# Questions tagged [dirac-delta-distributions]

Distributions are generalized functions, such as, e.g., the Dirac delta function. DO NOT USE THIS TAG for statistical probability distributions, profiles, graphs, plots, etc.

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### How are Schwinger functions defined as moments if they are actually operators?

Let $\mu$ be a measure on the space of tempered distributions. Assuming $\mu$ satisfies some other properties, then it is the measure of a quantum field in a Euclidean framework. The Schwinger ...
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### Given Green's function, can I find the corresponding operator? [migrated]

Green's function is the solution to the equation $L G(x;x') = \delta(x-x')$, where $L$ is a linear differential operator. Usually, we want to find the Green's function of a given $L$. Instead, if we ...
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### Normal Base for Hilbert Space of delta Potential Well

I'm interested in the problem of an attractive $\delta$ potential. The Hamiltonian is given by $$H = - \frac{\partial_x^2}{2m} - V \delta(x).$$ Solving this typically entails looking at scattering ...
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### Trick for integration into a plane wave basis

I'm reading the article https://arxiv.org/abs/hep-th/9705200 and part of it has left me very confused. In order to speak about their equation (1) the authors make the following statement: \begin{align*...
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### Lorentz-invariant phase space integral

Consider the following Lorentz invariant integral associated to a $2\to 2$ scattering: \begin{equation*} I = \int \frac{d^3\mathbf{p_3}}{(2\pi)^3 2E_3} \int \frac{d^3\mathbf{p_4}}{(2\pi)^3 2E_4} \...
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### Schrödinger equation, 2D delta function potential, and confusion

Apropos of nothing in particular, I thought I would play around with the Schrödinger equation in 2D with a delta function potential. To keep things simple I thought I would concentrate on the bound ...
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### A simple question in quanum mechanics on position and momenum eigenstates

The eigenfunctions (eigenstates) for the momentum of a particle are given by the plane waves $$\phi(x,t) = \sin(kx - \omega t)$$ If we sum a large number of these waves in a range from $0$ to $k_m$, ...
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### Diverging Scattering Amplitudes and Transmission/Reflection Coefficients

I am currently studying scattering theory from Sakurai and Griffiths and I have noticed that for the 1D Dirac potential, the transmission and reflection coefficients diverge when the energy ...
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### Manipulation of functions inside a Dirac Delta function [closed]

It is not clear to me how this derivation proceeds through the steps. Could someone help me understand how to arrive at this result or point me towards a resource that explains these algebraic ...
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### Grassmann variables and orthogonality of coherent fermionic states

Let a coherent fermionic state $$\left|\phi\right> := \left|0\right> + \left|1\right> \phi,\tag{0}$$ where $\phi$ is a Grassmann number (i.e. it anticommutes with other Grassmann numbers). ...
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### How does the Green's function related the wavefunctions at different space-time points in Schrödinger's equation?

I have been trying to study Quantum Field Theory and have come across Green's Functions for the first time. While referring to Tom Lancaster's book Quantum Field Theory for the Gifted Amateur, the ...
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### How to understand this Dirac delta function?

I am reading this paper about quantization of the electromagnetic field, and there is a point where the author imposes the fundamental commutation relation between the vector potential and its ...
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### How do we know Schwinger functions exist?

Let $\mathcal{D}'(\mathbb{R}^n)$ denote the dual of $C^\infty_C(\mathbb{R}^n)$, that is distributions on the set of infinitely differentiable functions with compact support. If $d\mu$ is a probability ...
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### Continuum bases: why do we use dirac delta function? [duplicate]

In discrete bais, we can express a vector as $$|\psi\rangle=\sum_{i} c_i|e_i\rangle$$ with orthonormality $$\langle e_i|e_j\rangle=\delta_{ij}.$$ $\delta_{ij}$ is usual kronecker delta. If we ...
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### How to standardize the energy of a Dirac delta function relative to sample rate (width) and amplitude?

Background I was instructed that a Dirac delta function (impulse from $0$ to $A$ then back to $0$ at short duration) has a white noise audio frequency type excitation distribution here ie. It should ...
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### Neutral Hydrogen Atom Time-Averaged Potential

I'm self-studying the 3rd edition of Jackson's Classical Electrodynamics and I have a question about a problem. The fifth problem of Chapter 1 asks the reader to determine the volume charge ...
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### Green Function for Poisson Equation Derivation

I've read the Green function derivation for Poisson Equation (electrostatics) in this document. There are some points which are not clear for me. On page 10, the document starts with the Poisson ...
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### Coulomb potential from QFT in the external field approximation [closed]

In eq. (13.6.8) at page 558 of the first volume of the Quantum Theory of Fields by Weinberg, the following identity is given: \begin{align} \left[\frac{1}{(q_1\cdot p +i \varepsilon)((q_1+q_2)\cdot p +...
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### Position operator action on a wavefunction [closed]

In a 1 dimensional infinite potential well with width $a$, the ground state wave-function is given by $$\psi(x) = \sqrt{\frac{2}{a}}\sin(\frac{\pi}{a}x)$$ The action of the position operator in the ...
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### Solving Wave Equation in 1+1D via Fourier Transforms with Dirac Delta function initial condition

I'm trying to use the Fourier transform method to solve the following PDE: This is a an infinite string with a pulse for it's initial condition. (At $t=0$, the string is stricken sharply so that the ...
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### A tricky derivation accompanied by delta function

I have been reading a book on Thermal Field theory by Michel Le Bellac During the reading I have come into a seemingly trivial but indeed tricky derivation. On page 26(2.47), we are supposed too prove ...
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### Proof that $-\partial^2 G(x, y) = \delta(x-y)$ for free field propagator
In the appendix A of this paper by Braaten et al., the authors try to compute the divergences of two integrals that come from an expansion of an action $I$ in $\langle e^{iI} \rangle$, via dimensional ...