# Tagged Questions

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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### 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 ...
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### What restrictions on time boundary conditions does it have to use Fourier transform to solve wave equation?

The wave equation can be solved using Fourier transform, by assuming a solution of the form of $$\mathbf{E}(x,y,z,t)~=~\mathbf{E}(x,y,z)e^{j\omega t}$$ and then reducing the equation to the Helmholtz ...
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### What's a good textbook to learn about waves and oscillations?

I'm taking a course on waves and oscillations using Crawford from the Berkeley series (out of print excluding international copies), and would like to know if anyone has any suggestions for a better ...
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### Position operator in QFT

My Professor in QFT did a move which I cannot follow: Given the state $$\hat\phi|0\rangle = \int \frac{d^3p}{(2\pi)^3 2 E_p} a^\dagger_p e^{- i p_\mu x^\mu}|0\rangle,$$ he wanted to show that this ...
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### Derivation of plane wave from inner product of position ket and momentum ket

In textbooks it seems to be taken for granted that $$\langle \mathbf{r}|\mathbf{k}\rangle ~=~ \frac{1}{\sqrt{\Omega}}\exp(i\mathbf{k}\cdot\mathbf{r}).$$ I'm sure it's obvious but is there a ...
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### Why are sine/cosine always used to describe oscillations?

