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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Derivation of the wave field from an impulsive planar source

The figure shows the geometry for deriving the wave field from an impulsive planar source. The impulse is approximated by a rectangle $c\epsilon$ wide and $\alpha=\frac{1}{c\epsilon}$ high so the ...
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Can the impulse response, Green's function, et al, be defined for each of the two initial conditions of the wave equation? [migrated]

{NOTE--This question has been re-posted on SE math in response to the hold put on it. Because of the answer I can not delete it.} The 1D homogeneous wave equation is $$u_{xx}(x,t)-\frac{1}{c^2} u_{...
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How to prove scaled delta function relation mathematically? [migrated]

I am working through Shankar's Introduction to Quantum Mechanics. I have come across exercise 1.10.1, which asks the reader to show that: $$\delta(ax)=\frac{\delta(x)}{|a|}.$$ I can understand it ...
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Transition probability derivation: How to prove $\lim_{\alpha\rightarrow\infty} \frac{\sin^2\alpha x}{\alpha x^2} ~=~\pi\delta(x)$?

How to prove $$\lim_{\alpha\rightarrow\infty} \frac{\sin^2\alpha x}{\alpha x^2} ~=~\pi\delta(x)~?$$ I have encountered this limit while learning time dependent perturbation and transition ...
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Limit of the $\sin^2$ function in the derivation of Fermi's golden rule

In the derivation of Fermi's golden rule one typically arrives at an expression of the form $$ \frac{\sin^2(\omega t)}{\omega^2} $$ which is then converted to $$ \pi t\delta(\omega). $$ I cannot ...
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Rationalizing the current operator

In eq. 1.191 of Mahan Many-Particle Physics, the current operator is defined as $$j(r)=\frac{1}{2}\sum_ie_i[v_i\delta(r-r_i)+\delta(r-r_i)v_i].$$ I'm trying to make some sense of this definition. I ...
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28 views

Double Sommerfeld Expansion

Consider the Fermi-Dirac expansion for an arbitrary function $f(\epsilon)$: $I(f)=\int_0^\infty d\epsilon\frac{f(\epsilon)}{e^{\beta(\epsilon-\mu)}+1}$ The large $\beta$ expansion of this quantity ...
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396 views

Understanding Heaviside and Dirac Delta for Quantum step function

Looking at the solution for from this site I'm a bit confused on how two quantities necessarily reduce. I'm given this wavefunction $$ \psi(x) = \begin{cases} Ax & 0<x<a/2 \\ ...
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1answer
144 views

Why Dirac delta function is considered as wave function?

I have read in my textbook that dirac delta function is regarded as wave function. But isn't it violating the condition required for a wave function ie . It becomes infinite $ δ(x-x_0)$ at $x=x_0$ ...
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1answer
131 views

Towards a matrix element definition of PDF

In Schwartz's book, 'Quantum Field Theory and the Standard Model' P.$696$, he starts to derive an expression for a parton distribution function in terms of matrix elements evaluated on the lightcone. ...
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2answers
187 views

Delta potential in terms of annihilation/creation operators

Let the Hamiltonian of a system on a discrete lattice be given by $$ \mathcal{H} = \gamma \sum_\vec{x} c^\dagger_\vec{x}c^\vphantom{\dagger}_{\vec{x}+\vec{y}} + \text{h.c.}, $$ where $\gamma$ is ...
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Constraints in path integral and the Lagrange multiplier

I was reading some references on the slave-particle approach to the Kondo problem and Anderson model. It is known that the slave-particle is introduced in the large Hubbard $U$ limit of the system so ...
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52 views

Colombeau Algebra for physics students

I have an undergraduate degree in physics, taken 2 years of calculus, and a rigorous course in linear algebra. I have not taken a math course in analysis, though have read a bit about it on my own. ...
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1answer
63 views

Covariant form of Green's function for wave equation

In J. D. Jackson's "Classical Electrodynamics", page 614 in the 3rd edition, he states that you can write the Green's functions for the wave equation in covariant form using the fact that \begin{align*...
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Confusion regarding the bound state of a Delta-function potential and Tunneling

I was reading (Griffith's QM book) about the Bound states for delta-function potential of the form $-\alpha \, \delta(x)$ where $\alpha > 0$. I feel a bit conceptually unclear. Few doubts I have ...
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60 views

${}$ Dirac delta potential

We know that the number of bound states for an attractive delta potential is one. If so what will the number of bound states for a particle in a repulsive delta potential? If $V(x)= +a \cdot \delta(x)$...
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127 views

Integration over phase space for a one-dimensional harmonic oscillator

The problem asks for a proof of the following equation, and I have no idea on how to approach this: $$\int dx dp \delta(E-\frac{p^2}{2m}-\frac{m \omega^2 x^2}{2})f(E) = \frac{2\pi}{\omega}f(E) ,$$ ...
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163 views

Quantum filter, potential barrier

I have been trying to teach myself quantum mechanics for quite a time now and I need help with a probably simple problem. We are looking at the Schrödinger equation of particles of mass $m$ in one ...
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55 views

Deriving Current density of a Moving Point Charge Using the Continuity-Equation

Problem: We know, a point charge at position $\mathbf{r}_q$ has the charge density $$\rho_q(\mathbf{r})=q\delta(\mathbf{r}-\mathbf{r}_q) \tag{2}$$ if it moves with the velocity $\mathbf{v}$, we get ...
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75 views

How does a Lagrangian with delta potential transform to a Hamiltonian?

