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.
533
questions
1
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1answer
24 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
1answer
44 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:
$$c_2(t) \propto \int \rho(\omega) \left( \frac{\sin(\frac{...
-1
votes
0answers
27 views
Proof that infinite square well gets split into two by a Dirac delta function in the middle
I was reading these lecture notes https://faculty.biu.ac.il/~barkaie/TA6.pdf ,
and at the end, they claim that $\tan{kL}=0$ implying $kL=n\pi$, thus resulting in the same condition as before, means ...
2
votes
2answers
71 views
D'Alembertian of a delta-function of a space-time interval (i.e. on the light-cone)
How one differentiates a delta-function of a space-time interval? Namely,
$$[\partial_t^2 - \partial_x^2 - \partial_y^2 - \partial_z^2] \, \delta(t^2-x^2-y^2-z^2) \, .$$
Somewhere I saw that the ...
4
votes
1answer
69 views
How to realize Heaviside $\theta(t-t')$ and Dirac $\delta(t-t')$ as matrices in numerics?
As is well known, single-particle Green's functions in the time domain might involve $$\theta(t-t')$$ for the retarded and advanced Green's functions. Sometimes, we also need $$\delta(t-t')$$ to ...
1
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0answers
31 views
Simplest possible solution to the Klein-Gordon field equation has a (KG) norm which is not constant in time
It is a fact that the Klein-Gordon inner product must be constant for all $t>0$, where the Klein-Gordon product is defined by
$$
\langle f, g \rangle \ := \ i \int d^3x \; \left[ f^{\ast}(t,\mathbf{...
4
votes
2answers
134 views
Is there a way to see linear and surface charge density as a āspecial caseā of volume charge density?
When deriving Gaussās law in differential form (GLDF), $$\nabla \cdot \mathbf E = \frac{\rho}{\epsilon_0},$$ from Gaussās law in integral form (GLIF) we get a tidier formula, which is however less ...
5
votes
2answers
184 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 ...
9
votes
1answer
90 views
Rigorous delta potential ā a formulation using distributions?
It is common in QM, even in mathematical physics, to consider Hamiltonians with a Dirac delta in the potential:
$$
\hat H = -\frac{\mathrm{d}^2}{\mathrm{d}x^2} + V(x) + \delta(x-a) \: .
$$
The most ...
3
votes
2answers
104 views
The gradient of the d'Alembertian Green's function
So I have to prove that the d'alembertian of the associated green's function $G(t,t',\vec{r},\vec{r}')$ is equal to zero when given that $\vec{r}\neq\vec{r}'$ $$\left(\frac{1}{c^2}\partial^2 t-\Delta\...
2
votes
1answer
68 views
Are these bra-ket manipulations correct?
I'm reading an old paper ("Wigner's Function and Other Distributions in Mock Phase Spaces," Balazs and Jennings, Phys. Rep. 104(6), 1984), and came across the following statement (in which $\...
2
votes
2answers
43 views
Question regarding a special identity for $2\pi\delta(E-\epsilon_\alpha)$
I am reading Datta's book about Quantum Transport at the moment and I stumbled over an identity for the Dirac delta distribution, which is correct since it fullfils all the requirements for the Dirac ...
2
votes
1answer
113 views
Question on how to actually use momentum space Feynman rules in $\phi^4$-theory
The momentum space Feynman rules state that we "integrate over all undetermined momenta" and "impose momentum conservation at each vertex". This is given for example on page 95 of ...
4
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0answers
77 views
Alternative representations of the momentum operator in position space
The fundamental relation between the position and momentum basis in quantum mechanics is summed up in the canonical commutation relation: $[x,p]=i\hbar.I$ From here, one can get to the matrix elements ...
1
vote
1answer
37 views
Potential well with 2 delta potentials
I have a problem to derive transcendental equation for eigenenergies of bounded states for potential:
$$
V(x)= \begin{cases}
-\lambda\delta(x+a/4)-\lambda\delta(a-a/4), & |x|<a/2 \\
...
1
vote
2answers
64 views
Can the vector potential be written as $\mathbf A = \nabla \chi$ for some singular function $\chi$?
Consider a magnetic field $\mathbf B= \Phi\delta(x) \delta(y) \hat z$. The corresponding vector potential becomes $\mathbf A = \frac{\Phi}{2\pi r} \hat\theta$ in the cylindrical coordinates. ...
1
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0answers
42 views
Dirac delta in 4 dimension - Continuity equation and energy momentum tensor
I had an exercise (Carroll Chapter 4, exercise 3) where I was asked to prove that writing the energy-momentum tensor for a single particle as a dust fluid, meaning
$$
T^{\mu \nu}\left(y^{\sigma}\right)...
0
votes
2answers
74 views
Why wave function does not vanish at a Dirac delta potential?
I have studied that a wave function should vanish at the location of an infinite potential. Consider a direct Delta delta potential at $x=0$. Why does does function not become zero here at $x=0$?
