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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Factor before Dirac delta in magnetic dipole field formula

I bumped into this formula for the magnetic induction field generated by a dipole, containing Dirac's delta, while studying hyperfine splitting: $$\textbf{B}(\textbf{r}) = \frac{2}{3}\mu_0 \textbf{m}\...
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Doubt in the completeness of wave function

I am reading about the completeness property of wave function. The following is given about it- The energy eigenstates are complete in the sense that any reasonable wave function $\psi(x)$ can be ...
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Delta Functions In Energy and Momentum Variables

I am confused about some integral formulae presented in An Introduction to Quantum Field Theory by Peskin and Schroeder, specifically formulae 4.81 and 4.82. Formula 4.81 is $\int \frac{p_1^2 dp_1 d\...
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Delta Function Identity and Path Integral containing Delta Function

Context I'm currently working through a paper where we utilize the identity $$1=\int\mathcal{D}\sigma\delta(\sigma-\phi^2)=\int\mathcal{D}\sigma\mathcal{D}\zeta e^{i\int\zeta(\sigma-\phi^2)}$$ This is ...
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How can the Fourier delta function produce different matrices in single-pixel imaging?

The pattern in the masks used for single-pixel imaging are created applying equation $(1)$, $$P_\phi (x,y) = \frac{1}{2} \left[ 1 +| F^{-1} \{\delta_H (u,v) e^{i\phi}\}|\right], \tag{1}$$ in which the ...
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Visualizing Electron Behavior in a Steel Plate

Start with a neutral iron plate, voltage measurement to ground is zero, number of electrons = number of protons. Now start adding electrons to the plate. Negative voltage to ground will increase in ...
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Singular Integrand vs Diverging integral

I am reading Jackson's Electrodynamics and came across this part that I'm not sure I understand. Specifically what is Jackson referring to when he says "it turns out that the resulting integrand ...
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What is the one-sided Fourier transform of a constant? [migrated]

A definition of the Fourier transform commonly used is (I always forget which convention of normalization to use) \begin{align}f(\omega)=\int_{-\infty}^\infty e^{i \omega t}f(t) dt\end{align} For a ...
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Delta functional representation in response field formalism

A general way of obtaining a field-theoretical description of Langevin dynamics is via the Martin-Siggia-Rose (MSR) response fields. This is essentially just representing the identity - up to some ...
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Dirac force impulse and energy

Modelling a force impulse i.e, a high amount of force for a very short time to yield a finite change in momentum, can be modelled as $$ F = p\ \delta(t) $$ Integrating this force acting on a body with ...
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Double and triple sum into integral with density of states

I asked this on mathematical forum section. I'm triying to expand the results of certain calculation, where the author has the following kind of sums: $$ \sum_{j} A(\omega_j) n(\omega_j), $$ where $A$ ...
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Dealing with $\delta^{(3)}(0)$ using normal ordering

On page 109 of David Tong's lecture notes on QFT, equations (5.11) and (5.12) read: $$ H = \int \frac{d^{3}p}{(2\pi)^{3}} E_{\vec{p}}[(b_{\vec{p}}^{s})^{\dagger}b_{\vec{p}}^{s}-c_{\vec{p}}^{s}(c_{\vec{...
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How do I calculate the inverse Fourier transform of the delta function? [migrated]

In the context of single-pixel imaging, the following statement is given: "A Fourier basis pattern $P_F (x,y) $ can be obtained by applying an inverse Fourier transform $\delta_F (u, v, \phi)$to ...
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How can you approximate number of bound states in a harmonic oscillator potential $V$ and also for a Dirac delta function using uncertainty principle?

How can you approximate the number of bound states in a harmonic oscillator potential $V$ and also for a Dirac delta function using the uncertainty principle?
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Some Questions about Formulae

I am currently studying Quantum Field Theory from the textbook Overview of Quantum Field Theory and I am confused by two formulae presented in chapter 2 (2.39) and (2.40). The first is $$(1)_{1-...
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Why do we factor out a momentum-conserving delta function in a 1-loop 1PI diagram in $\phi^4$ theory?

I do not understand the meaning of factoring out $(2\pi)^4 \delta ^4(p_1 - p_2)$, surely the delta function will have argument zero ($p_1 = p_2$) and should consume one integration and $(2\pi)^4$ ...
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How come that $\int \delta(H(p,q)-E)dpdq=\Omega(E)$ not infinity?

