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.
573
questions
2
votes
1answer
51 views
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: $$\textbf{B}(\textbf{r}) = \frac{2}{3}\mu_0 \textbf{m}\...
0
votes
2answers
92 views
Doubt in the completeness of wave function
I am reading about the completeness property of wave function. The following is given about it-
The energy eigenstates are complete in the sense that any reasonable wave function $\psi(x)$ can be ...
0
votes
0answers
33 views
Delta Functions In Energy and Momentum Variables
I am confused about some integral formulae presented in An Introduction to Quantum Field Theory by Peskin and Schroeder, specifically formulae 4.81 and 4.82. Formula 4.81 is
$\int \frac{p_1^2 dp_1 d\...
1
vote
2answers
94 views
Delta Function Identity and Path Integral containing Delta Function
Context
I'm currently working through a paper where we utilize the identity $$1=\int\mathcal{D}\sigma\delta(\sigma-\phi^2)=\int\mathcal{D}\sigma\mathcal{D}\zeta e^{i\int\zeta(\sigma-\phi^2)}$$
This is ...
0
votes
0answers
91 views
How can the Fourier delta function produce different matrices in single-pixel imaging?
The pattern in the masks used for single-pixel imaging are created applying equation $(1)$,
$$P_\phi (x,y) = \frac{1}{2} \left[ 1 +| F^{-1} \{\delta_H (u,v) e^{i\phi}\}|\right], \tag{1}$$
in which the ...
0
votes
0answers
10 views
Visualizing Electron Behavior in a Steel Plate
Start with a neutral iron plate, voltage measurement to ground is zero, number of electrons = number of protons. Now start adding electrons to the plate. Negative voltage to ground will increase in ...
2
votes
0answers
59 views
Singular Integrand vs Diverging integral
I am reading Jackson's Electrodynamics and came across this part that I'm not sure I understand. Specifically what is Jackson referring to when he says "it turns out that the resulting integrand ...
-1
votes
0answers
28 views
What is the one-sided Fourier transform of a constant? [migrated]
A definition of the Fourier transform commonly used is (I always forget which convention of normalization to use) \begin{align}f(\omega)=\int_{-\infty}^\infty e^{i \omega t}f(t) dt\end{align}
For a ...
1
vote
2answers
63 views
Delta functional representation in response field formalism
A general way of obtaining a field-theoretical description of Langevin dynamics is via the Martin-Siggia-Rose (MSR) response fields. This is essentially just representing the identity - up to some ...
0
votes
0answers
19 views
Dirac force impulse and energy
Modelling a force impulse i.e, a high amount of force for a very short time to yield a finite change in momentum, can be modelled as
$$
F = p\ \delta(t)
$$
Integrating this force acting on a body with ...
1
vote
0answers
28 views
Double and triple sum into integral with density of states
I asked this on mathematical forum section.
I'm triying to expand the results of certain calculation, where the author has the following kind of sums:
$$
\sum_{j} A(\omega_j) n(\omega_j),
$$
where $A$ ...
-1
votes
1answer
48 views
Dealing with $\delta^{(3)}(0)$ using normal ordering
On page 109 of David Tong's lecture notes on QFT, equations (5.11) and (5.12) read:
$$ 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{...
0
votes
0answers
77 views
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 ...
0
votes
0answers
18 views
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?
0
votes
1answer
40 views
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
$$(1)_{1-...
1
vote
2answers
39 views
Why do we factor out a momentum-conserving delta function in a 1-loop 1PI diagram in $\phi^4$ theory?
I do not understand the meaning of factoring out $(2\pi)^4 \delta ^4(p_1 - p_2)$, surely the delta function will have argument zero ($p_1 = p_2$) and should consume one integration and $(2\pi)^4$ ...
4
votes
1answer
103 views
How come that $\int \delta(H(p,q)-E)dpdq=\Omega(E)$ not infinity?
In microcanonical ensembles we have (for one particle in 1 dimension)
$$\int \delta(H(p,q)-E)dpdq=\Omega(E)$$
I am not convinced and believe that this integral diverges. Take for example a harmonic ...
1
vote
1answer
104 views
Why does it follow that the Dirac delta function is a scalar “because determinant of the Lorentz transformation is 1”
In Weinberg's Gravitation and Cosmology it's stated that
$\delta^4(x - x(\tau))$ is a scalar (because Det $\Lambda$ = 1)
(just after eqn 2.6.5, for reference.)
I cannot make sense of this statement. ...
7
votes
2answers
272 views
Why is the wavefunction not equal to 0 at the spike of the Dirac delta potential, but it is 0 for the infinite square well boundaries?
Let an infinite square well have its left corner at origin and right corner at $x=L$. We then have the following boundary conditions for a wavefunction $\Psi(0)=\Psi(L)=0$. Now for a different system ...
0
votes
0answers
30 views
How to prove Ampère's Circuital Law rigorously with the help of Dirac delta function
When reading this proof of Ampère's Circuital Law in Emilio Pisanty's answer to this question, I had some difficulties in understanding the usage of the Dirac delta function. I have learned that the ...
1
vote
1answer
40 views
Why a dirac delta function makes the dimension of elements in a density matrix inconsistent?
