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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Different expression of $\Delta(x, m^2)=(2\pi)^{-3}\int e^{ip \cdot x}\theta(p^0)\delta(p^2+m^2)d^4p$

Let $$\Delta(x, m^2)=(2\pi)^{-3}\int e^{ip \cdot x}\theta(p^0)\delta(p^2+m^2)d^4p.$$ Here $\theta$ is the step function at $0$. I would like to show that this is the same as $$\Delta(x, m^2)=(2\pi)^...
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What is a delta potential?

I know how a delta potential is described mathematically but how can a delta potential be a 'well'? Does it have particles outside the 'well' and 'bind' it or does it somehow have particle inside it? ...
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1answer
56 views

Dirac delta, Fourier transform & exponentials

Consider the following equation/identity: $$ \int d^3x e^{i(\vec{p}+\vec{q})\cdot\vec{x}}=(2\pi)^3\delta^{(3)}(\vec{p}+\vec{q}). $$ I am trying to calculate some commuters I'm encountering in my ...
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Propagators and Green functions for general fields

In my QFT class we have defined the Feynman propagator of a field $\phi^r$ (where $r$ could be a vector or spinor index, or even a multiindex if $\phi$ is a tensor field etc.) as $$ \Delta^{rs}_F(x - ...
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Delta function in Fermi's Golden rule [duplicate]

I am currently trying to understand the Fermi's golden rule. We consider a system with Hamiltonian: $$\hat H = \hat H_0 + \hat Ue^{i \omega t},$$ where the expectation value of $\hat U$ i much ...
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27 views

Books about distribution theory? [duplicate]

I need tips on (elementary) books about distribution theory, with applications to physics and engineering. There is a one and a half page introduction (on page 910) to the Dirac delta function in ...
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60 views

QFT and measures on distributions

Recently I came across the following slogan: ,,constructing quantum field theory on a space $X$ means constructing a measure on the space of all (Schwarz?) distributions $\mathcal{S}'(X)$''. I would ...
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449 views

Green's function on torus

I have a question about the Green's function $G(z,w)$ on torus which takes the form (for example the first equation in the paper https://annals.math.princeton.edu/wp-content/uploads/annals-v172-n2-p03-...
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3answers
72 views

Verify that the electrostatic potential satisfies the Poisson equation [closed]

I'm reading Sect1.7 of Jackson's classical electrodynamics but I have trouble following his argument. Could someone help explain how exactly the Laplacian is evaluated in 1.30? Is it calculated with ...
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3answers
58 views

Product of projectors of a observable with continuous spectrum

Consider a hermitian quantum mechanical observable $\hat{N}$ with discrete non-degenerate eigenvalues $n_{i}$, and eigenstates $\left | N_{i} \right>$, thus $$\hat{N}\left | N_{i} \right>=n_{i} ...
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1answer
38 views

$r$-representation of Operator

I am watching this video https://www.youtube.com/watch?v=sYgX5pdncG8 at 14:30, it has $\langle r|H|r'\rangle = H(r) \delta(r-r') $ Can you help me to understand why it is so? I thought it should ...
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How can I find the normaliztion constant of probabilty amplitude for small space-time path?

For free particle with $V=0$ case we get $$<x_n,t_n;x_{n-1},t_{n-1}>=\frac{1}{w(\Delta t)}\exp \left [\frac{im(x_n-x_{n-1})^2}{2\hbar\Delta t}\right ] \tag{6.42}$$ given in eqn. 6.42 of ...
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Fourier transform and commutation of integral and laplacian

While determining the Breit-Fermi potential, I carry out a Fourier transform by using the following identity: $$\int\frac{d^3q}{(2\pi)^3}e^{-i\vec{q}\cdot\vec{r}}\frac{1}{|\vec{q}|^2}=\frac{1}{4\pi r}$...
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51 views

Second quantised momentum operator in position basis

The Problem I was trying to show that the momentum operator is given by $\sum_r c^{\dagger}_r (-i\hbar)\nabla c_r$.This is how I tried showing it $$\hat{P}=\sum_{r,r'}c_r^{\dagger} \langle r|\hat{P}|...
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2answers
64 views

Book recommendations for Fourier Series, Dirac Delta Function and Differential Equations?

