Questions tagged [approximations]
The approximations tag has no usage guidance.
499
questions
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How is Schwarzschild metric asymptotically flat for large $r$?
The Schwarzschild line element is:
$$ds^2 = -\left(1-\frac{2M}{r}\right)dt^2 + \left(1-\frac{2M}{r}\right)^{-1}dr^2 +r^2(d\theta^2 +\sin^2\theta d\phi^2). $$
As $r \to \infty$, this is supposed to ...
4
votes
1answer
133 views
Taylor Series of a logarithmic function
I was reading Intro to Modern Statistical Mechanics by David Chandler, on page 63. He states the following:
we can expand $\ln\Omega(E-E_v)$ in the Taylor series $$\ln\Omega(E-E_v) = \ln\Omega(E) - ...
0
votes
0answers
26 views
Understanding the relationship between Interaction, Schrodinger Picture, and Time Dependent Perturbation Theory
(I feel like this has been asked already but I wasn't able to find it.)
I've been reading through Sakurai in hopes of understanding the interaction picture and Time Dependent Perturbation Theory. I ...
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0answers
20 views
Applying Time Dependent Perturbation Theory as spatial perturbations?
From a problem solving perspective, I'm trying to understand how to deal with spatial perturbations that are time dependent.
For clarification, I'm imagining a case of a square well or harmonic ...
-2
votes
2answers
52 views
Changing derivative to difference quotient
Can differential be changed to Delta or difference? In high school education, in the acceleration section of Newton's formula 2, acceleration is a change velocity (velocity difference) divided by a ...
1
vote
0answers
50 views
Understanding the adiabatic and sudden/diabatic approximations?
I’m trying to build a stronger understanding of what the adiabatic and sudden/diabatic approximations mean and imply through examples. Specifically, I want to know how energy is transferred based on ...
2
votes
2answers
271 views
How can integration be used in deriving radioactive decay formula?
I recently learnt the derivation of radioactive decay formula and I am quite surprised about using integration to derive the formula.
But $N$ (the number of atoms) can only be discrete numbers (like ...
0
votes
1answer
56 views
How do you derive non-relativistic momentum using energy equations?
I know that $E_k=\frac{p^2}{2m}$ and $E^2=p^2c^2+m^2c^4$
$$p=\dfrac1c \sqrt{E^2-m^2c^4}$$
how do I derive non-relativistic momentum so that the final equation looks like
$$p=\dfrac1c\sqrt{2mE_k}$$ ...
4
votes
1answer
90 views
Is there a Lorentz invariant approximation to General Relativity?
Since General Relativity is the most accurate description of gravity is there any possible way to derive a Lorentz invariant theory from: $$R_{\mu\nu}-\frac{1}{2} g_{\mu\nu}R+\Lambda g_{\mu\nu}=kT_{\...
0
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1answer
42 views
What is the Difference between $F = mg$ and law of universal gravitation? [duplicate]
is (F=mg) equal to (F=GmM/r^2)? And what's the difference between them?
1
vote
3answers
56 views
Is spring constant really a constant value? ( Assume the spring is not changed )
l just encountered a problem that is about a string in harmonic motion. The question states that the cord is elastic and gives a table like this
The question didn't states that the cord changes, only ...
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0answers
16 views
Why it is not possible to get exact solution for cubic potential perturbation for 1D SHO and we have to use perturbation theory? [duplicate]
Can anyone help me in providing the process of finding exact solution in case of cubic perturbation in 1D SHO, or any suitable resource?
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0answers
57 views
Why it is not possible to get exact solution for cubic potential perturbation in 1D SHO and we have to use perturbation theory?
For $x$ and $x^2$ we get exact solution easily without applying perturbation theory, but I read that above order perturbation can not be solved exactly. Can anyone explain clearly why?
0
votes
2answers
49 views
Can we ignore air resistance?
Is there any case in real life we can get the right (correct) "equations of motion" for object with ignoring air resistance? In any object condition (size or shape of the object we are studying).
2
votes
2answers
100 views
When can a function $f(x_0 - x)$ be approximated as $f(x_0 - x) = f(x_0) - f'(x_0) x$?
When can a function $f(x_0 - x)$ be approximated as $f(x_0 - x) = f(x_0) - f'(x_0) x$? In Reif's statistical mechanics it is said that when $x$ is much smaller than $x_0$ then the approximation can be ...