Suppose that $\left | m \right >=\int \frac {dk}{2\pi} G(k)\left | k \right >$ and $G(k)$ has the dimension of $[L]$. So both sides of the equation are dimensionless.
A density matrix is defined ...
2
votes
1answer
53 views
Units of Dirac delta potential constant
Suppose we have a Dirac delta potential correction of the form $$ V(x)=V_0\delta(x-x_0)$$ What would be the units of $V_0$ ? I think it should be units of energy as calculating first order ...
0
votes
1answer
37 views
Charge Density for a Rod
In Jackson's Classical Electrodynamics, there's a line charge distribution on a rod of length 2b. Jackson gives the charge density in Eq. (3.132) by
$$\rho\left(\mathbf{x}\right)=\frac{Q}{2 b} \frac{1}...
1
vote
1answer
48 views
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.
$$\partial_\mu\delta^{(4)}(x-q_a(\tau_a))=\frac{\partial}{\partial x^...
0
votes
0answers
28 views
Time dependent spinning rod; Dirac Deltas
Suppose I have a uniform spinning rod of length $L,$ mass $M$ spinning about the perpendicular axis through it's center.
I describe its mass distribution:
$$\rho(\vec{r}) =\rho\delta(z)\delta\left(\...
1
vote
0answers
16 views
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 ...
0
votes
1answer
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
$$\frac{\delta^2 S[x]}{\delta x(t1)\delta x(t2)}=-m(...
1
vote
1answer
71 views
What is the solution to the Schrodinger equation for a dirac delta potential next to a hard wall?
Consider the Schrodinger equation for a particle confined in the following potential:
$V(x) = - V_{0}\delta(x)$ $\space$ $ x>-d $
$V(x) = \infty$ $\space$ $ x< -d $
When i solve the equation ...
0
votes
1answer
53 views
What is the charge distribution of an Electric Quadrupole?
I'm trying to compute the charge distribution corresponding to the (point) quadrupole moment to gain some intuition (I wanted to know if there can be a quadrupole surface layer like there can be ...
2
votes
0answers
46 views
Can current density $J$ for a thin wire by written in terms of the current $I$ and Dirac delta function?
If there is a thin wire with current $I$ flowing through it, could I write the current density at all points in space of a horizontal 2D slice of the wire as $I \cdot \delta^2(\vec r)$ ?
I'm a bit ...
0
votes
1answer
58 views
Normalization of Hamiltonian Eigenfunctions for Free Particle [closed]
I am trying to prove that given
$$\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(...
0
votes
1answer
33 views
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 \...
0
votes
0answers
55 views
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 ...
0
votes
1answer
49 views
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
$$E=...
0
votes
1answer
24 views
Transforming the equation of motion of a cubic scalar theory to momentum space
I was reading a paper in which the equation of motion of a cubic scalar theory is Fourier transformed to momentum space. The Lagrangian of the theory is given by
$$\mathcal{L}=\frac{1}{2}(\partial \...
3
votes
1answer
74 views
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
$$K(x_b,t_b;x_a,t_a) = \left( \frac{m\omega}{2 \pi i \sin(\omega t)} \right)^{1/2}
\exp({...
1
vote
0answers
44 views
Can someone show how to reduce the general integral of $\vec{A}$ to the one used by the author in the first line?
Example 10.2 of 3rd edition Griffith [electrodynamics]
click here to read this question
So I thought to convert I into $\vec{J}$ as follows :
$$\vec{J}(\vec{r},t)= I_o\theta(t)\delta(x)\delta(y)\hat{z}...
3
votes
0answers
46 views
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 ...
0
votes
1answer
57 views
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 ...
2
votes
1answer
83 views
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 ...
3
votes
1answer
181 views
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 ...
0
votes
1answer
46 views
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 ...
0
votes
1answer
70 views
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 ...
1
vote
1answer
28 views
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 ...
1
vote
2answers
92 views
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:
$$c_2(t) \propto \int \rho(\omega) \left( \frac{\sin(\...
2
votes
2answers
111 views
D'Alembertian of a delta-function of a space-time interval (i.e. on the light-cone)
How one differentiates a delta-function of a space-time interval? Namely,
$$[\partial_t^2 - \partial_x^2 - \partial_y^2 - \partial_z^2] \, \delta(t^2-x^2-y^2-z^2) \, .$$
Somewhere I saw that the ...
4
votes
1answer
73 views
How to realize Heaviside $\theta(t-t')$ and Dirac $\delta(t-t')$ as matrices in numerics?
As is well known, single-particle Green's functions in the time domain might involve $$\theta(t-t')$$ for the retarded and advanced Green's functions. Sometimes, we also need $$\delta(t-t')$$ to ...
1
vote
0answers
38 views
Simplest possible solution to the Klein-Gordon field equation has a (KG) norm which is not constant in time
It is a fact that the Klein-Gordon inner product must be constant for all $t>0$, where the Klein-Gordon product is defined by
$$
\langle f, g \rangle \ := \ i \int d^3x \; \left[ f^{\ast}(t,\mathbf{...
4
votes
2answers
147 views
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 ...
5
votes
2answers
219 views
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 ...
