# Tagged Questions

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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### 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 ...
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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(...
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### Unfamiliar Notation in Sakurai

In chapter 5 section 9 of Sakurai, 2nd edition, he uses some notation that I am unfamiliar with. This may be suited for Math.se but I figured it could be peculiar physicist notation. Anyways it is ...
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### Bound states of the $V(x)=\pm \delta'^{(n)}(x)$ potential?

The $\delta(x)$ Dirac delta is not the only "point-supported" potential that we can integrate; in principle all their derivatives $\delta', \delta'', ...$ exist also, do they? If yes, can we look for ...
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### Possible ambiguity in using the Dirac Delta function

When doing integration over several variables with a constraint on the variables, one may (at least in some physics books) insert a $\delta\text{-function}$ term in the integral to account for this ...
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### What really is a Dirac delta function?

Yesterday a friend asked me what a Dirac delta function really is. I tried to explain it but eventually confused myself. It seems that a Dirac delta is defined as a function that satisfies these ...
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### Maxwell Equations don't give unique Electric Field?

Consider the class of electric fields given by $$\mathbf{E}=\begin{cases} \ln (Cr)\hat{z} & 0\leq r < R\\ 0 & r> R \end{cases}$$ where $C$ is a constant and $r$ is the polar-distance ...
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### Mathematical interpretation of Poisson Brackets

Lets say we are working in a classical scalar field theory and we have two functional $F[\phi, \pi](x)$ and $G[\phi, \pi](x)$. In most of the references, starting with two functional the Poisson ...
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### Hilbert space of a free particle: Countable or Uncountable?

