Questions tagged [approximations]
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929 questions
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Radial coordinate transformation in General Relativity
I've been studying a paper by Alexander Vilenkin regarding Domain Walls and Cosmic Strings. Currently, I'm trying to reproduce the following coordinate change: We start we cylindrical coordinates and ...
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Difference between the pear-shaped Earth, ellipsoid, and the geoid
About 32 years ago, when I was a university student, I took a course called Advanced Mechanics that also covered some astronomy topics.
I vaguely remember a derivation showing that the shape of the ...
- 3,067
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When is the 3D Dirac delta function a good approximation for scattering?
When discussing X-ray or neutron scattering, it is usually assumed that the scattering potential is of the form
$$V_{\mathrm{eff}}(\mathbf{r}) = \sum_i b_i \delta^3(\mathbf{r}-\mathbf{r}_i)$$
with $...
- 1,403
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Use of Thomas-Fermi approximation for 2D fermionic gas with series of potential barriers
For a 2D electron gas of spinless fermions we can easily compute the density profile $n(x,y)$. If now I add a series of square barriers along only one direction, say $V(y)$, I can factories my density ...
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Invariant condition of weak gravitation field
Given metric tensor $g_{\mu\nu}(x)$ is there a coordinate-invariant criteria that would tell me that grav. field described by this metric is weak, in a sense that there exists some coordinate system $...
- 1,716
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Feynman's approximation of electric field far away from a moving charge
Feynman in his lectures, Vol 1, chapter 28 gives formula for electric field at point $P$ which is generated by a moving charge $q$ as
$$E = \frac{-q}{4\pi\epsilon_0}[\frac{e_{r'}}{r'^2}+\frac{r'}{c}\...
- 1,685
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WKB at second order
I have a similar question to this forum that was asked 6 years ago... Transmission coefficient of second-order WKB approximation
I am working on solving a differential equation using the WKB method up ...
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Geometric mean used to calculate gravitational work $L_{AB}=U(A)-U(B)$
We know that
$$
\mathbf{F}^{G}(r) = -\frac{GMm}{r^2} \, \mathbf{\hat{r}}
$$
where $\mathbf{\hat{r}}$ is the unit vector in the radial direction.
I want to calculate the work $L_{AB}$ done by this ...
- 3,067
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Why can we ignore the effect the electron has on the EM field in electron orbitals in quantum mechanics?
It is a well known fact in the history of quantum mechanics that the standard classical orbital model produced issues, since an accelerating charge would release radiation causing it to lose energy ...
- 209
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$E$-field at large distance from finite sources not coming out as I expected
The equation for the $E$-field (on the axis) of a finite sheet of charge is
$$E=\frac{\sigma}{2\varepsilon_0}\left[1-\frac{1}{\sqrt{\frac{R^2}{x^2}+1}}\right].$$
Where $R$ is the radius of the ...
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Approxiamte Atom-wise energy from Hartree-Fock calculation
Is there a way to get atom-wise energies from an ab initio calculation like Hartree-Fock?
I will describe some ideas we had and would appreciate any critique, corrections, or alternative suggestions.
...
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Assumptions on acceleration for kinematic interpolation
I have a sequence $[(s_0,v_0),\,\dots,\,(s_n,v_n)]$ of pairs of distances and speeds which I want to interpolate with a twice differentiable function in order to also have continuous acceleration.
I ...
- 141
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When is the Difference Between Truncating Operator Exponentials via BCH Formula vs. Direct Taylor Expansion?
When dealing with operator exponentials for small parameters we can use Baker-Campbell-Hausdorff (BCH) formula to approximate the operator exponential as a product of exponentials for example:
$$
e^{\...
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Why spatial $g^{mn}$ has the diagonal elements approximately $-1$ in the Newtonian approximation?
Picture below is from the 28th page of Dirac's General Theory of Relativity. I can't understand the red line.
Let $t=x^0, x=x^1, y=x^2, z=x^3$. We have
$$
g_{m0}=0~~~~~~(m=1,2,3)
$$
and $g_{uv}$ ...
- 471
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Landau&Lifshitz's comment on WKB formula
In L&L's quantum mechanics textbook $\S 46$, they derived the WKB approximated wave function at the classically forbidden region:
$$
\psi = \frac{C_1}{\sqrt{|p|}} \exp\left(-\frac{1}{\hbar} \int |...
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