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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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205 views

Wave-Function Normalization in Momentum Space Not Possible

Hello, I just have a question about this passage; specifically, I do not understand why the result of the inner product (the integral of u_k* and u_k') being the delta function defies conventional ...
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1answer
33 views

Non-resting initial value problem with impulsive input

Consider a hypothetical model of an extended mechanical system (in which a derivatives of higher order than acceleration may exist d) as bellow: $$\sum_{n=0}^N {a_n x^{(n)}}= f_0 \delta(t-t_0)$$ ...
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2answers
76 views

Multiplying Distributions in finite-temperature Keldysh/Thermo-field field theory

In the real-time finite temperature formalisms (Schwinger-Keldysh or Thermo-field), the free propagators are often defined with terms like: $$ \mathrm{Dirac\ Delta}\ \times \ \mathrm{Thermal\ ...
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0answers
24 views

Shifting the derivative outside the integral [closed]

In page 62 of Shankar's Principles of Quantum Mechanics, the author conveys the following: $$\int \delta'( x-x') f(x') dx' = \int \frac{d\delta(x-x')}{dx}f(x')dx'= \frac{d}{dx} \int \delta(x-x') f(x') ...
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1answer
82 views

Quantum mechanics perturbation and the orthogonality of energy states [closed]

Consider the following question and its solution: My question is concerning the solution of $a_{nm}$. Surely if the energy eigenstates are orthogonal then $a_{nm}$ must be equal to zero. WHy is this ...
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2answers
30 views

probability distribution of dependent random variables

If we have dependent random variables, then what how is the distribution the pdf look like? Can it be a normal distribution? For example, additive white Gaussian noise (AWGN) has a normal distribution,...
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1answer
130 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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1answer
85 views

1D delta potential hamiltonian

I have two concerns regarding the delta potential hamiltonian: Is my understanding that in quantum mechanics we use self-adjoint operators. However I cannot figure it out if the hamiltonian that ...
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0answers
98 views

is it true that $\lim\limits_{\epsilon \to 0^{+}} \ln( x \pm i \epsilon ) = \mathscr{P}\ln|x| \pm i \pi \Theta(-x)$?

Is the following statement true? $$ \lim\limits_{\epsilon \to 0^{+}} \ln( x \pm i \epsilon ) = \mathscr{P}\ln|x| \pm i \pi \Theta(-x) $$ where $\mathscr{P}$ is the Cauchy principal value. The above ...
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2answers
62 views

Calculating $\langle p|x\rangle$ and $\langle x|\hat{p}|x'\rangle$ - does one result from the other?

In showing that $$\langle x|\hat{p}|x'\rangle = -i \hbar \frac{dδ(x-x')}{dx}$$ I've seen many solutions doing something similar to Can I replace eigenvalue of p operator with position space ...
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1answer
261 views

Dirac Delta potential

As we know a particle in attractive Dirac delta potential has discontinuity in the derivative of its wavefunction. I have two questions in this regard: Can a second order differential equation be ...
1
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1answer
469 views

Solution of Schrödinger equation for Dirac delta potential $V(x) = \sum_{i=1}^P \sigma_i\delta(x-x_i)$

So, I am trying to solve Schrödinger Equation for Dirac delta potential. The Schrödinger equation: $$ -\frac{\hbar^2}{2m}\frac{d^2\Psi(x)}{dx^2} + V(x)\Psi(x) = E\Psi(x) $$ And, the potential looks ...
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2answers
95 views

What's the acceleration of an object if we applied delta dirac function?

If we applied a delta Dirac function as a force, how can we obtain the acceleration of that object? I know that this is called impulse that changes the velocity, but since there is a change in ...
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1answer
71 views

Continuous limit of discrete position basis

Say we have a $1D$ lattice with spacing $a$ between two sites. How does one formally map the discrete position basis of the lattice to a continuous one in the limit $a\to 0$. For instance how does ...
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1answer
205 views

Mean value with delta function [closed]

How do I compute this matrix element $$\langle 1|\delta(\hat x-b)|1\rangle$$ that models a 1-D harmonic oscillator? I have done the same for the ground state (by seting delta function as the Fourier ...
3
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3answers
3k views

Divergence of Electric Field Due to a Point Charge

I am trying to formally learn electrodynamics on my own (I only took an introductory course). I have come across the differential form of Gauss's Law. $$ \nabla \cdot \mathbf E = \frac {\rho}{\...
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1answer
98 views

Inhomogeneous wave equation by fourier in analysis

$$\nabla^2\psi_\omega+\frac{\omega^2}{c^2}\psi_\omega=-g\omega,\tag{14-16}$$ which is similar to Poisson's equation. We may synthesize the solution of Eq. (14-16) by the superposition of unit ...
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1answer
204 views

Why position operator is non-degenerated?

