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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.

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.

857 questions
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Hilbert space of harmonic oscillator: Countable vs uncountable?

Hm, this just occurred to me while answering another question: If I write the Hamiltonian for a harmonic oscillator as $$H = \frac{p^2}{2m} + \frac{1}{2} m \omega^2 x^2$$ then wouldn't one set of possible basis states be the set of…
Lagerbaer
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What's wrong with this derivation that $i\hbar = 0$?

Let $\hat{x} = x$ and $\hat{p} = -i \hbar \frac {\partial} {\partial x}$ be the position and momentum operators, respectively, and $|\psi_p\rangle$ be the eigenfunction of $\hat{p}$ and therefore $$\hat{p} |\psi_p\rangle = p |\psi_p\rangle,$$ where…
ganzewoort
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Don't understand the integral over the square of the Dirac delta function

In Griffiths' Intro to QM [1] he gives the eigenfunctions of the Hermitian operator $\hat{x}=x$ as being $$g_{\lambda}\left(x\right)~=~B_{\lambda}\delta\left(x-\lambda\right)$$ (cf. last formula on p. 101). He then says that these eigenfunctions…
Peter4075
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What are the units or dimensions of the Dirac delta function?

In three dimensions, the Dirac delta function $\delta^3 (\textbf{r}) = \delta(x) \delta(y) \delta(z)$ is defined by the volume integral: $$\int_{\text{all space}} \delta^3 (\textbf{r}) \, dV = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}…
Andrew
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How exactly is the propagator a Green's function for the Schrodinger equation?

Sakurai mentions (in various editions) that the propagator is a Green's function for the Schrodinger equation because it solves $$\begin{align}&\left(H-i\hbar\frac{\partial}{\partial t}\right)K(x,t,x_0,t_0) \cr=…
Kasper
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The Dirac-delta function as an initial state for the quantum free particle

I want to ask if it is reasonable that I use the Dirac-Delta function as an initial state ($\Psi (x,0) $) for the free particle wavefunction and interpret it such that I say that the particle is exactly at x=0 during time t=0? If I use this initial…
Kurome
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Can the momentum operator have an imaginary expectation value?

I'm making examples of wave functions to incorporate in a QM exam. I came up with the following wave function, which gives me some troubles: $$\psi(x,0) = \begin{cases} A(a-x), & -a \leq x \leq a,\\ 0& \text{otherwise}. \end{cases}$$ This is…
Nicolas
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Divergence of a field and its interpretation

The divergence of an electric field due to a point charge (according to Coulomb's law) is zero. In literature the divergence of a field indicates presence/absence of a sink/source for the field. However, clearly a charge is there. So there was no…
Subhra
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What is the square root of the Dirac Delta Function?

What is the square root of the Dirac Delta Function? Is it defined for functional integrals? Can it be used to describe quantum wave functions? \begin{align} \int_{-\infty}^{\infty} f(x)\sqrt{\delta(x-a)}dx \end{align}
linuxfreebird
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Dirac Delta in definition of Green function

For a inhomogeneous differential equation of the following form $$\hat{L}u(x) = \rho(x) ,$$ the general solution may be written in terms of the Green function, $$u(x) = \int dx' G(x;x')\rho(x'),$$ such that $$\hat{L}G(x;x') = \delta(x-x') .$$ In…
WoofDoggy
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To what extent can we use the informal version of the Dirac delta function in Physics?

Apparently expressions such as $$ \int \delta (x) f(x)dx = f(0)\tag{1}$$ are widely used in Physics. After a little discussion in the Math SE, I realized that these expression are absolutely wrong from the mathematical point of view. My question is:…
niobium
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3D Delta Potential Well

The 1D delta potential well $V(x) = -A\delta(x - a)$ always has exactly one bound state. The same is true for the 3D delta potential well $V(\vec{r}) = -A\delta(\vec{r}-\vec{a})$. I can show this for $\ell = 0$, I don't know how to do the…
countunique
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How is Gauss' Law (integral form) arrived at from Coulomb's Law, and how is the differential form arrived at from that?

On a similar note: when using Gauss' Law, do you even begin with Coulomb's law, or does one take it as given that flux is the surface integral of the Electric field in the direction of the normal to the surface at a point?
Meow
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Derivatives of distributions in general relativity

I am having some trouble when trying to reproduce some calculations involving the description of distributions (mostly used in spacetime junction conditions). I am trying to reproduce the calculations in this paper. In the appendix, I can't derive…
DisStudent
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Dirac Delta and Sloppy Notation

I am an undergraduate neuroscientist and recently I have been studying probability distributions in relation to information theory, and came across the definition of the Dirac Delta as a singular distribution. My mathematical maturity is relatively…
nguzman
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