Suppose the Lagrangian was given as: $$L = \frac{1}{2}\int_{-\infty}^{\infty} \underbrace{\left(\dot{A(}z)^2-A(z)^2\right)+\left(\dot{Q(}z)^2\delta(z)-Q(z)^2\delta(z)\right)+2\dot{Q(}z)\cdot A(z) \...
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210 views

Continuity equation involving vectors

A time dependent point charge $q\left ( t \right )$ at the origin $\rho\left ( \vec{r},t \right )=q\left ( t \right )\delta ^{3}\left ( \vec{r} \right ) $, is fed by a current $$\vec{J}\left ( \vec{r},...
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535 views

Hamiltonian with Dirac Delta function

I've to compute this expression $$ \hat{H} ~=~\frac{1}{4}g_2\int d^3R\int d^3r\ \bar{\Psi}(\vec{R}+\frac{\vec{r}}{2})\bar{\Psi}(\vec{R}-\frac{\vec{r}}{2}) $$$$ \times \left[ \delta(\vec{r})\nabla_{\...
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52 views

How to get the imaginary part from the Källén-Lehmann propagator

During field theory course the Källén-Lehmann propagator was defined as follows: $$D_F(p^2) = \frac{i}{p^2-m^2+i\epsilon} + \int^{\infty}_{4m^2}ds\rho(s)*\frac{i}{p^2-s+i\epsilon} \tag{1}$$ ...
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Interpretation of induced force between two Dirac delta potential wells

This problem is based on MIT OCW course 8.04 problem set 6 question 5(e). https://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2013/assignments/MIT8_04S13_ps6.pdf Consider two Dirac ...
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1answer
26 views

Deriving the form of radial distribution function in molecular dynamic simulation?

I am trying to derive the equation 8.16 in the attached image from the definition of the radial distribution function. However, I am unable to derive it with proper constants of $\frac{2}{N(N-1)}$ My ...
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223 views

Divergence of $\frac{ \hat {\bf r}}{r^2}$ , what is the 'paradox'?

I just started in Griffith's Introduction to electrodynamics and I stumbled upon the divergence of $\frac{ \hat r}{r^2}$ , now from the book, Griffiths says: Now what is the paradox, exactly? ...
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1answer
38 views

Writing a two variable function$ f(x,t)$ in terms of Dirac-Delta $δ(x)$ function and a function $P(t)$?

How to write a two variable function $f(x,t)$ in terms of Dirac-Delta $\delta(x)$ function and a function $P(t)$? For example; I read something in a book. You can find the following picture. But ...
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1answer
194 views

Singularity of $B$-field in a Dirac String

I was assigned this question related to Dirac strings: Given a vector potential $\vec{A}= \frac{1-\cos(\theta)}{r\sin(\theta)}\hat{\phi}$, show that there is a singularity in the B field ...
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59 views

What is the relationship between wave function and classical distribution function?

The question is a bit weird since the wave function is quantum mechanical and the distribution function in phase space is really something classical. But I would still like to know if I take the ...
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1answer
57 views

Itzykson and Zuber: Reduced $T$-matrix

Could someone help me understand the reduced $T$-matrix mentioned in Itzykson and Zuber, eq. $$\langle{f}| T|p_1p_2\rangle=(2\pi)^4\delta^4(P_f-p_1-p_2)\langle f|\mathcal{T}|p_1p_2\rangle. \tag{5-7}$...
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Green function derived from the idea of position representation in quantum mechanics

I am trying to study green's function by using a lecture by NPTEL its link is https://www.youtube.com/watch?v=ZJ7v6VZQ32k&t=439s I don't get this step by him at the minute of ten. $$D_x G(x,x^{'...
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2answers
151 views

Mathematical rigorous definition for an electrical dipole

I've been reading Laurent Schwartz's Mathematics for the physical sciences, and in the chapter on distributions he makes many cool examples of ways to define in a mathematical rigorous way certain ...
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1answer
57 views

Position representation of an operator

$$\langle\ x\rvert M\lvert\ x'\rangle=M(x)\langle\ x\lvert\ x'\rangle=M(x)\delta(x-x')$$ I know this is true for if $M$ is a momentum operator or position operator, is this is true for a general ...
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679 views

Simple question about Dirac delta-function [closed]

