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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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3
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1answer
449 views

Justification for smeared fields in the Wightman axioms?

I just started reading PCT, Spin and Statistics, and All That. Can someone explain why we use operator valued distributions to describe fields? I read somewhere that it would take infinite energy to ...
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2answers
1k views

What is the most general expression for the coordinate representation of momentum operator?

I have a question about deriving the coordinate representation of momentum operator from the commutation relation, $[x,p]= i$. One derivation (ref W. Greiner's Quantum Mechanics: An Introduction, 4th ...
8
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1answer
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Particle density operator in second quantization form

The particle-density operator is given by $n(\mathbf{x})=\sum_{\alpha}\delta^{(3)}(\mathbf{x}-\mathbf{x}_{\alpha})$, then how to derive its representation in terms of creation and annihilation ...
2
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1answer
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Infinite Potential Well Energy for Piece-wise Constant Wave Function

I'm trying to compute the expectation value of energy for a certain state in an infinite potential well but I'm getting contradictory answers. The well has potential \begin{align} V(x) = \left\{ ...
10
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1answer
1k views

Hilbert space of a free particle: Countable or Uncountable?

This is obviously a follow on question to the Phys.SE post Hilbert space of harmonic oscillator: Countable vs uncountable? So I thought that the Hilbert space of a bound electron is countable, but ...
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4answers
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Hilbert space of harmonic oscillator: Countable vs uncountable?

Hm, this just occurred to me while answering another question: If I write the Hamiltonian for a harmonic oscillator as $$H = \frac{p^2}{2m} + \frac{1}{2} m \omega^2 x^2$$ then wouldn't one set of ...
5
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2answers
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Translation Operator and Position Basis

In Modern Quantum Mechanics by Sakurai, at page 46 while deriving commutator of translator operator with position operator, he uses $$\left| x+dx\right\rangle \simeq \left| x \right\rangle.$$ But for ...
5
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1answer
940 views

Getting rid of double delta function in Feynman rules

[1] A very simple example of feynman rule for scalar fields. After computing the diagram i have got the following: $$ -i(2\pi)^4g^2\int d^4q \frac{i}{q^2 -m^2c^2}\delta^{(4)}(p_1 - p_3 -q) \...
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2answers
1k views

What is the most natural value of Heaviside step function at zero argument?

In many physical applications, the Heaviside step fuction is defined as $$H(x) = \left\{\begin{eqnarray} 1, \quad x>0 \\ 0, \quad x<0 \end{eqnarray}\right.$$ The value $H(0)$ is left undefined. ...
6
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4answers
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Is the momentum operator diagonal in position representation?

The matrix elements of the momentum operator in position representation are: $$\langle x | \hat{p} | x' \rangle = -i \hbar \frac{\partial \delta(x-x')}{\partial x}$$ Does this imply that $\langle x |...
6
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1answer
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Using $\frac{1}{A+i\epsilon} = PV\frac{1}{A}-i\pi\delta(A)$ in Feynman Integrals

Are the following operations O.K.? This is related to the Feynman parameter trick. $$F:= \int_0^1 \mathrm{d}x\int_0^{1-x}\mathrm{d}y \frac{1}{f(x,y)+\mathrm{i}\epsilon}.$$ Now using $$\frac{1}{A+i\...
3
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1answer
454 views

Crazy Dirac Deltas

I'm not expecting any rigor in the following and the answers...since we're dealing with Dirac deltas in the context of QFT. Consider the integral $$ \int d^4q\ \Theta(q_0)\Theta(p_{3,0}+q_0)\ \...
6
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3answers
2k views

Schrodinger equation in term of Fokker-Planck equation

From Wikipedia on the Fokker-Planck equation: $$\tag{1}\frac{\partial }{\partial t}f\left( x^{\prime },t\right) ~=~\int_{-\infty}^\infty dx\left( \left[ D_{1}\left( x,t\right) \frac{\partial }{\...
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2answers
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The definition of Density of States

The density of states (DOS) is generally defined as $D(E)=\frac{d\Omega(E)}{dE}$, where $\Omega(E)$ is the number of states. But why DOS can also be defined using delta function, as $$D(E)~=~\sum\...
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4answers
2k views

Possible ambiguity in using the Dirac Delta function

When doing integration over several variables with a constraint on the variables, one may (at least in some physics books) insert a $\delta\text{-function}$ term in the integral to account for this ...
3
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1answer
2k views

Deriving the Sommerfeld expansion by contour integration (Le Bellac p. 277)

In Le Bellac's statistical physics book he derives the Sommerfeld expansion by a contour integral. The idea is to expand integrals of the type $I(\beta)\equiv \int_{0}^{\infty}d\epsilon\, \frac{\...
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3answers
10k views

Matrix elements of momentum operator in position representation

I have two related questions on the representation of the momentum operator in the position basis. The action of the momentum operator on a wave function is to derive it: $$\hat{p} \psi(x)=-i\hbar\...
2
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1answer
501 views

Differentiation and delta function

Need help doing this simple differentiation. Consider 4 d Euclidean(or Minkowskian) spacetime. \begin{equation} \partial_{\mu}\frac{(a-x)_\mu}{(a-x)^4}= ? \end{equation} where $a_\mu$ is a constant ...
23
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3answers
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Don't understand the integral over the square of the Dirac delta function

