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.
1
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1answer
99 views
Under what conditions is a wavefunction $\psi(x)$ equal to the probability amplitudes $a(x)$?
For context, consider a general expansion of a wavefunction into continuous eigenstates of position, $\phi(x_m,x)$, multiplied by continuous probability amplitudes, $a(x_m)$
$$\begin{align}\psi(x) &...
0
votes
1answer
45 views
Delta function from poles of Green's function
In quantum mechanical scattering theory, we often use Green's functions which contain poles. For example, in Schroedinger quantum mechanics the free Green's function is given by
$$
G_0(\vec{p}) = \...
0
votes
0answers
26 views
Green's function regularization and delta distribution
I have a free Green's function which is proportional to a $2\times 2$ matrix:
$$
G_0 = \frac{1}{E^2-E_k^2}\begin{pmatrix} a & b \\ c & d \end{pmatrix}
$$
The total Green's function after ...
0
votes
0answers
18 views
To prove the Lorentz invariance of density distribution functions for massless particles in phase space
One defines the density distribution function of a collection of $N$ particles in phase space as follows, $$f(\vec{x},\vec{p},t)=\sum_{i=1}^N\delta^{(3)}(\vec{x}-\vec{x}_i)\delta^{(3)}(\vec{p}-\vec{p}...
0
votes
3answers
143 views
What is principal value in delta function integral? [closed]
The delta function may have different forms of definition. One related to Fourier transform is shown below,
$$\int_{-\infty}^{\infty}\!dt ~e^{i\omega t}~=~2\pi\delta(\omega).$$
then I wonder what if ...
0
votes
0answers
46 views
Does it matter whether or not $\delta(x)$ is a valid wave function for a particle on the real line?
We model the wave-functions of a particle on the line by vectors $\psi \in L^2(\mathbb{R})$, and the position operator $X:D(X) \rightarrow L^2(\mathbb{R})$ as the operator such that $X\psi(x) = x\psi(...
1
vote
2answers
81 views
Is $\delta(r-ct)/4\pi r$, the 3D wave equation elementary solution, a transverse or longitudinal wave?
Background:
https://en.wikipedia.org/wiki/Longitudinal_wave
'Longitudinal waves are waves in which the displacement of the medium is in the same direction as, or the opposite direction to, the ...
2
votes
2answers
147 views
Derivative of delta function
I am reading and following along the appendices of "The Physical Principles Of The Quantum Theory", and trying to learn how he derives Schrödinger's Equation from his Matrix Mechanics, but I have run ...
2
votes
0answers
57 views
How can the propagator be written in the below integral form?
I’am finding it difficult to understand as to how the delta function is written as a product of many delta functions with the integral
1
vote
1answer
30 views
How to prove that divergence of the current density is equal to the minus time derivative of the charge density?
Namely, in Weinberg's book (Graviation and Cosmology...) on p. 40 after eq. 2.6.5 we see:
$$\begin{align}\nabla\cdot \vec J(\vec x,t) = \sum_n e_n \frac{\partial}{\partial x^i} \delta^3(\vec x-\vec ...
2
votes
0answers
37 views
Half of Dirac Delta within spectral integration
I was taking a course in Quantum Information along this last semester, and apart from some mathematical details it all made sense to me. One of these mathematical tricks that I haven't been able to ...
3
votes
1answer
71 views
Triple Delta Potential in Quantum Mechanics
I am facing a problem of Quantum Mechanics, and I gently need your help in continuing to solve it.
The problem is the old usual problem of a particle subject to a potential, which this time has the ...
-1
votes
1answer
29 views
Collision and impulsive forces: a formal approach
Consider two bodies $m$ and $M$. Suppose that $m$ is moving with constant velocity $v_0 > 0$ along a certain axis (e.g., it is moving on the right on the $x$-axis), and at a certain time, it ...
-1
votes
1answer
33 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 ...
0
votes
1answer
30 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:
$$\sum_{n=0}^N {a_n x^{(n)}}= f_0 \delta(t-t_0)$$
...
3
votes
2answers
61 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:
$$
\mathrm{Dirac\ Delta}\ \times \ \mathrm{Thermal\ ...
1
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0answers
22 views
Shifting the derivative outside the integral [closed]
In page 62 of Shankar's Principles of Quantum Mechanics, the author conveys the following: $$\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') ...
0
votes
0answers
59 views
What is the normalized Maxwell-Boltzmann velocity distribution for 2D lattices?
The Maxwell Boltzmann velocity distribution is most of times deduced in 3D on books. How does it looks like in 2D?
0
votes
1answer
29 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 ...
0
votes
2answers
20 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,...
0
votes
0answers
20 views
Use advection diffusion equation to define an initial-boundary problem via programming
Considering the advection-diffusion equation:
$$
u_t=-au_x+\mu u_{xx}
$$
for $x\in[-\infty,\,\infty]$, $0\leq t\leq T$ and $u\left(t=0,\,x\right)=u_0\left(x\right)$.
A solution to the initial value ...
0
votes
0answers
60 views
Spherical Harmonics: Physical Meaning behind the Mathematics: Dirac
Hello Physics Exchange,
I have a series of concept questions. Mostly to prove or disprove my understanding. This will be my first post, so if you have any suggestions to improve my formatting - I ...