In microcanonical ensembles we have (for one particle in 1 dimension) $$\int \delta(H(p,q)-E)dpdq=\Omega(E)$$ I am not convinced and believe that this integral diverges. Take for example a harmonic ...
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Why does it follow that the Dirac delta function is a scalar “because determinant of the Lorentz transformation is 1”

In Weinberg's Gravitation and Cosmology it's stated that $\delta^4(x - x(\tau))$ is a scalar (because Det $\Lambda$ = 1) (just after eqn 2.6.5, for reference.) I cannot make sense of this statement. ...
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Why is the wavefunction not equal to 0 at the spike of the Dirac delta potential, but it is 0 for the infinite square well boundaries?

Let an infinite square well have its left corner at origin and right corner at $x=L$. We then have the following boundary conditions for a wavefunction $\Psi(0)=\Psi(L)=0$. Now for a different system ...
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How to prove Ampère's Circuital Law rigorously with the help of Dirac delta function

When reading this proof of Ampère's Circuital Law in Emilio Pisanty's answer to this question, I had some difficulties in understanding the usage of the Dirac delta function. I have learned that the ...
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Why a dirac delta function makes the dimension of elements in a density matrix inconsistent?

Suppose that $\left | m \right >=\int \frac {dk}{2\pi} G(k)\left | k \right >$ and $G(k)$ has the dimension of $[L]$. So both sides of the equation are dimensionless. A density matrix is defined ...
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Units of Dirac delta potential constant

Suppose we have a Dirac delta potential correction of the form $$ V(x)=V_0\delta(x-x_0)$$ What would be the units of $V_0$ ? I think it should be units of energy as calculating first order ...
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Charge Density for a Rod

In Jackson's Classical Electrodynamics, there's a line charge distribution on a rod of length 2b. Jackson gives the charge density in Eq. (3.132) by $$\rho\left(\mathbf{x}\right)=\frac{Q}{2 b} \frac{1}...
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4D Dirac delta differential relation

I'm self teaching from the Anthony Zee Book "Einstein Gravity in a Nutshell", and I came across the following expression. $$\partial_\mu\delta^{(4)}(x-q_a(\tau_a))=\frac{\partial}{\partial x^...
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Time dependent spinning rod; Dirac Deltas

Suppose I have a uniform spinning rod of length $L,$ mass $M$ spinning about the perpendicular axis through it's center. I describe its mass distribution: $$\rho(\vec{r}) =\rho\delta(z)\delta\left(\...
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Local density of a deformed lattice to lowest order in displacement

I am trying to understand the derivation from appendix A in the paper https://arxiv.org/abs/cond-mat/9501087. The idea is to calculate the local density of a deformed lattice of a particle as a ...
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71 views

Integration involves derivative of delta function

This appears in explicitly calculating the path integral of harmonic oscillators: First note the second functional derivative of classical action is $$\frac{\delta^2 S[x]}{\delta x(t1)\delta x(t2)}=-m(...
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71 views

What is the solution to the Schrodinger equation for a dirac delta potential next to a hard wall?

Consider the Schrodinger equation for a particle confined in the following potential: $V(x) = - V_{0}\delta(x)$ $\space$ $ x>-d $ $V(x) = \infty$ $\space$ $ x< -d $ When i solve the equation ...
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What is the charge distribution of an Electric Quadrupole?

I'm trying to compute the charge distribution corresponding to the (point) quadrupole moment to gain some intuition (I wanted to know if there can be a quadrupole surface layer like there can be ...
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Can current density $J$ for a thin wire by written in terms of the current $I$ and Dirac delta function?

If there is a thin wire with current $I$ flowing through it, could I write the current density at all points in space of a horizontal 2D slice of the wire as $I \cdot \delta^2(\vec r)$ ? I'm a bit ...
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Normalization of Hamiltonian Eigenfunctions for Free Particle [closed]

I am trying to prove that given $$\phi_E(x) = \left(\frac{m}{2E}\right)^{1/4} \frac{1}{\sqrt{2 \pi \hbar}} e^{i \sqrt{2mE}x/\hbar}$$ Then, $$\int dx \phi^*_E \left(x\right) \phi_{E^\prime}(x) = \delta(...
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Proportionality constant between eigenvectors in 1D free quantum particle

My textbook proposes to find the proportionality constant $\lambda$. Given a free particle ($V = 0$), we know that the Hamiltonian is $$\mathcal{H} = \frac{\mathcal{P}^2}{2m}.$$ We know that if $| p \...
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A question on Dirac delta function

If $$\int f(x)\frac{\mathrm{d}}{\mathrm{d}x}\delta(x-x’)\mathrm{d}x=-f’(x’)$$ What happens when I switch the integration and differentiation to $x’$ instead of $x$? Will I just get the negated result ...
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Why do we not include contributions from $\delta(x)$ in the energy eigenvalue of the delta potential bound state?