2
votes
0answers
55 views
Influence of delta potential (particle in a box) [duplicate]
I have a question about the problem with a particle in a box and an additional delta potential:
$V(x)=g\delta(x)+V_B(x)$
with $V_B(x)=0$ for $|x|\leq L$ and $V_B(x)=\infty$ elsewhere.
From this I want ...
3
votes
2answers
72 views
Is there a sense in which one can interpret $\delta'(0) = 0$?
I wasn't sure whether to post this under physics or math (and landed on physics due to fear of being crucified for lack of rigor on math.stackexchange).
In field theory, when we encounter divergences ...
1
vote
1answer
30 views
Plane wave basis in a box. Inconsistent terminology about Fourier series and transform
I will refer to the one-dimensional equivalent of the problem in the picture for simplicity. At the end of the picture, the author argues that the Fourier transform of the function $\psi(x)$ is the ...
2
votes
2answers
106 views
$\frac{0}{0}$ from Curvilinear Dirac Delta
The definition of the Dirac Delta in an arbitrary curvilinear coordinate:
$$\delta(\vec{r})=\frac{\delta(x^1-x^1_0)\delta(x^2-x^2_0)\cdot \cdot\cdot \delta(x^N-x^N_0)}{h_1h_2\cdot\cdot\cdot h_N}$$
...
7
votes
3answers
361 views
Is the Dirac $\delta$-function necessarily symmetric?
The Dirac $\delta$-function is defined as a distribution that satisfies these constraints:
$$ \delta (x-x') = 0 \quad\text{if}\quad x \neq x' \quad\quad\text{and}\quad\quad \delta (x-x') = \infty \...
3
votes
2answers
239 views
Dirac-delta-functions as eigenbasis of the position operator - pure nonsense? Or can more be said?
I remember overthinking equations like
\begin{equation}
\mathbf{1}=\int dx\ |x\rangle\langle x|\tag{1}
\end{equation}
and
\begin{equation}
X=\int dx\ |x\rangle\langle x|x\tag{2}
\end{equation}
when I ...
2
votes
1answer
38 views
What is the correct cosine-like integral representation of Dirac delta?
exploring the integral representations of the Dirac delta I found this in terms of an integral of cosine function (from wolfram's database, https://functions.wolfram.com/GeneralizedFunctions/...
1
vote
1answer
118 views
Dirac delta in spherical coordinates. What I'm doing wrong?
I must show that the integral
$$\frac{1}{(2\pi)^{3}}\int_{\vec{k}}d^{3}k\frac{\cos(\vec{k}\cdot\vec{x})}{\left({\sqrt{|\vec{k}|^2+m^{2}}}\right)^{s}}=\delta^{3}(\vec{x})$$
when $s=0$ by using ...
0
votes
1answer
37 views
Why does this simple integral in electromagnetism involving the delta function seem to both vanish and not vanish depending on how you evaluate it?
This relates to David Tong's notes on electromagnetism, specifically page 18.
We have as our charge density a delta function:
$$\rho(\vec x)=Q\delta^{(3)}(\vec x).$$
We want to solve:
$$\nabla^2\phi=-\...
4
votes
2answers
75 views
Quantum expressions involving Dirac delta function
I want to find the following quantum expression:
$$ \langle x|PX|x'\rangle.$$
A. If I use $X|x'\rangle = x'|x'\rangle $, I will get:
$$ \langle x|PX|x'\rangle = \langle x|Px'|x'\rangle = x'\langle x|...
3
votes
2answers
101 views
Understanding Maxwell's Equations in a Box
Maxwell's equations for $n$ charged particles each with charge $e_j$ are known to be (in cgs) $$\begin{align}
\nabla\cdot\textbf{E}(t,x)&=4\pi\rho(t,x)\\
\nabla\cdot\textbf{B}(t,x)&=0\\
\...
1
vote
1answer
70 views
Approximation for the square of a Dirac delta function
I am working in the appendix to Section II.6 in Zee's QFT book, 2nd Ed. I am trying to compute the cross section for a meson to decay to two mesons as $\varphi\to\eta+\xi$ with three respective ...
-2
votes
1answer
44 views
Using Gauss' theorem [duplicate]
I am working through some questions and I am stuck on the working of this one:
Where the working to the answer is here:
Can someone explain the highlighted section as I can't see where it comes from?...
1
vote
2answers
89 views
Evaluating general solution of Dirac equation
In Ashok Das Lecture on QFT book, pg. 40, the solution of the Dirac equation for the general motion of a free particle with mass $m$ along an arbitrary direction is given by
$$\psi (x)=\int d^4p \ a(...
1
vote
1answer
49 views
Completeness relation of spherical harmonics
In spherical coordinates, the resolution of the identity can be written as $$ 1=\int_0^{2\pi}d\phi\int_0^{\pi}\sin\theta\, d\theta\, |\theta,\phi\rangle\langle\theta,\phi| \equiv \int d\Omega |\Omega\...
1
vote
1answer
56 views
Bound States in Dirac Delta Potential [closed]
Consider a particle of mass m in the potential:
$$V(x)=\left\{\begin{array}{rll}\infty & \text { for } & x \leq 0 \\ -V_{0} \delta(x-a) & \text { for } & x>0\end{array}\right.$$
...