0
votes
0answers
77 views
Scattering states in Dirac Delta potential
I was reading the solution to the Dirac delta potential well in Griffiths and for the case of scattering, the book mentioned is possible to generate normalizable functions with this eigenstates, but ...
0
votes
1answer
36 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:
$\int dx dp \delta(E-\frac{p^2}{2m}-\frac{m \omega^2 x^2}{2})f(E) = \frac{2\pi}{\omega}f(E)$ , for ...
0
votes
0answers
61 views
Dirac-delta orthonormality
Is my understanding that eigenfunctions of position operator and momentum operator exhibit, Dirac-delta orthonormality. I wanted to ask if all self-adjoint operators in quantum mechanics exhibit this ...
0
votes
1answer
64 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 ...
1
vote
0answers
47 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 ...
2
votes
2answers
51 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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votes
1answer
90 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
vote
1answer
217 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 ...
0
votes
0answers
44 views
symmetric and antisymmetric solution for delta potential in periodic boundaries
I got the following question in which I got a little confused about:
Assume we got periodic boundaries like 1-dimensional "ring" in the length of $a$ with the following potential:
$v(x)=g\delta(x)$...
-1
votes
2answers
58 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 ...
0
votes
1answer
50 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 ...
-2
votes
1answer
116 views
Mean value with delta function
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 ...
1
vote
3answers
497 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.
$$ \nabla \cdot \mathbf E = \frac {\rho}{\...
-2
votes
1answer
63 views
Inhomogeneous wave equation by fourier in analysis
$$\nabla^2\psi_\omega+\frac{\omega^2}{c^2}\psi_\omega=-g\omega,\tag{14-16}$$
which is similar to Poisson's equation.
We may synthesize the solution of Eq. (14-16) by the superposition of unit ...
1
vote
1answer
101 views
Why position operator is non-degenerated?
In quantum mechanics one can assume position operator $\hat{X}$ must have continuous spectrum, as experiments say it is possible to find a quantum particle at any point of the space. The question is ...
0
votes
1answer
159 views
How do you solve the Schrödinger equation with a position space delta function potential in momentum space? [closed]
I am solving the Schrodinger equation in position space with an attractive delta function potential energy,
$$
-\frac{h^2}{2m} \frac{d^2}{dx^2} \psi(x)-\lambda \delta(x) \psi(x)=E \psi(x),
$$
for a ...
0
votes
0answers
142 views
Kronecker delta commutation relations for QFT
Setup:
In many textbook treatments of canonical quantization (e.g., Peskin and Schroeder), one imposes canonical (equal time) Dirac delta commutation relations on the conjugate field operators. e.g., ...
0
votes
0answers
77 views
Product of two one dimensional Dirac delta function of two functions
I had a question pertaining to the product of two one dimensional Dirac delta function of two functions. Let two there be two functions $h(x, \chi, \eta)$ and $g(y, \chi, \eta)$ and a product of two ...
0
votes
0answers
32 views
Confusions in discretizing a momentum delta integral
I have an integral of the following form:
$$\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)$$
There are two ways to convert it into discrete ...
-1
votes
1answer
92 views
Step involving delta-function in the Klein-Gordon equation solving
The solution to the equation
\begin{equation}
\int d^3k \; e^{i\mathbf{k}.\mathbf{x}}(k^2-m^2)\phi(\mathbf{k})=0
\end{equation}
(which appears in the Klein-Gordon equation solving) is said to be
\...
3
votes
1answer
121 views
Massless $m=0$ 4D Fourier transform of $(p^2 + i \epsilon)^{-2}$
This question is related to this one. I'm assuming that we're in or on the the light-cone $s \leq 0$ in what follows.
Suppose I'm interested in computing the following Fourier transform, in the ...
-1
votes
1answer
165 views
Solution of differential equation (Dirac delta function)
I have been given the following:
$$y''(x)+\omega^2y(x)=s(x),$$
$$s(x)= \delta(x)-\delta\left(x-\frac{1}{2}\right)$$ for $-\frac{1}{4}<x<\frac{3}{4}$.
(Periodically repeating for $x$ ...
3
votes
1answer
565 views
Physical meaning of the Jacobian in relation to Dirac delta function
Is there a physical meaning to the equation
$$\delta(x-a)=\dfrac{\delta(\xi-\alpha)}{|J|} \, ?$$
In non-rectangular coordinate systems where the transformation is non-singular, what is the ...
-1
votes
1answer
110 views
Ambiguous Use of Dirac Delta Function [closed]
Shankar (in his book Principle of Quantum Mechanics book,page 64) mentions that instead of integrating with respect to dx' in
$$\int \delta '(x-x') f(x')dx'=\frac{df(x)}{dx},$$
where
$$ \delta '(x-x'...
5
votes
0answers
108 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 ...
0
votes
0answers
55 views
Matsubara space Dirac delta distribution
I have three questions regarding the following line of reasoning.
In the context of thermal quantum field theory, more specifically the Matsubara a.k.a. imaginary time formalism, quantum fields at ...
1
vote
1answer
78 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\...
1
vote
1answer
224 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 ...
