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.
865 questions
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Possible typo in Griffiths Particle Physics 2nd Ed. pag 218
I am studying Griffiths book on Particle Physics and on pag 218 he has a derivation that ends on equation 6.62 (page 218):
It appears to me that there is a typo as the factor $[(p_4-p_2)^2-m^2_c c^2]$...
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How to implement constraints via delta functions?
I have a question regarding the implementation of constraint equations as delta functions in integrals.
My confusion can best be illustrated with a quick example:
Consider a Gaussian integral of the ...
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Derivative of density of state
I came across a reference here where in Eq. 2.12 one seems to be concerned with the derivative of the density of state which is given by
$$\begin{aligned}
-\operatorname{Tr} \delta^{\prime}(H-\mu) &...
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Looking for a textbook which provides a good introduction to operator-valued distributions
I'm working on a quantum computing problem and have realized I need to develop a solid understanding of operator-valued distributions. So far the only textbooks I've seen in relation to the topic are ...
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Magnitude of basisvectors [duplicate]
We know that the inner product of a basis vector of an observable or operator with itself should be 1 and should be 0 when inner producted with any other basis vector of the same observable is $0$.But ...
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Can someone show that the thermodynamic limit $N\to\infty$ is a singular limit, and thus the limits are non-commuting?
Can someone work out the limits and show indeed that the limit is singular?
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Trivial Shift in the momentum operator definition
Let's assume the Dirac postulates that $[x,p_x]=i\hbar$. Given the position basis $|x\rangle$ , we have then that: $ \langle q|[x,p_x]|q'\rangle= i\hbar \delta(q-q')$.
Hence we have the equation for ...
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Correct treatment of singular distributions in a hydrogenic expectation value
I am trying to derive the following expectation value for an arbitrary $S$ state of hydrogen:
$$
\left\langle\vec{p}\frac{1}{r^2}\cdot\vec{p}\right\rangle_{n,0}=
\frac{8}{3n^3}-\frac{2}{3n^5} \ ,
$$
...
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What is $\lim_{z \rightarrow 0} \left( \frac{z}{\vec{x}^2 + z^2}\right)^{d/2}$? In particular the potential contact term?
It is known that
$$\lim_{z \rightarrow 0} \left( \frac{z}{\vec{x}^2 + z^2}\right)^{\Delta} = z^{d-\Delta} \frac{\pi^{d/2}\Gamma(\Delta - d/2)}{\Gamma(\Delta)}\delta^{(d)}(\vec{x}) + z^\Delta \frac{1}{(...
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Intuition on the delta function as a basis
I'm following Littlejohn's notes on quantum mechanics, where it gave some examples of what can happen for operators acting on an infinite-dimensional Hilbert space such as 1d wavefunctions.
My ...
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Curl of singular phase using Dirac's delta function
I am investigating topological defects using a complex-valued order parameter of the form
$A=|A|e^{i\theta}$.
A defect is located wherever the phase $\theta$ is not defined and can be spatially ...
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Dirac delta on a Manifold
I'm currently trying to understand scalar field quantization on curved spacetimes and i'm stuck at the choice of commutation relations.
The equal time commutator of the field and it's conjugate it's ...
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How to interpret these "energy states"? [duplicate]
I just solved the Schrödinger equation with the potential:
$$
V(x) =
\begin{array}{cc}
\ \{ &
\begin{array}{cc}
\alpha\delta(x) & -a \leq x\leq a \\
\infty & \text{...
user522770
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Stern-Gerlach Experiment with Initial Position Space Measurement
Suppose at time $t = t_0$ I have a spin-1/2 particle in the superposition state
$$
\chi(r,t_0) \otimes (\alpha|\uparrow\rangle + \beta|\downarrow\rangle),
$$
where $\chi$ is the position wavefunction ...
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How to interpret the equations of motion after second quantization?
Consider the classical Klein-Gordon equation with a quartic interaction:
$$\partial_{tt}\phi - \Delta \phi + m^2\phi + \lambda\phi^3 = 0. \tag{1}$$
After second quantization the field $\phi$ is ...
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