In quantum mechanics one can assume position operator $\hat{X}$ must have continuous spectrum, as experiments say it is possible to find a quantum particle at any point of the space. The question is ...
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1answer
401 views

How do you solve the Schrödinger equation with a position space delta function potential in momentum space? [closed]

I am solving the Schrodinger equation in position space with an attractive delta function potential energy, $$ -\frac{h^2}{2m} \frac{d^2}{dx^2} \psi(x)-\lambda \delta(x) \psi(x)=E \psi(x), $$ for a ...
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296 views

Kronecker delta commutation relations for QFT

Setup: In many textbook treatments of canonical quantization (e.g., Peskin and Schroeder), one imposes canonical (equal time) Dirac delta commutation relations on the conjugate field operators. e.g., ...
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32 views

Confusions in discretizing a momentum delta integral

I have an integral of the following form: $$\int dk_{x}dk_{y}dk_{z}\frac{1}{(2\pi)^{3}}\delta\left(\sqrt{k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}-p\right)f(k)$$ There are two ways to convert it into discrete ...
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1answer
183 views

Step involving delta-function in the Klein-Gordon equation solving

The solution to the equation \begin{equation} \int d^3k \; e^{i\mathbf{k}.\mathbf{x}}(k^2-m^2)\phi(\mathbf{k})=0 \end{equation} (which appears in the Klein-Gordon equation solving) is said to be \...
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1answer
249 views

Massless $m=0$ 4D Fourier transform of $(p^2 + i \epsilon)^{-2}$

This question is related to this one. I'm assuming that we're in or on the the light-cone $s \leq 0$ in what follows. Suppose I'm interested in computing the following Fourier transform, in the ...
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1answer
337 views

Solution of differential equation (Dirac delta function)

I have been given the following: $$y''(x)+\omega^2y(x)=s(x),$$ $$s(x)= \delta(x)-\delta\left(x-\frac{1}{2}\right)$$ for $-\frac{1}{4}<x<\frac{3}{4}$. (Periodically repeating for $x$ ...
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1answer
1k views

Physical meaning of the Jacobian in relation to Dirac delta function

Is there a physical meaning to the equation $$\delta(x-a)=\dfrac{\delta(\xi-\alpha)}{|J|} \, ?$$ In non-rectangular coordinate systems where the transformation is non-singular, what is the ...
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1answer
143 views

Ambiguous Use of Dirac Delta Function [closed]

Shankar (in his book Principle of Quantum Mechanics book,page 64) mentions that instead of integrating with respect to dx' in $$\int \delta '(x-x') f(x')dx'=\frac{df(x)}{dx},$$ where $$ \delta '(x-x'...
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0answers
172 views

$D$-dimensional Schrodinger's equation with a Dirac delta potential

I know that for $D\geq 2$, there is no bound state for a Dirac potential $V=-\alpha \delta(\textbf{x})$ unless we use an ultraviolet cutoff $k_{max}=1/a$. I showed this by solving the Schrodinger's ...
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1answer
104 views

Using integrals to expand a vector in continuous basis

I am new to quantum mechanics. I have been trying to understand why when we want to represent a function $$\psi(x)$$ as a ket in continuous basis |x> we us the integral: $$\vert \psi(x)\rangle =\int\...
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1answer
568 views

Normalization of eigenfunction to Dirac-delta function

In the first chapter of Principles of Quantum Mechanics by R. Shankar, he describes finding the eigenvalues and eigenfunctions of the operator $K=-iD=-i\frac{d}{dx}$. For context, he does this: What ...
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1answer
187 views

Wigner function of position eigenket?

I am trying to derive the Wigner function of position eigenket $\rho = |x'\rangle \langle x'|$. One method is to use the formal expression for the Wigner function and then solve: $$ W(q,p) = \frac{1}...
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1answer
169 views

Delta function potential and continuity of the derivative

In Gasiorowicz Quantum Mechanics, 3rd ed, pg.81, he finds the bound states for a delta function potential the following way: I have the following delta potential: $V(x)=-\frac{\hbar^2\lambda}{2ma}\...
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1answer
160 views

Scalar commutation relations with equal times enforced by a delta function

(I'm following these notes by Vadim Kaplunovsky titled "Feynman Propagator of a Scalar Field," specifically asking about equation 10.) When calculating the time derivative of the Feynman scalar ...
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1answer
143 views

Regarding the commutator of ladder operator in QFT

I am trying to verify the computation of the commutator of the ladder operator for Klein-Gordon solutions, but it seems like I am unable to do it properly. Here is what I do: For, $$ \varphi(x^\mu)=\...
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1answer
355 views

Normalization of states and bracket notation

In Peskin & Schroeder's QFT, if we set $$|p\rangle = \sqrt{2E_p} a^{\dagger}_p|0\rangle \tag{2.35}$$ as in equation 2.35, then how do I get to the next equation 2.36: $$\langle p|q\rangle = 2E_p (...
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1answer
96 views

Is the static electric field $\vec{E}$ defined and finite where charges are located?

