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 ...
10
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3answers
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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 ...
7
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5answers
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Laplacian of $1/r^2$ (context: electromagnetism and poisson equation)

We know that a point charge $q$ located at the origin $r=0$ produces a potential $\sim \frac{q}{r}$, and this is consistent with the fact that the Laplacian of $\frac{q}{r}$ is ...
7
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1answer
462 views

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 ...
3
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3answers
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Matrix elements of momentum operator in position representation

I have two related questions on the representation of the momentum operator in the position basis. The action of the momentum operator on a wave function is to derive it: $$\hat{p} ...
6
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4answers
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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 ...
6
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1answer
329 views

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 ...
4
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2answers
531 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 ...
3
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3answers
547 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 ...
2
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2answers
407 views

What is the most natural value of Heaviside step function at zero argument?

In many physical applications, the Heaviside step fuction is defined as $$H(x) = \left\{\begin{eqnarray} 1, \quad x>0 \\ 0, \quad x<0 \end{eqnarray}\right.$$ The value $H(0)$ is left undefined. ...
0
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2answers
724 views

Normalization of Momentum Eigenfunctions: the number of particles

After finding the eigenfunctions $u_p(x)=Ce^{ipx/\hbar}$ of the momentum operator just like in this UCSD lecture notes, one seeks to normalize them, so one first tries: ...
7
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5answers
639 views

How to interpret the derivative of the Dirac delta potential?

I met a Hamiltonian containing the derivative of the Dirac delta potential: In order to do it we use a method described in [9]. We define a formal Hamiltonian $$ ...
4
votes
3answers
591 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 ...
6
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2answers
174 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. ...
0
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1answer
575 views

State normalization in Dirac's formulation of quantum mechanics

Let us divide the time $T$ into $N$ segments each lasting $δt = T/N$. Then we write $\langle q_F | e^{−iHT} |q_I \rangle = \langle q_F | e^{−iHδt} e^{−iHδt} . . . e^{−iHδt} |q_I \rangle $ Our ...
4
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1answer
306 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.
3
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1answer
120 views

Can operators be argument of Dirac Delta function

In one part of Marc Bee's book on Quasielastic Neutron Scattering, he defines the pair correlation function $$ G(\textbf r,t) = \frac{1}{(2\pi)^3}\int I(\textbf Q,t)\text e^{-i\textbf Q.\textbf r}\ ...
3
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1answer
127 views

Yang-Mills constraints and Poisson brackets

Let's have constraints for Yang-Mills theory: $$ \varphi_{a} = \partial_{i}\pi^{i}_{a} - f_{abc}\pi^{b}_{i}A^{c}_{i}. $$ I have read the statement that $$ \tag 1 [\varphi_{a}(\mathbf x), ...
3
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1answer
338 views

Getting rid of double delta function in Feynman rules

[1] A very simple example of feynman rule for scalar fields. After computing the diagram i have got the following: $$ -i(2\pi)^4g^2\int d^4q \frac{i}{q^2 -m^2c^2}\delta^{(4)}(p_1 - p_3 -q) ...
2
votes
2answers
232 views

A rigorous treatment of distributions in quantum mechanics

In many introductory courses to quantum mechanics, we see $\delta$-functions all over the place. For example when expressing an arbitrary wave function $\psi(x)$ in the basis of eigenfunctions of the ...
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0answers
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Poisson-like green functions

How can I verify that equation $$\nabla ^2 f (r) = - \frac{e}{4 \pi \epsilon ^2} \delta (r-\epsilon)$$ in 3D has a solution of the form $$f (r) = a - \frac{e}{4 \pi r} \theta (r-\epsilon) ...
1
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2answers
64 views

Expectation value with plane waves [duplicate]

Hey guys Im a little confused with the concept of plane waves and how to perform an expectation value. Let me show you by an example. Suppose you have a wave function of the form ...
1
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1answer
235 views

Could two different bases of a Hilbert space have different cardinality? [duplicate]

Here is a quote from http://en.m.wikipedia.org/wiki/Hilbert_space#Hilbert_dimension (accessed: Nov. 22, 2013) : As a consequence of Zorn's lemma, every Hilbert space admits an orthonormal basis; ...
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2answers
413 views

Detecting the presence of a delta potential

Suppose you have a particle in a box, and there may or may not be a Dirac delta potential somewhere in the box. How could one detect whether or not the potential is present? Furthermore: If there's ...