Questions tagged [approximations]
The approximations tag has no usage guidance.
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Higher order corrections in relativistic hamiltonian
This is a question I came to while reading Srednicki. My overall goal is to understand quantum/QFT in the end so that doing math research in physics-adjacent topics seems more meaningful. I am a ...
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Energy Density of a Boson Star in non-relativistic Newtonian approximation
I'm reading a paper called Gravitational Atoms: Gravitational Radiation from excited boson stars (RG) and in the paper after applying the weak field approximation to the Einstein Field equations given ...
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The difference in path lengths for waves in the double slit experiment
Fig.1
I don't quite understand the diagram, because it shows $L_1$ and $L_2$ as parallel, even though they are supposed to meet at the same point. I believe the idea is that $\Delta L$ approaches $d \...
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Why does the paraxial approximation work?
Paraxial approximation is a small angle approximation (about 5 degrees), where we study the light rays only near the center of curvature of the spherical mirror/lens. If the rays near the center are ...
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Why can we ignore other charged particles when dealing with the Hydrogen atom?
Introductory treatments of the Hydrogen atom use the following Hamiltonian:
$$H=\frac{|\textbf{p}_p|^2}{2m_p} +\frac{|\textbf{p}_e|^2}{2m_e}-\frac{Ke^2}{|\textbf{r}_p-\textbf{r}_e|}$$
However, given ...
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Fermi's golden rule approximations
The derivation of Fermi's golden rule uses the
following approximations:
Transition time is small.
Photon frequency $\omega\approx\omega_f-\omega_i$.
The question is, are there any experiments ...
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Under what (mathematical) condition can a quantum interaction potential be ignored?
Suppose there are two quantum systems. Suppose that they are subject to an interaction potential which has a limited spatial range, as in, it is exactly zero outside of a certain spatial range. Thus ...
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How is the method of stationary phase used? [closed]
Many path integrals involve complex exponentials
that are highly oscillatory.
These integrals can often be solved using the method of stationary phase.
Please explain how to use the method of ...
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How does asymptotic analysis work with quantum mechanics?
I am trying to wrap my head around asymptotic analysis when it is applied to quantum mechanics. The best source I have found so far is from MIT (https://ocw.mit.edu/courses/8-04-quantum-physics-i-...
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Newton-Cartan from GR
How does EFE reduce to Newton-Cartan Field Equation $R_{tt}=4\pi G \rho$ in Newtonian Limit? I understand its direct derivation from geodesics in weak field, what I am curious about is how EFE reduces ...
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Why is Coulomb's law still relevant?
There is no scenario where we can use Coulomb's law. There is no static charge. Even if we consider the local charge density to be constant for a system of charges, the individual charges are still ...
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For Two Bodies At Rest, How Many $g$'s Before General Relativity Matters?
I'm an EE, so please forgive if this question is dumb.
It is my understanding that in GR, acceleration can curve spacetime, even if the velocity between two bodies is not comparable to the speed of ...
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Why doesn't the ratio of inertial to viscous forces ($R_e$) depend on the object shape - frontal vs side area?
Why doesn't $R_e$ depend on frontal and side area? - Understanding Reynolds number in the context of varying object geometries.
I have not taken a fluid dynamics course, but have been reading up on ...
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How to do the non-relativistic limit of an Energy expression?
I solved for the bound state energies of a system using the Dirac equation, I was told that to compare my result to the energies obtained with the Schrodinger equation I need to do the non-...
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Why do we take this approximattion in optics?
I am studying optics, particularly diffraction by circular lenses. I am following Pedrotti's book. To derive the lateral magnification for such lenses, the author makes the following drawing, and uses ...
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Connection between a saddle point approximation and plain perturbation theory
I am currently studying functional integration in the context of classical and quantum equilibrium thermodynamics. However one thing puzzles me:
In the book "Phase Transitions and Renormalization ...
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Time-Independent Perturbation: Correction Components are Orthogonal to the Original State (Townsend)
I am reading Townsend's A Modern Approach to Quantum Mechanics, Second Edition. While developing the first order correction to the eigenstate (in time independent, non-degenerate Perturbation theory), ...
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What is the significance of a reference point in calculating the potential?