0
votes
0answers
45 views
What does happen if we use a delta function for density on the De BroglieāBohm theory?
I was reading Pilot wave theory and De BroglieāBohm theory pages on Wikipedia that I found how similar they are comparing to classical physics and I wondered what happens if we just replace the ...
2
votes
1answer
39 views
Dirac delta and quantum mechanics
I want to know why is this equality true:
$$P(o)P(o') = \delta(o-o')P(o)$$
Where P is the projector operator
I could see that:
$$P(o)P(o') = |o\rangle\langle o|o'\rangle\langle o' | = |o\rangle\delta^{...
0
votes
1answer
38 views
Two-Body Decay Conservation of Energy
I was trying to derive transition rate for a two-body decay process. In one of the reference I'm following, it consider $a\rightarrow1+2$ decay, and said the daughter particles in center-of-mass ...
0
votes
1answer
24 views
$N$-body phase space for Fermi golden rule
I was following along Mark Thomson's Modern Particle Physics, and stumble upe the derivation of d$n$ of Fermi golden rule on page 62:
"...
For the decay of a particle to a final state consisting ...
1
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2answers
96 views
Dirac-delta function in Derivation of Fermi's Golden Rule
I was following along Mark Thomson's Modern Particle Physics, and got stuck on this book's derivation of Fermi's Golden Rule (On page 53):
"...
If there are d$n$ accessible final states in the ...
4
votes
2answers
205 views
Dirac Notation and Coordinate transformation of a function
Lets say we have a 1 dimensional system with coordinate $x$ and the associated operator $\hat x$ with eigenstates $|x\rangle$. A function of $x$ is defined as
$$
f(x) = \langle x |f\rangle \tag{1}
$$
...
2
votes
1answer
71 views
The Dirac comb with one $\delta$-function removed
Has anyone ever encountered the Dirac comb/Shah function with one removed $\delta$-function,
$$
V(x)=\frac{\hbar^2\kappa}{m}\sum_{n\neq0}\delta(x-an),
$$
in any literature? I want to find the solution ...
0
votes
1answer
40 views
How to handle a delta-function divergence from an infinite-dimensional trace of a quantum operator with a general continuous parameter?
Say I am working in the standard Schrƶdinger-style Hilbert space that corresponds to square-integrable functions on 3-space. Some normal operator $\hat{A}$ therefore has an eigenbasis indexed by 3 ...
2
votes
1answer
98 views
How to use principal value in propagator definition?
The propagator for a scalar particle can be written as
$$
\frac{1}{x + i\epsilon} = {\rm PV}\left( \frac{1}{x} \right) - i\pi\delta(x), \quad x = p^2 - m^2, \tag{1}
$$
where $p, m$ are the momentum ...
2
votes
0answers
46 views
Hilbert Space and distribution theory in QFT
When developing Quantum Field Theory, we usually refer to the Minkowski coordinates which cover the whole space in the case it is flat. I don't know how to set distribution theory and Hilbert space ...
7
votes
4answers
1k views
Is continuity of the wavefunction “put in by hand” for the Dirac delta potentials?
In 1d, for $V(x) = g\delta(x)$, integrating the TISE yields (assuming that $\psi$ is bounded$^\dagger$, so as to suppress the term containing $E$)
$$
-\frac{\hbar^2}{2m} \left( \psi'(\varepsilon) - \...
2
votes
2answers
92 views
What is $\langle x_1 |\hat V(\hat x)| x_2 \rangle$?
So I've become rusty in Quantum Mechanics. What is $\langle x_1 |V(\hat x)| x_2 \rangle$? Where $V$ is the potential and $|x \rangle$ is the postion eigenket?
$$ \langle x_1 |\hat V(\hat x)| x_2 \...
0
votes
0answers
21 views
Do the free particle energy eigenfunctions satisfy the closure relation?
The energy eigenfunction of the free particle ($V(x)=0$) are given by $\psi_{E, \pm}=A(E)e^{\pm ik_Ex}$ where $A(E)=\left( m/\left(8\pi^2\hbar^2 E\right) \right)$ and $k_E=\sqrt{2mE}/\hbar$ for each $...
2
votes
2answers
49 views
Is $E=0$ included in the energy spectrum of the free particle in 1d?
In finding the eigenfunctions, $\psi_E$'s, of the free-particle Hamiltonian in 1d,
$$
H = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2},
$$
with eigenvalues $E$'s, subject to the conditions that they are ...
5
votes
4answers
362 views
Schwarzschild solution, stress-energy side of Einstein's equation
The Schwarzschild solution in GR only has a singularity at the origin $r=0$: otherwise there is no matter content. The right-hand side of Einstein's equation is hence almost everywhere zero, but I ...
0
votes
1answer
119 views
Finding the Fourier Transform of a Plane Wave
Hello I am trying to find the fourier transform of a plane wave of the form $$\psi(x) = \frac {1}{\sqrt{2\pi \hbar}}\exp\left(\frac {i}{\hbar}p_0 x\right)$$ where $p_0$ is fixed and Real
I've worked ...