I'm a second-year undergrad and currently taking a course in Mathematical Physics which covers the topics of Dirac delta functions, Fourier series, Fourier transforms and Differential equations. They ...
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38 views

Dirac Delta Function and Schwarzschild Singularity [duplicate]

Pardon my naive question. I recently found out about Dirac Delta function. It is interesting to note that Schwarzschild singularity gives the infinity values of the General relativity field equations ...
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1answer
60 views

Expression of Dirac Delta Correlation

spatio-temporal white noise $\xi(x,t)$ is often expressed as $$\langle\xi(x,t)\rangle=0,$$ $$\langle\xi(x_1,t_1)\xi(x_2,t_2)\rangle=\delta(t_2-t_1)\delta(x_2-x_1).$$ Now I understand that the first ...
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The Dirac function derivative potential [duplicate]

Does the Dirac delta function derivative potential have a solution in one dimension?
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42 views

Scattering cross section from sum of delta functions in 3D

we had the following question in our exam: I know basic scattering concepts like partial waves, born approximation etc. and the solution of common potentials like coulomb or hard spheres but have ...
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40 views

Integration by parts with Dirac delta function in deriving the Lienard-Wiechert fields

Lienard-Wiechert fields can be derived by directly differentiating the Lienard-Wiechert potentials. But for convenience many textbook authors choose to differentiate under the integration sign of the ...
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102 views

Which integration is right? (Integration of operators and delta functions)

Let us consider following integral $$\int \int dx dy f(x)g(y)\delta(x-y).\tag{1}$$ We can bring integral with respect to $x$ to the front or $y$ to the front and integrate out to get the same answer $$...
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102 views

Classical motion in delta potential

The question about classical motion in delta potential may seem artificial, but it makes sense it you try to calculate the propagator for particle in delta-potential, because you usually need to know ...
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53 views

Proof of commutator in Tong's notes on QFT

I am following David Tong's notes on QFT: http://www.damtp.cam.ac.uk/user/tong/qft.html . In equation 2.21, he tries to prove $$[\phi(\vec{x}),\pi(\vec{y})] = i\delta^{(3)}(\vec{x}-\vec{y}).$$ Here, $\...
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quantum work distribution from sudden quench

I'm trying to calculate the quantum work distribution P(W) from a sudden quench, which the expression is $P(W) = \sum_{n,m}p_n^0p_{m|n}^\tau \delta[W - (\epsilon_m^\tau - \epsilon_n^0)],$ where $p_n^...
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1answer
105 views

Delta Function of a Curve

I need to evaluate the integral \begin{equation} \int_0^1\mathrm dt\,f\left(t\right)\delta^{\left(3\right)}\left(\vec r\left(t\right)-\vec r_0\right) \end{equation} where there is only one $0\leq t_0\...
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Delta Potential Boundary Conditions on the wavefunction

I'm reading over how the delta function potential problems are solved and I can't really understand the origin of these boundary conditions: $(1) \,\,\psi \,\,$ is always continuous $(2) \,\, \dfrac{...
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1answer
127 views

Integrating the derivative of a Dirac delta function [closed]

I need to prove that $\rho=p_0{d\over dx}\delta(r-r_0)$ is the charge density of a single electric dipole in the point $r_0$ which direction is the $x$ axis. for that i should find the charge , dipole ...
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1answer
48 views

Why are $E > 0$ particle waves influenced by a Delta-function well potential?

One of the simple examples of potential wells we learn about in Quantum Physics is the Dirac delta function well $$V(x) = -\alpha\delta(x)$$ and we learn that this function has a single bound state, ...
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What's the difference between repulsive Delta function and Attractive Delta Function?

What's the difference between repulsive Delta function and Attractive Delta Function? I need to clear the concept. Please note that I know what's the Delta function.
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Expression for the (1+1)-dimensional retarded Dirac propagator in position space

Where an expression for the (1+1)-dimensional retarded Dirac propagator in position space can be found, especially including the generalized funcion supported on the light-cone? In particular, is it ...
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1answer
32 views

Can't verify solution to Debye-Huckel equation

In the derivation of Debye length (https://en.wikipedia.org/wiki/Debye_length), the Debye-Huckel equation for a neutral system is $\nabla^2 \Phi(r) = \frac{\Phi(r)}{\lambda_D^2} - \frac{Q \delta^3(\...
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1answer
111 views

Coherent state basis of (relativistic) particle Fock space

For a neutral scalar bosonic particle of mass $m$, I consider a Fock space with an orthonormal basis of momenta eigenstates \begin{equation}\label{Fock-p-states} \left|p_1p_2\cdots p_n\right\rangle=\...
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Do Equal Time Commutation / Anticommutation relations automatically also apply for spatial derivatives?

The question is basically in the title. My naive thought was that when a commutation relation holds for all field operators $\Psi(\vec{x})$ (by "all" I mean "at all positions $\vec{x}$") on a fixed ...
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1answer
129 views

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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1answer
36 views

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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52 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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1answer
223 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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110 views

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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62 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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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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1answer
91 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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99 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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111 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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85 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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1answer
75 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

My question 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
38 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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4answers
483 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
39 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
72 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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