The gravitational potential is given as $$U(r)=-\frac{GMm}{R}$$ where $G$ is the universal gravitational constant $M$ is the mass of the earth and $m$ is the mass of an arbitrary object and $R$ is the ...
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Intermediate step in Floquet-Magnus expansion
I am reading The Magnus expansion and some of its applications by Blanes et al. and I have a question about one equation regarding the Floquet-Magnus expansion.
I. Defintions
I use the same ...
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Potential energy of a one-dimensional mass
The potential energy of a one-dimensional mass $m$ at a distance $r$ from the origin is $$U(r) = U_0 \left(\frac{r}{R} + \lambda^2\frac{R}{r} \right)$$ for $0 < r < \infty$, with $U_0$, $R$, $\...
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Coupled-mode theory and slowly varying envelope approximation
I am facing a situation where I have the following coupled-system equation:
$ \dot{U}(z) = i \; M(z) \cdot U(z) \quad ,$
where U is a N-vector and M is a NxN matrix.
Now, the diagonal elements of M ...
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How does Einstein's theory of relativity prove Newton's laws of gravitation (or is it incorrect)?
I have heard that at the speed of light the Newton's laws of gravitation stop working. Why does that happen or does that even happen? Is there any proof to it? From what I have heard It's related to ...
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Why does Rayleigh-Jeans law agree with Planck's at long wavelengths?
Why does Rayleigh-Jeans law agree with the Plancks distribution at high wavelengths? I know mathematically, that at higher wavelengths, we can approximate the exponential in the denominator by $1+ \...
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Raising and lowering indices to second order
If I consider a metric perturbed to second order, $$g_{\mu\nu}= \eta_{\mu\nu} + \lambda h_{\mu\nu}^{(1)} + \lambda^2 h_{\mu\nu}^{(2)},\tag{1}$$ how should I raise and lower indices for a generic ...
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Binomial And Trigonometric Approximations Give Different Answers?
So I was solving a question that is based on Simple Harmonic Motion. The question is as follows: There is a spring of spring constant $k$ and natural length $2a$ (it is in its natural length initially)...
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Approximations in physics
I have a very abstract at the same time awkward question. In many formulas across physics we need to take several approximations and often we derive formula from previous formula which had certain ...
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How to use saddle point approximation with path integrals?
i would like to evaluate $$\int\mathcal{D}x\ e^{-\int\limits_{-\infty}^{\infty} dt\ (\dot x+\alpha x)^2}\tag{1}$$
and it is my understanding that the way to do so is using the saddle point ...
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How to find the distance of an light source where it arrives parallel on the optic lens?
This question is related to the question:
"how-to-measure-the-strength-of-a-prescription-eyeglass-lens"
That was asked in the following link:
https://rb.gy/lgp99
Han-Kwang Nienhuys explained ...
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Explanation for the approximation of the annulus for calculating the resistance with the skin-effect
I am currently researching the skin effect within a wire for my project. In the Wikipedia entry for the skin effect, an approximation for the effective DC resistance is given. I understand the second ...
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"To order $n$ of" arguments
Often one finds in physics textbooks that arguments will be made "to order $n$". I am not sure on what the procedure or argument ought to be when we have some denominator dependence though. ...
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Linearization of an expression
I'm doing a physics lab and I have a question that asks me to linearize the expression of time (it gives me the equation for time) in function of the mass.
I don't want any solution, just want to know ...
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Approximation of Small Perturbation [closed]
From Morin's Classical Mechanics, on the chapter of Small Oscillations in Lagrangian Mechanics, he does this approximation on the last equality, I don't understand what happened there.
I get the first ...
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Why can't we use the time-dependent Schrödinger equation twice in the adiabatic approximation derivation?
In the standard derivation of the adiabatic approximation (see Sakurai in Modern Quantum Mechanics, Wikipedia) a differential equation for the coefficients is reached as
$$
i\hbar \dot{c}_m(t) + i\...
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Is there a Noether's theorem for approximate conservation and approximate symmetries?
It is my impression that verifying a symmetry in physical laws often takes the form of verifying the corresponding conservation law instead. But since experimental verification must allow for ...
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Recovering the continuity equation in the Newtonian limit for a perfect fluid
I have been trying to show that $T^{0\nu}_{;\nu} = 0$, where $T$ is the stress-energy tensor, reduces to the flat spacetime continuity equation in weak gravitational field.
