of motion: $$G_{i j} = g_i \delta_{i j} + g_i \sum\limits_{k \ne i} W_{i k} G_{k j}$$ where $G_{i j}$ are matrix elements of ...
0
votes
0answers
150 views

How do you fourier transform a tight binding hamiltonian numerically?

The task is to do a fourier transformation of a tight binding hamiltonian of a 1D-chain with unit cell size 2, but even after many tries and googling I still don't have a idea how to do it correctly. ...
1
vote
1answer
71 views

Why do physicists use a positive sign for the Fourier kernel / outward propagating waves? [closed]

I am not a physicist but rather an engineer / mathematician, so I've always wondered why is it that physicists use the positive sign convention in the forward Fourier transform. That is, in all of my ...
2
votes
1answer
98 views

Is the wavelet transform utilized at all in QM?

Excuse any ignorance, but something was on my mind today and my professor didn't give me a very clear answer... Obviously the Fourier Transform is used pretty constantly in QM. What about the wavelet ...
2
votes
2answers
67 views

Physical interpretation of Fourier $[x(t)]$ where $x(t)$ is the position of mass $m$ as a function of time?

If a macroscopic body of mass $m$ moves according to a certain law of motion like, for example, $$x(t)=A\cos(2\pi ft)$$ then what physical interpretation can be attributed to the Fourier transform of ...
1
vote
0answers
164 views

Transforming the Schrodinger equation from momentum-space to position-space [closed]

I am working on a problem right now that states the following: A particle of mass $m$ is subjected to a force $\mathbf{F}(\mathbf{r}) = - \nabla V(\mathbf{r})$ such that the wave function ...
1
vote
1answer
101 views

Does “sum over all paths” in the path integral imply “sum over all paths” in momentum space when one Fourier-transforms?

How is the Fourier-transformed-field path integral interpreted? Is it still a "sum of all paths" in momentum space? Just that with another action? Consider for instance the (Euclidean) partition ...
0
votes
1answer
46 views

Fourier transform for $W(J)$ in a free QFT

In chapter I.3 and I.4 of A. Zee's QFT in a Nutshell, he starts with the theory $$ \mathcal{L}(\varphi) = \frac 12[(\partial\phi)^2-m^2\varphi^2]+J\varphi $$ Using the path integral approach, he ...
1
vote
1answer
103 views

A question on using Fourier decomposition to solve the Klein Gordon equation

Given the Klein Gordon equation $$\left(\Box +m^{2}\right)\phi(t,\mathbf{x})=0$$ it is possible to find a solution $\phi(t,\mathbf{x})$ by carrying out a Fourier decomposition of the scalar field ...
0
votes
1answer
84 views

Motion of string fixed at both ends

I was reading about the Fourier analysis from Waves by Frank S Crawford Jr. But I got trapped at the very beginning; this is the excerpt that troubled me: Motion of string fixed at both ends. ...
1
vote
1answer
116 views

Switch from the position representation to the momentum representation

If we use Fourier Transform, we can switch from the position representation to the momentum representation, like the following formula here comes the problem, if we use dirac notation we can see it ...
2
votes
2answers
84 views

How to detect “noisiness” of sound wave?

Some phonemes like "ssss" are basically white noise. How would you determine which parts of a wave are white noise? From frequency analysis the white noise will have no tones so just using this would ...
1
vote
0answers
62 views

Coulomb potential of a periodic crystal in reciprocal space

Usually the Coulomb potential (electron-electron interaction) can be Fourier transformed (aside from prefactors) like that: $$ \frac{1}{|\vec r_1 -\vec r_2|} = \int \frac{\text d ^3 k}{(2\pi)^3} ...
2
votes
0answers
66 views

$i\epsilon$ versus $2i \epsilon E_k$ in the propagator

The Fourier Transform of the propagator can be written as $$\tilde{\Delta}(k) = \frac{i}{k^2-m^2+i\epsilon} \tag{1} $$ which is then "factored" into $$ = \frac{i}{\left( k^0-E_k ...
0
votes
1answer
38 views

In quantum Fourier transform, why can any controlled $R_{k}$ gate be formed by two controlled-Not gate?

Controlled $R_{k}$ gate is implemented in quantum Fourier transform like this: Each of the $R_{k}$ on a qubit is in this matrix form: My question is: Why each of these controlled $R_{k}$ gates, no ...
3
votes
0answers
67 views

Linear KDV eq. asymptotics

The question arises from the book Solitons by P. G. Drazin about the linearized KDV eq. $$ u_t+u_{xxx}=0 $$ My first step was to take a Fourier transform of the equation, find that the dispersion ...
1
vote
2answers
390 views

Normalized wave functions in position and momentum space

Using the following expression for the Dirac delta function: $$\delta(k-k')=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i(k-k')x}\mathrm{d}x$$ show that if $\Psi(x,t)$ is normalized at time $t=0$, ...
2
votes
0answers
29 views

How to mathematically model a realistic aperture illumination?