Consider a delta function potential $V(x)=-\alpha \delta(x)$. There is a bound state $$\psi(x) = \frac{\sqrt{m\alpha}}{\hbar} \exp\left(-\frac{m\alpha|x|}{\hbar^2}\right),$$ with energy eigenvalue $$E=...
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Transforming the equation of motion of a cubic scalar theory to momentum space

I was reading a paper in which the equation of motion of a cubic scalar theory is Fourier transformed to momentum space. The Lagrangian of the theory is given by $$\mathcal{L}=\frac{1}{2}(\partial \...
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Behaviour of the path integral kernel $K(x_b,t_b;x_a,t_a)$ for the harmonic oscillator when $t_b \to t_a$

The exact propagator for the harmonic oscillator in atomic units in the path integral formulation is given by $$K(x_b,t_b;x_a,t_a) = \left( \frac{m\omega}{2 \pi i \sin(\omega t)} \right)^{1/2} \exp({...
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Can someone show how to reduce the general integral of $\vec{A}$ to the one used by the author in the first line?

Example 10.2 of 3rd edition Griffith [electrodynamics] click here to read this question So I thought to convert I into $\vec{J}$ as follows : $$\vec{J}(\vec{r},t)= I_o\theta(t)\delta(x)\delta(y)\hat{z}...
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Free massive propagator as an OPE?

Consider a free massive propagator $$G(p)\equiv\frac{1}{p^2+m^2}$$ There is a 'naive' expansion in terms of the mass $$G(p)\sim\sum^\infty_{n=0} (-1)^n \frac{m^{2n}}{p^{2(n+1)}}$$ This expansion might ...
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How to determine charge density using Dirac deltas in advance? — not after the fact

Context I have already asked one question regarding charge densities, Diracs, and Heavisides [0]. At the time of writing, that question remains open. More importantly, I still remain unclear regarding ...
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Transition rate derivation in non-relativistic quantum scattering

I am reading Principles of Quantum Mechanics by Shankar, here's a derivation I am puzzled. To evaluate probability of particle entering detector in some solid angle, using $S$-matrix and Fermi's ...
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Dirac delta, Heaviside step, and volume charge density

Context There are many questions on this web cite related to the question at hand. None of them meet my needs. While reading [1], I came across the following: "A ring of charge of radius $a$ and ...
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Scalar product between two kets using eigenkets (QM)

I'm reading Dirac's "Principles of QM" and I reached the point (page 39), when he introduces the concept of expressing the scalar product of two kets using the complete set of eigenkets of ...
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Why can I not use the Dirac delta function in the classical potential expansion?

I would like to verify my understanding on why it is that we can represent a charge density $\rho(\mathbf{r})$ as delta functions. If we start from $$ \nabla^2V=-4\pi\rho(\mathbf{r})\tag{1} $$ then we ...
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What is a 'delta function optical pulse'?

What is a 'delta function optical pulse'? I suspect this means that the optical pulse can be modelled as having the properties of the Dirac delta function, but then what does it mean to say that an ...
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Why do we need large time assumption for energy conservation in electron transitions?

For electron absorption calculations (with an electric field perturbation $\Delta H = eE_0x \cos(\omega t)$) we end up with an integral like: $$c_2(t) \propto \int \rho(\omega) \left( \frac{\sin(\...
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D'Alembertian of a delta-function of a space-time interval (i.e. on the light-cone)

How one differentiates a delta-function of a space-time interval? Namely, $$[\partial_t^2 - \partial_x^2 - \partial_y^2 - \partial_z^2] \, \delta(t^2-x^2-y^2-z^2) \, .$$ Somewhere I saw that the ...
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How to realize Heaviside $\theta(t-t')$ and Dirac $\delta(t-t')$ as matrices in numerics?

As is well known, single-particle Green's functions in the time domain might involve $$\theta(t-t')$$ for the retarded and advanced Green's functions. Sometimes, we also need $$\delta(t-t')$$ to ...
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Simplest possible solution to the Klein-Gordon field equation has a (KG) norm which is not constant in time

It is a fact that the Klein-Gordon inner product must be constant for all $t>0$, where the Klein-Gordon product is defined by $$ \langle f, g \rangle \ := \ i \int d^3x \; \left[ f^{\ast}(t,\mathbf{...
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Is there a way to see linear and surface charge density as a “special case” of volume charge density?

When deriving Gauss’s law in differential form (GLDF), $$\nabla \cdot \mathbf E = \frac{\rho}{\epsilon_0},$$ from Gauss’s law in integral form (GLIF) we get a tidier formula, which is however less ...
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How to solve double delta potential bound states by “brute force”

I just solved a problem in Griffiths' Intro to QM, where one had to find the bound states given the potential: $$V(x)=-\alpha [\delta (x-a)+\delta(x+a)]$$ In order to solve it, one had to exploit the ...

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