What I am really asking is are there other functions that, like $\sin()$ and $\cos()$ are bounded from above and below, and periodic? If there are, why are they never used to describe oscillations in ...
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Using Minkowski signature $(+,-,-,-)$, for the Lagrangian density $${\cal L}=\partial_{\mu}\phi\partial^{\mu}\phi^{\dagger}-m^2\phi \phi^{\dagger}$$ of the complex scalar field, we have the field $... 4answers 4k views ### Intuitive explanation of why momentum is the Fourier transform variable of position? Does anyone have a (semi-)intuitive explanation of why momentum is the Fourier transform variable of position? (By semi-intuitive I mean, I already have intuition on Fourier transform between time/... 3answers 2k views ### What is the relation between position and momentum wavefunctions in quantum physics? I have read in a couple of places that$\psi(p)$and$\psi(q)$are Fourier transforms of one another (e.g. Penrose). But isn't a Fourier transform simply a decomposition of a function into a sum or ... 1answer 2k views ### Is there a relation between quantum theory and Fourier analysis? I found that some theories about quantum theory is similar to Fourier transform theory. For instance, it says "A finite-time light's frequency can't be a certain value", which is similar to "A finite ... 0answers 296 views ### Action of Parity operator on Impulse representation Is my derivation of the action of the parity operator$\mathbb{P}$on the$|p\rangle$representation correct? $$\left( \mathbb{P}\tilde\psi \right)(p)= - \tilde\psi (p).$$ Obtained from $$\left( \... 0answers 134 views ### Edge states of Kitaev chain [closed] I am reading paper about Kitaev chain of electrons, which can exhibit famous Majorana fermions at ends of wire. The Hamiltonian (his Eq. (6)) reads H = \frac{i}{2} \sum_j - \mu c_{2j-1}c_{2j} +(w+|\... 3answers 1k views ### Why use Fourier expansion in Quantum Field Theory? I have just begun studying quantum field theory and am following the book by Peskin and Schroeder for that. So while quantising the Klein Gordon field, we Fourier expand the field and then work only ... 1answer 6k views ### How does the Fourier Transform invert units? I don't really understand how units work under operations like derivation and integration. In particular, I am interested in understanding how the Fourier transform gives inverse units (i.e. time ... 1answer 276 views ### Amplitude and phase in vector wave field Is it possible to make some separation of amplitudes and phase for a general vector-wave field? For example, like a paraxial approximation of a complex scalar field of the form$$\Phi(x,y,z) = A(x,y,... 7answers 5k 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 ... 3answers 1k views ### Evaluating propagator without the epsilon trick Consider the Klein–Gordon equation and its propagator: $$G(x,y) = \frac{1}{(2\pi)^4}\int d^4 p \frac{e^{-i p.(x-y)}}{p^2 - m^2} \; .$$ I'd like to see a method of evaluating explicit form of$G$... 4answers 616 views ### Reconstruction of “wavefunction” phases from$|\psi(x)|$and$|\tilde \psi(p)|$Consider a "wavefunction"$\psi(x)$, which has a Fourier transform$\tilde \psi(p)$Suppose that we know, for each$x$,$|\psi(x)|^2$, and that we know, for each$p$,$|\tilde \psi(p)|^2$. Have we ... 2answers 294 views ### Schrödinger equation in momentum space In literature on an introduction to quantum mechanics which I am working through, there is a section which explains that a vector has different representations based on the basis you choose. It then ... 2answers 263 views ### Why there is no Gibb's phenomenon in QM? Why we don't see any Gibb's phenomenon in quantum mechanics? EDIT At sharp edges (discontinuities), we usually find ringing. This can be observed in many physical phenomenon (eg. shock waves). ... 4answers 1k 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$p$, ... 2answers 758 views ### The poles of Feynman propagator in position space This question maybe related to Feynman Propagator in Position Space through Schwinger Parameter. The Feynman propagator is defined as: $$G_F(x,y) = \lim_{\epsilon \to 0} \frac{1}{(2 \pi)^4} \int d^4p ... 3answers 377 views ### The ubiquitous Planewave Ansatz In physics, the planewave ansatz (meaning: an educated solution guess) is very ubiquitously used, when solving differential equations, in different domains of physics. E.g. to solve the dispersion ... 1answer 359 views ### How is Green function in many-body theory introduced? Normally, for a (linear) operator L and a DE$$ Lu(x) = f(x) $$the Green function is defined as$$ LG(x,s) = \delta(x-s) $$and it is found that$$ u(x) = \int G(x,s) f(s) ds $$is the ... 3answers 1k views ### The Dirac-Delta function as an initial state for the quantum free particle I want to ask if it is reasonable that I use the Dirac-Delta function as an intial state (\Psi (x,0) ) for the free particle wavefunction and interpret it such that I say that the particle is ... 1answer 635 views ### Parseval's Theorem on a Random Signal NB - I'm re-posting this question in physics because I haven't had any luck getting a response from the maths StackExchange site - it's a rather applied problem so is probably better suited here ... 2answers 745 views ### Was uncertainty principle inferred by Fourier analysis? I would like to know: did Heisenberg chance upon his Uncertainty Principle by performing Fourier analysis of wavepackets, after assuming that electrons can be treated as wavepackets? 3answers 499 views ### Physics of a guitar I understand that when you pluck a guitar string, then a bunch of harmonic frequencies are produced rather than just the frequency of the desired note. If this is true, why does C2 sound so different ... 3answers 135 views ### Can a physical wavefunction be non-smooth (its first derivative is discontinuous)? Here's an argument that might support the statement that such a non-smooth wavefunction is not physical: You cannot add a finite number of smooth functions to get a non-smooth function. By fourier ... 3answers 370 views ### No well-defined frequency for a wave packet? There are similar questions to mine on this site, but not quite what I am asking (I think). The de Broglie relations for energy and momentum$$ \lambda = \frac{h}{p}, \\ \nu = E/h .$$equate a ... 2answers 629 views ### A four-dimensional integral in Peskin & Schroeder The following identity is used in Peskin & Schroeder's book Eq.(19.43), page 660:$$\int\frac{d^4k}{(2\pi)^4}\,\frac{1}{(k^2)^2}e^{ik\cdot\epsilon}=\frac{i}{(4\pi)^2}\log\frac{1}{\epsilon^2},\... 3answers 5k views ### What is the significance of negative frequency in Fourier transform? What is the significance of negative frequency in Fourier transform? Why we include the band widths of the negative frequency also while calculating band width of the signal. 1answer 2k 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 ... 3answers 525 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 ... 1answer 211 views ### Diffraction and$k$-space Regarding diffraction I am a little bit lost reading about reciprocal space and the space of$k$'s. As I understand it the Fourier relationship between a wavepacket$\Psi(\vec r,t)$and the complex ... 2answers 218 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 (... 1answer 120 views ### Replacing fermionic operators with their Fourier transform and boundary conditions In the section 4.1 of Quantum Computation by Adiabatic Evolution, Farhi et al proposes a quantum adiabatic algorithm to solve the$2$-SAT problem on a ring. To compute the complexity of the algorithm ... 3answers 327 views ### Fourier Transforms Related to Green's Functions I'm reading a text on field theory where there are a number of assertions made about Fourier transforms that I'm finding confusing. For example, let$G^R = -i \theta(t - t')e^{-i \omega_0 (t - t')}$. ... 5answers 12k views ### Why are AC quantities represented by sine waves always? Usually we use a sinusoidal wave form to represent a alternating quantity. Why not a cosinusoidal wave or a ramp wave form? In sine wave forms we can indicate the maximum and minimum amplitude and ... 1answer 68 views ### What does$σ\$ equal to zero mean?

Consider the Laplace transform of an RC filter. For those who can't immediately summon it, refer equation (46) at this link: http://web.mit.edu/2.151/www/Handouts/FreqDomain.pdf for a refresher. In ...
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### What is the advantage of using exponential function over trigonometric function in analyzing waves?

A.P.French in his book Vibrations and Waves writes: . . . Why should the exponential function be such an important contribution to the analysis of vibrations? The prime reason is the special ...
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### Initial condition for Fourier transformed Schrödinger equation

I asked in this thread Time-dependet Schrödinger equation how to solve the Time-dependent Schrödinger equation. One of JamalS' recommendations was the Fourier transform, which is why I want to quote ...
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### 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 ...