This is obviously a follow on question to the Phys.SE post Hilbert space of harmonic oscillator: Countable vs uncountable? So I thought that the Hilbert space of a bound electron is countable, but ...
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So the virial theorem tells us that: $2\langle T\rangle = \langle \textbf{r}\cdot\nabla V\rangle$. Now I was wondering what would happen if V has te form: $V(\textbf{r}-\textbf{r}') = V_0\delta^{(D)... 4answers 6k views ### How do you do an integral involving the derivative of a delta function? I got an integral in solving Schrodinger equation with delta function potential. It looks like $$\int \frac{y(x)}{x} \frac{\mathrm{d}\delta(x-x_0)}{\mathrm{d}x}$$ I'm trying to solve this by ... 4answers 6k views ### 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. ... 1answer 149 views ### Question on doing the integral for Fermi golden rule Today in the lecture, my professor did something which confused me As an example, we consider the photoelectric effect, in which an electron bound in a Coulomb potential is ionized after ... 2answers 214 views ### Eigenstates of a Hermitian field operator Consider a Hermitian field operator$\phi(x)$with eigenstates satisfying $$\phi(x) |\alpha\rangle = \alpha(x) | \alpha \rangle$$ I'm trying to determine the inner product between the eigenstates. ... 2answers 644 views ### Basis in quantum mechanics My quantum mechanics textbook (Primer of Quantum Mechanics, by Marvin Chester) says that both the momentum space and the position space are basis spaces. It also says that the momentum space is ... 2answers 472 views ### Resources for theory of distributions (generalized functions) for physicists I am looking for tutorials, articles or books containing theory of distributions in context of mathematical physics. Please suggest. 1answer 3k views ### The definition of Density of States The density of states (DOS) is generally defined as$D(E)=\frac{d\Omega(E)}{dE}$, where$\Omega(E)$is the number of states. But why DOS can also be defined using delta function, as $$D(E)~=~\sum\... 1answer 550 views ### Using \frac{1}{A+i\epsilon} = PV\frac{1}{A}-i\pi\delta(A) in Feynman Integrals Are the following operations O.K.? This is related to the Feynman parameter trick.$$F:= \int_0^1 \mathrm{d}x\int_0^{1-x}\mathrm{d}y \frac{1}{f(x,y)+\mathrm{i}\epsilon}.$$Now using$$\frac{1}{A+i\... 3answers 784 views ### Schrodinger equation in term of Fokker-Planck equation From Wikipedia on the Fokker-Planck equation: $$\tag{1}\frac{\partial }{\partial t}f\left( x^{\prime },t\right) ~=~\int_{-\infty}^\infty dx\left( \left[ D_{1}\left( x,t\right) \frac{\partial }{\... 1answer 204 views ### Functional derivatives as distributions I have asked this on math stack exchange, due to its mostly mathemtical content, but aside from one upvote and minimal views it has not garnered any attention, so I am trying here as well. This isn't ... 4answers 2k views ### Is the momentum operator diagonal in position representation? The matrix elements of the momentum operator in position representation are:$$\langle x | \hat{p} | x' \rangle = -i \hbar \frac{\partial \delta(x-x')}{\partial x}$$Does this imply that \langle x |... 2answers 5k views ### Limit of Lorentzian is Dirac Delta I have a quick question that just came up in my research and I could not find an answer anywhere so I thought I'd try here. So one of the definitions of the Dirac Delta is the limit of the Lorentzian ... 6answers 2k views ### What is the origin of the Dirac delta term in the dipole electric field? I am a bit lost how one has deduced the formula for electric field with electric dipole because of some inconsistency between different sources. The Wikipedia article contains a delta function in the ... 3answers 491 views ### Units INSIDE of a Dirac Delta function I know that the units of a Dirac Delta function are inverse of it's argument, for example the units of \delta(x) if x is measured in meters is \frac{1}{meters}. But, my question is what are ... 2answers 3k views ### Translator Operator In Modern Quantum Mechanics by Sakurai, at page 46 while deriving commutator of translator operator with position operator, he uses$$\left| x+dx\right\rangle \simeq \left| x \right\rangle.$$But for ... 1answer 219 views ### Simple question regarding the Green's function for the diffusion equation The differential operator for diffusion in three dimensions is given by \partial_t - k \nabla^2 where k is a constant. The Green's function is (according to Wikipedia)$$\theta(t)\left( \frac{1}{4\... 1answer 2k views ### Principal value of 1/x and few questions about complex analysis in Peskin's QFT textbook When I learn QFT, I am bothered by many problems in complex analysis. 1) $$\frac{1}{x-x_0+i\epsilon}=P\frac{1}{x-x_0}-i\pi\delta(x-x_0)$$ I can't understand why$1/x$can have a principal value ... 1answer 191 views ### Is this a valid proof that the four-current is conserved? The four-current of a particle moving along a worldine$X^\nu(s)$is defined as $$j^\mu(x^\nu) = ec \int u^\mu(s)\, \delta^4(x^\nu - X^\nu(s)) \, ds$$ So here's my proof that this is conserved: \... 1answer 223 views ### How to make rigorous the idea of a continuous complete set? In Quantum Mechanics, when using Dirac's formalism one of its features is the expansion of state vectors into continuous basis of eigenvectors of unbounded self-adjoint operators. Let$\mathcal{H}$be ... 1answer 166 views ### Does regularity of distributions have anything to do with definiteness of their product? Recently I've gone through some literature concerning causal perturbation theory (CPT). As is well known, it deals with UV divergences in QFT by defining products of (operator-valued) distributions ... 2answers 4k views ### How exactly is the propagator a Green's function for the Schrodinger equation Sakurai mentions that the propagator is a Green's function for the Schrodinger equation because it solves $$\left(H-i\hbar\frac{\partial}{\partial t}\right)K(x,t,x_0,t_0) = -i\hbar\delta^3(x-x_0)\... 3answers 641 views ### The magnetic field of a magnetic monopole Let us define the magnetic field$$\vec{B} = g\frac{\vec{r}}{r^3}$$for some constant g. How can we show that the divergence of this field correspond to the charge distribution of a single magnetic ... 3answers 1k views ### The Dirac-Delta function as an initial state for the quantum free particle I want to ask if it is reasonable that I use the Dirac-Delta function as an intial state (\Psi (x,0) ) for the free particle wavefunction and interpret it such that I say that the particle is ... 3answers 2k views ### Can the expectation value of the square of momentum be negative? I've been solving a problem in quantum mechanics, and I was deriving the standard deviation of P, knowing that \langle P\rangle=0. Because \Delta P=\sqrt{\langle P^2 \rangle - \langle P \rangle ^... 3answers 786 views ### Multivariable Dirac Delta and Faddeev-Popov Determinant From this mathstack page and in particular Qmechanic's answer: There exists an n-dimensional generalization$$\tag{1} \delta^n({\bf f}({\bf x})) ~=~\sum_{{\bf x}_{(0)}}^{{\bf f}({\bf x}_{(0)})... 2answers 113 views ### Picture of supports This questions stems from Axiomatic Quantum Field Theory and is mathematical in nature. However, I feel that an answer from physicists is more in line with what I will be asking. Let$\phi$be a ... 1answer 4k views ### 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 ... 1answer 197 views ### George Green's definition of Green's function This is a curious question about the way George Green could have defined his Green's function. All the definitions I see have only Dirac-delta$\delta(x−x′)$function as their source on the RHS. But ... 2answers 783 views ### What is the most general expression for the coordinate representation of momentum operator? I have a question about deriving the coordinate representation of momentum operator from the commutation relation,$[x,p]= i$. One derivation (ref W. Greiner's Quantum Mechanics: An Introduction, 4th ... 2answers 60 views ### Heisenberg EOM for$\langle x \rangle\$ in momentum eigenstate - where is my error?

Equation of motion for expectation value of a quantum particle in a momentum eigenstate: $$\frac{d}{dt} \langle x \rangle = \frac{1}{i h} \langle [x,H] \rangle$$ and since it's in a momentum ...
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### Imaginary Part of the Free Energy - Sohotski Plemenj theorem

I have posted this question already on Math Stack Exchange and I hope not to annoy the community if I post it here again, looking maybe for a better suited audience. I need to understand how the ...
I have come across the expression $$\int f(x) \delta(x-a) \delta''(x-a) \mathrm dx$$ where the prime represents the derivative. Usually with derivatives of the delta distribution I'd partially ...