I can not undestand this mathematical formula: $$ \large \int_{a-\epsilon}^{a+\epsilon}f(x)\delta(x-a)dx=f(a) $$ I understand that it is the derivative of an integral evaluated in a, but still can ...
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56 views

Is it true that any electrostatic charge density can be represented by Dirac Delta function? Give me a general example

My college professor said that any charge density can be replaced by Dirac Delta function but how I want to know.
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44 views

Finding normalization constant of a wave function with definite momentum

I try to read Sakurai's Modern Quantum Mechanics but I stuck at this point, $$\delta(x^{'}-x^{''})=|N|^{2}\int dp^{'}\exp\Biggl({ip^{'}(x^{'}-x^{''})2\pi\over h}\Biggr)$$ This is an expression for ...
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Expected value of the current density operator

In Ullrich's TD-DFT book, the paramagnetic current-density operator is defined as $$\hat{\mathbf{j}}(\mathbf{r})=\frac{1}{2i}\sum_{a=1}^{N}\left[\nabla_a\delta(\mathbf{r}-\mathbf{r}_a)+\delta(\mathbf{...
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74 views

Why do Candelas and Howard say that $\sum_{n=1}^\infty \cos\left( n \kappa \epsilon \right) \ = \ - \frac{1}{2}$?

In the paper Vacuum $\langle \phi^2 \rangle$ in Schwarzschild Spacetime by Candalas and Howard, they say that for each non-zero $\epsilon$ it is true that $$ \sum_{n=1}^\infty \cos\left( n \kappa \...
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1answer
63 views

Problem in the continuum limit of a Kronecker delta

I am having troubles in understanding how to correctly perform the continuum limit of a double sum containing a Kronecker delta. Imagine to integrate a function depending on $t$ and $t'$, both ...
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2answers
64 views

Factorising a 4D Dirac delta function in a rest frame

I'm working through a QFT problem and at one stage in the solutions we have this step: $$\delta^{(4)}(p - q_1 - q_2) = \delta(E_1 +E_2 - M)\delta^{(3)}(\bf{q_1} - \bf{q_2}).$$ We are working in the ...
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54 views

What is dipolar charge distribution?

An electric dipole is a system of two opposite point charges when their separation goes to zero and their charge goes to infinity in a way that the product of the charge and the separation remains ...
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1answer
49 views

Properties of Dirac delta function in Integral

I was reading commutation relation of canonical momentum in KG Field from Lectures of Quantum Field Theory by Ashok Das. In page 179, He has used Integration to derive the result where he expressed ...
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1answer
67 views

What is the volume charge density (in spherical coordinates)?

What is the volume charge density (in spherical coordinates) of a uniform, in finitesimally thin spherical shell of radius $R$ and total charge $Q$, centered at the origin? Give your answer in terms ...
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1answer
56 views

$\delta^{(2)}$ convention

In this note: https://arxiv.org/abs/hep-th/0410165 at page 12 there is a delta-function constraint written as: \begin{align} \delta \left( ^ { U } M \right) = \prod _ { i < j } \delta ^ { ( 2 ) } \...
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1answer
68 views

A question about $\delta(x) \star \delta(p)$

Using the Moyal product between two delta functions in $(x,p)$-space one gets $$ \delta(x) \star \delta(p) = \frac{1}{\pi} e^{2ixp}. $$ However, $\delta(-x)=\delta(x)$ and last time I checked $e^{...
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1answer
87 views

Dirac delta function mathematical expression proof

In a discussion of the second order transitions in graphene this mathematical expression is used. $$ \left|\frac{1}{\varepsilon + i \Gamma/2}\right|^2 = \frac{2\pi}{\Gamma}\delta(\epsilon) $$ And I'm ...
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254 views

What is the definition of $\delta^{m|n}$ and $\delta^{m}$?

I am reading the paper. What is the definition of $\delta^{m|m}$ and $\delta^{m+k}$ in (1.1) and (1.3) on pages 2,3? Are they some kind of delta function?
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55 views

What is the symmetry behind this degeneracy?

I was working on a quantum mechanics problem, involving the perturbation of the 3D cubical potential well: Suppose we perturb the infinite cubical well \begin{equation} V(x,y,z)=\begin{cases} 0, \...
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1answer
72 views

Delta function eigenstate for non-zero potential

Consider the potential $V(x)=\frac{2}{x^2}$ and let $\frac{\hbar^2}{2m}=1$ for convenience. Now consider the function $\psi(x)=\delta(x)$. According to Griffiths (electrodynamics book) problem 1.45(a),...
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1answer
75 views

Derivation of the QFT Propagator

I don't understand how we get from the RHS to the last line. \begin{eqnarray} \left[ \hat{H}_x - i \frac{\partial}{\partial t_x} \right] G^+(x,t_x,y,t_y) &=& -i \delta (t_x - t_y) \sum_n{\...