In Griffiths' Intro to QM [1] he gives the eigenfunctions of the Hermitian operator $\hat{x}=x$ as being $$g_{\lambda}\left(x\right)~=~B_{\lambda}\delta\left(x-\lambda\right)$$ (cf. last formula on ...
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4answers
12k views

Delta Dirac Charge Density question

I have to write an expression for the charge density $\rho(\vec{r})$ of a point charge $q$ at $\vec{r}^{\prime}$, ensuring that the volume integral equals $q$. The only place any charge exists is at $...
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1answer
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State normalization in Dirac's formulation of quantum mechanics

Let us divide the time $T$ into $N$ segments each lasting $δt = T/N$. Then we write $\langle q_F | e^{−iHT} |q_I \rangle = \langle q_F | e^{−iHδt} e^{−iHδt} . . . e^{−iHδt} |q_I \rangle $ Our ...
8
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4answers
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How do you do an integral involving the derivative of a delta function?

I got an integral in solving Schrodinger equation with delta function potential. It looks like $$\int \frac{y(x)}{x} \frac{\mathrm{d}\delta(x-x_0)}{\mathrm{d}x}$$ I'm trying to solve this by ...
11
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4answers
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How is Gauss' Law (integral form) arrived at from Coulomb's Law, and how is the differential form arrived at from that?

On a similar note: when using Gauss' Law, do you even begin with Coulomb's law, or does one take it as given that flux is the surface integral of the Electric field in the direction of the normal to ...
3
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2answers
2k views

Is the braket notation of the Dirac delta function symmetric?

I have a book saying, $\int \delta(x-x')\psi(x)dx = \psi(x')$ where $\psi(x) = \langle x\lvert\psi\rangle$, so our definition of delta function would be $\langle x'\lvert x\rangle = \delta(x-x')$. ...
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1answer
132 views

mathematical explanation for UV divergences and $ \delta ^{(n)}(0) $

is there any mathematical explanation for the UV divergences ?? i have read that in the framework of Epstein-Glser theory :D these UV divergences appear from the product of distributions anyone does ...
26
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2answers
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What are the units or dimensions of the Dirac delta function?

In three dimensions, the Dirac delta function $\delta^3 (\textbf{r}) = \delta(x) \delta(y) \delta(z)$ is defined by the volume integral: $$\int_{\text{all space}} \delta^3 (\textbf{r}) \, dV = \int_{-...
2
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1answer
123 views

When does the “norm of quasi-eigenvectors” matter in calculations? For which physical results are these even used?

Which physical system in nonrelativistic quantum mechanics is actually described by a model, where the norm of the "position eigenstate" (i.e. the delta distribution as limit of vectors in the Hilbert ...
13
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1answer
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3D Delta Potential Well

The 1D delta potential well $V(x) = -A\delta(x - a)$ always has exactly one bound state. The same is true for the 3D delta potential well $V(\vec{r}) = -A\delta(\vec{r}-\vec{a})$. I can show this for $...
18
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2answers
8k views

How exactly is the propagator a Green's function for the Schrodinger equation

Sakurai mentions that the propagator is a Green's function for the Schrodinger equation because it solves $$\left(H-i\hbar\frac{\partial}{\partial t}\right)K(x,t,x_0,t_0) = -i\hbar\delta^3(x-x_0)\...
3
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2answers
825 views

Derivatives of Dirac delta function and equation of continuity for a single charge

For a single charge $e$ with position vector $\textbf R$, the charge density $\rho$ and and current density $\textbf{j}$ are given by: \begin{equation} \rho(\textbf{r},t)= e\,\delta^3(r-\textbf{R}(t))...
4
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1answer
212 views

$\frac{1}{(1-x)_+}$ type distributions and parton distribution functions

I am trying to get to grips with Altarelli-Parisi-type equations. In chapter 17 of Peskin/Schroeder, they first develop the equations for a similar problem in QED. Equation $(17.123)$ introduces the ...
29
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4answers
5k views

What's wrong with this derivation that $i\hbar = 0$?

Let $\hat{x} = x$ and $\hat{p} = -i \hbar \frac {\partial} {\partial x}$ be the position and momentum operators, respectively, and $|\psi_p\rangle$ be the eigenfunction of $\hat{p}$ and therefore $$\...
5
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6answers
5k views

What is the origin of the Dirac delta term in the dipole electric field?

I am a bit lost how one has deduced the formula for electric field with electric dipole because of some inconsistency between different sources. The Wikipedia article contains a delta function in the ...
5
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2answers
10k views

Limit of Lorentzian is Dirac Delta

I have a quick question that just came up in my research and I could not find an answer anywhere so I thought I'd try here. So one of the definitions of the Dirac Delta is the limit of the Lorentzian ...
12
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5answers
4k views

Laplacian of $1/r^2$ (context: electromagnetism and Poisson equation)

We know that a point charge $q$ located at the origin $r=0$ produces a potential $\sim \frac{q}{r}$, and this is consistent with the fact that the Laplacian of $\frac{q}{r}$ is $$\nabla^2\frac{q}{r}~...