Taking the metric to be $ds^...
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Will my Green's function approximation be any good?
I would like to use the Greens Function $G(t,t')$, satisfying \begin{equation} \int G^{-1}(t,s)G(s,t') ds = \left( \begin{array}{cc} \delta(t-t') & 0 \\ 0 & \delta(t-t') \end{array} \right) \...
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In physical calculations, is the elimination of higher-order small quantities an approximation or a strict equality in mathematics?
Physics sometimes uses a technique called the method of differentials, which seems magical and not very systematic. This makes me unsure which variable I should take the differential of, and sometimes ...
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Is $F=mg$ derived from Newton's law of universal gravitation $F=Gm_1m_2/r^2$?
If so, that means gravity is only 9.8 m/s^2 at the surface of the earth?
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General relativity when can we approximate to Newtonian gravity?
Lets consider this scenario in deep void of space where other curvatures of large objects are negligible in this case and we bring 2 objects lets say $A$ and $B$.
We give it a force slightly lower ...
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Second-order Trotter error involving an unbounded Hamiltonian
I have an Hamiltonian of this form:
\begin{equation}
H = \frac{p^2}{2m} + V(x),
\end{equation}
I would like to approximate the time evolution for a time $\tau$ of a known initial Gaussian state $|\...
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Derivation of Gravitational Lensing in Schutz's, "A First course in General Relativity" textbook
On page 310 of the third edition of the textbook, Schutz writes:
Suppose now we assume $M u \ll 1$ but is not entirely negligible. Then if we define
$$
y:=u(1-M u), \qquad u=y(1+M y)+\mathrm{O}\left(...
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Why can the term $k/a^2$ in the Friedmann equations be neglected for inflation models?
I am currently reading the TASI Lectures on Inflation 1. Deriving the Friedmann equations one gets (with $c = 1$)
$H^2 = \frac{4\pi G}{3}\rho - \frac{k}{a^2}$.
But when talking about inflation with ...
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In this time dilation explanation, shouldn't gravity and mass affect the example?
recently i've started "exploring" general relativity, and found this example explaining time dilation caused by velocity:
The example was taken from https://en.wikipedia.org/wiki/...
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"Fermi golden rule" approximation in the form $\int_0^\tau dt_1 \int_0^{t_1} dt_2 e^{i\lambda(t_1-t_2)} \approx \tau \frac{i}{\lambda+i0}$
In Eq. (40) of the paper PRB 74, 125319 (2006) (arXiv), the "golden rule approximation" is stated in the form of
$$\int_0^\tau dt_1 \int_0^{t_1} dt_2 \: e^{i\lambda(t_1-t_2)} \approx \tau \...
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What are the limitatios of angular displacement for the period of oscillation formula $T=2π\sqrt{\ell/g}$ for simple pendulums?
I'm currently working on my physics school investigation, where I try to determine the limitations of the formula for period od oscillation for simple pendulum. I didn't know that this investigation ...
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ONLY non-relativistic limit of general relativity
From my study of GR I learnt that to reach the “Newtonian” limit of the Einstein field equation we have to assume:
weak field $g_{\mu\nu} = \eta_{\mu\nu} + \epsilon h _{\mu\nu}$ with $\epsilon <&...
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A calculation question about Taylor expansion in Altland and Simons p 106, the gutter like potential
I have a question regarding the book condensed matter field theory by Altland Simons p 106. In their a gutter like potential is given and it is required to calculate the fluctuation $\delta V_{tension}...
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Is there a general math term for the idea behind the WKB and similar methods that assume slowly varying sources?
Many different physics techniques for approximately solving differential equations seem to follow the same basic pattern. One starts with some differential equation $Df(x) = s(x)$ (or $s(x) f(x)$), ...
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Is Dyson Series a unitary operator?
I am currently study time dependent perturbation theory. If I understand correctly Dyson series help us to approximate time evolution of initial state.
However, I am confused about the case when we ...
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Nonequilibrium Greens function via a DMFT-mapped Anderson model?
Let $H = \sum_{<i,j,\sigma>}t_{i,j}c^\dagger_{i,\sigma} c_{j,\sigma} + h.c. + U \sum_i n_{i,\uparrow}n_{i,\downarrow}$ be a standard Fermi-Hubbard Hamiltonian. Let us single out one site of the ...
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