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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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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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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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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 ...
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,...
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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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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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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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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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 ...
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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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 ...
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 ...
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 ...
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$D$-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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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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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 $\rho = \frac {Q}{\pi a^3 \frac {4}{3+n}} (\frac{r}{a})^n$, ...
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 ...
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 ...
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 ...
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 ...