I want to know a mathematical expression that I can use to model a realistic aperture illumination to produce the primary beam of an antenna so that the radial distribution of this aperture ...
-1
votes
1answer
29 views

Having trouble understanding the proof for Fourier Transform Scaling Property [closed]

Starting from Plancherel's Theorem: $$ f(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}F(k)e^{ikx}dk ...(1) $$ $$ F(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(x)e^{-ikx}dx ...(2) $$ I need to ...
1
vote
2answers
57 views

Characteristics of an Optical System in the Fourier Domain

An imaging system can be characterized by its point spread function (PSF), which in most cases is space-variant. The final image is the result of the convolution of the PSF with the object (2d ...
1
vote
0answers
49 views

Discrete Fourier transform for periodic signal?

From the Signal and System textbook, by Oppenheim, I learned that the discrete-time Fourier transform can be written as $$ x[n]=\frac{1}{2\pi}\int_{2\pi}X(e^{j\omega})e^{j\omega n}d\omega $$ $$ ...
1
vote
1answer
64 views

Finding $\phi(k)$

If you have $\Psi(x,0) = c(\psi_1 + \psi_2)$ where $\psi_n$ is an Energy eigenfunction for a quantum number $n$. I'm supposed to find $\phi(k,t)$ at $t$ = 0. This is for an infinite square well from ...
-1
votes
2answers
124 views

Fourier Transform with Branch Cuts [closed]

I want to compute the Inverse Fourier transform of the following function (it appears as a certain correlation function in a physical model I am interested it): $$ \widetilde{f}(\omega) = ...
3
votes
1answer
88 views

A question about Fourier Transformation [closed]

Recently I try to evaluate a integral in a paper: $$ \Gamma(x,y)=\int_{-\infty}^{\infty} \frac{dk}{2\pi} \sqrt{k^2+m^2} e^{ikx} $$ This is the Fourier Transform of: $$ f(k)=\sqrt{k^2+m^2} $$ The ...
1
vote
1answer
137 views

Wavefunction interpretations in QM

From two-slit electron-interference experiment we can infer that there is a wave $\psi(x,t)$ that can be associated with electron. The amplitude at some point is the sum of amplitudes reaching that ...
0
votes
0answers
38 views

Estimation of the autocorrelation for data on finite size interval

Let's consider we have a continuous random signal ${ t \in ] - \infty \,;\, + \infty [ \mapsto b (t)}$. We assume this signal to be stationary, so that when ensemble-averaged, one may introduce the ...
0
votes
0answers
25 views

Units of a fourier transform end energy density of the time dependent force

I have a signal, which is time dependent force F(t) (obtained from the Atomic Force Microscope) I wanted to estimate the energy content in the signal for given frequency band. I have calculated Energy ...
0
votes
1answer
80 views

Fourier transform & asymptotic expansion of Klein-Gordon equation

I am looking for an approximate analytical solution to the generalized Klein-Gordon equation \begin{equation} \frac{\partial^2{\phi}}{\partial{t^2}}+\frac{\partial^2{\phi}}{\partial{x^2}}+\phi=0 ...
1
vote
2answers
218 views

Why can you only measure velocity or location in a particle?

I was talking to a family friend in the field of optics at a quantum scale (not sure the proper name for this) and he was explaining to me why you can only determine either the velocity or location of ...
0
votes
0answers
108 views

Band Structure from Fourier Transform Solution of Dirac Comb

I have use Fourier transforms to solve the Schrödinger equation for an attractive Dirac-$\delta$ comb potential of the form $$ V(x) = -\alpha \sum_{j = -\frac{N}{2}}^{\frac{N}{2}} \delta \left( x - ...
1
vote
0answers
76 views

Riemannian generalization/adaption of the Hubbard–Stratonovich transformation

I'd like to write the Hubbard–Stratonovich (HS) transformation of a scalar function on a Riemannian manifold. This transformation is quite simple in Euclidean space. One can consider it as a Fourier ...
0
votes
1answer
177 views

Superposition of waves with different initial phase in Quantum Mechanics [closed]

In Quantum Mechanics, if a particle's state is a superposition of many states, then we say that its position is well-defined (by the Heisenberg uncertainty principle, because here we have ill-defined ...
1
vote
1answer
64 views

Books on waves with Fourier Transforms [duplicate]

There are many waves and oscillations books out there that also include Fourier analysis but very few give the subject a thorough treatment, they just pass it in a few pages. If anybody has any ...
0
votes
1answer
47 views

Fourier expansion and transform - what about the phase of the waves that i am adding?

Say we have a wave on the surface of the water and we want to describe it as a sum of other waves. So we use Fourier expansion to add waves of different wavelengths. For simplicity, say we have to ...
3
votes
1answer
388 views

Fourier Transforms of position and momentum space in Quantum Mechanics

Fourier transformations: $$\phi(\vec{k}) = \left( \frac{1}{\sqrt{2 \pi}} \right)^3 \int_{r\text{ space}} \psi(\vec{r}) e^{-i \mathbf{k} \cdot \mathbf{r}} d^3r$$ for momentum space and ...