Suppose you have a continuous charge distribution, e.g. a wire, a disk, or a plane, represented by a subset $D\subset \mathbb{R^3}$. The charge density is $\rho: \mathbb{R^3}\rightarrow \mathbb{R}$. ...
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0answers
457 views

Relationship between Green's function and impulse response

In my field, electrical engineering, we frequently study linear time-invariant systems of the following form: $$ a_n\frac{d^ny}{dt^n} + a_{n-1}\frac{d^{n-1}y}{dt^{n-1}} + \ldots + a_1\frac{dy}{dt} + ...
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2answers
105 views

Basic confusion with quantum mechanical operators

Given a classical observable, $a(x,p),$ Weyl quantization gives the correspondent QM observable as: $$\langle x | \hat{A} | \phi \rangle=\hbar^{-3}\int \int a \left(\frac{x+y}{2},p\right)\phi(y) e^{2\...
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1answer
375 views

Off-diagonal elements of momentum operator in position representation

In another Phys.SE question, I've proposed the next-cited proof of this statement: the momentum matrix elements in position representation, $\langle x'|\hat{p}|x\rangle$, are all not null I'm ...
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1answer
738 views

What are the real life examples of Double Dirac-Delta Potential barrier/well?

My question is do we see in nature any potential which is close to Double Delta Potential barrier/well? If yes then which are those? Thanks in advance.
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2answers
191 views

Physics terminology about smeared and unsmeared fields

Let $M$ be a smooth manifold and denote $C^\infty_0(M)$ the space of smooth functions with compact support. In Mathematics a distribution is defined to be a continuous linear functional $\phi : C^\...
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0answers
238 views

Delta function potential, bound state energies

A lot of books find the bound state energies of the single delta function potential centred at $x=0$ by integrating the Schrodinger equation around $x=0$, using symmetric limits and making the width ...
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1answer
223 views

The charge density $\rho_{\infty}$ of the sphere

I have two charged hemispheres (which are very close to each other, we consider them now as a sphere) with the charge density given as $\rho = \frac {Q}{\pi a^3 \frac {4}{3+n}} (\frac{r}{a})^n$, ...
3
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3answers
693 views

Heat equation: Heat Kernel as $t\to0$

Consider heat flow on an infinite, 1D wire. The temperature T(x,t) obeys the diffusion equation, $$ \frac{∂T}{∂t} = D \frac{∂^2T}{∂x^2} $$ with initial condition $T(x,0) = δ(x)$. The heat kernel is ...
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0answers
291 views

Density of a thin disk by delta Dirac function

How to write a volume (charge for example) density using delta Dirac function in spherical coordinates in this case: There are thin charged disk (radius $a$) with surface charge density $\sigma$. The ...
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0answers
171 views

Classical field theory: marriage of differential geometry to functional analysis?

I have always wondered (and thoroughly - but perhaps not enough - searched for literature) how the purely geometrical formulation of classical field theory (take for example a hardcore text on this ...
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1answer
505 views

Units of a dirac delta function in quantum mechanics

In quantum mechanics the eigenfunctions of the position are dirac delta functions, $A\delta(x-x_0)$, where $A$ is some constant. Eigenfunctions of the position are usually normalized with a "Delta ...
0
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1answer
220 views

position state as a sum of momentum states

Leonard Susskind, in his lecture (minute 41) about QFT states that when a field acts on a vacuum state it gives a position state- $$\Psi^\dagger(x)=\sum_{k}e^{-ikx}a^\dagger(k)\left|0\right>=\sum_{...
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1answer
487 views

How to express Dirac delta-function in functional form?

In the paper written by Dmitry Bagrets, Alexander Altland and AlexKamenev (Sachdev–Ye–Kitaev model as Liouville quantum mechanics: http://www.sciencedirect.com/science/article/pii/S0550321316302206?...
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3answers
147 views

Obtaining the charge from the charge density using distribution theory

In electrostatics, for several reasons, it seems that the correct way to understand the charge density $\rho$ isn't as a function $\rho : \mathbb{R}^3\to \mathbb{R}$, but rather as a distribution $\...
4
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3answers
871 views

Generalising a Dirac Delta function formula in General Relativity

I'm currently stuck on a problem where I have to integrate on a particular set defined through a dirac delta function. If I understood correctly it all boils down to using the curved analogous of $$ ...