1
vote
1answer
55 views
Phase-ordering dynamics: numerical solution of the Mazenko equation in $D=2$
I'm considering the Mazenko equation as it's written in https://doi.org/10.1103/PhysRevB.46.10594 (eq. 7)
\begin{equation}
\label{a}
f''+\left(\frac{1}{x}+\frac x 4 \right)f'+\frac \lambda \pi \,\tan\...
5
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0answers
69 views
Question about the Newtonian limit of general relativity
I ran into something peculiar while attempting to carefully derive the Newtonian limit of general relativity, specifically for the geodesic equation.
To set it up, we assume that the curve $q:[a,b]\...
0
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2answers
56 views
Does the small angle approximation apply to torsional pendulums?
I learned that the period of a Torsional pendulum is equal to $$T = 2\pi\sqrt{\frac{I}{\kappa}}$$
I know the simple pendulum has a similar expression which assumes that $\sin\theta=\theta$ as long as ...
0
votes
1answer
25 views
Validity of the two-body system approximation in astrophysics
I'm taking an intro course in astrophysics and studying Kepler's Laws of planetary motion - all of which are built over the assumption that we can approximate our system to one where there's only two ...
9
votes
2answers
1k views
Is there an approximation for the Lorentz factor for very large velocities?
I am aware of the approximation generally used for low speeds to calculate the Lorentz factor, that being,
$$\gamma \approx 1 + \frac{1}{2} \left(\frac{v}{c} \right)^2$$
But I need the exact ...
1
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2answers
41 views
Why do springs have a linear relationship?
Why does:
F = k*(change in position)
Why can't the relationship be quadratic or higher ordered?
0
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1answer
58 views
Are these two weak field approximations to the Schwarzschild metric the same?
Given the Schwarzschild metric
$$ ds^{2} = -\Big(1-\frac{r_{s}}{r}\Big)c^{2}dt^{2} + \Big(1-\frac{r_{s}}{r}\Big)^{-1}dr^{2} + r^{2}(d\theta^{2} + \sin^{2}\!\theta \, d\phi^{2}), $$
we can apply the ...
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0answers
27 views
How to properly implement Newton Cooling in FTCS Heat Equation
I am attempting to model the temperature in 2D plate using the FTCS scheme for the heat equation. The plate is finite, and discretized squarely. I do not want the temperature fixed at the edges. ...
2
votes
1answer
92 views
When does the Post-Newtonian expansion break down?
In the book Gravitational waves Vol.1: theory and experiment by M.Maggiore, in chapter 5, page 236, the author discusses the Post-Newtonian (PN) expansion and says that it is valid for small speed and ...
0
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1answer
40 views
Why is the graph (Voltage vs Current) of an ohmic object not exactly linear?
What are the reasons for the graph Voltage vs Current of an Ohmic object being approximately and not exactly linear? In other words, we accept its lineality, but in reality, it is not exactly linear. ...
5
votes
1answer
108 views
If $\text{Tr}_B \rho^{AB}$ is almost pure, then $\rho^{AB}$ is almost a product state?
Let $\rho^{AB}$ be a bipartite state, and let $\rho^A$ denote the partial trace. Suppose
$$ \lVert \rho^A - |\sigma\rangle\langle\sigma|^A \rVert_1 \leq \varepsilon $$
for some pure state $|\sigma\...
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2answers
101 views
Simple harmonic motion amplitude of oscillation
I know that for a pendulum we need small amplitudes. But why is it necessary that a spring oscillator should have small amplitude of oscillation?
0
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1answer
56 views
When is a parameter considered small for perturbation and how does physical units affect that?
In perturbation theory procedures (not specific to any particular topic) we tend to have (or delibrately insert) some small variable $\epsilon$ in an equation that is otherwise difficult to solve if ...
0
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0answers
42 views
Riemann tensor in Locally Minkowskian Reference System (LMRS)
Riemann tensor is evaluated in Locally Minkowskian Reference Frame
$$
g_{\mu\nu}=\eta_{\mu\nu}+O(\varepsilon^2)
$$
$$
\Gamma^{\alpha}_{\mu\nu}=0+O(\varepsilon)
$$
$$
R_{\mu\nu\alpha\beta}=g_{\mu\...
3
votes
2answers
109 views
Why are we allowed to let $\hbar \ \rightarrow \ 0$ in the semi-classical regime?
I am currently studying the WKB approximation, and certain parts of the argument (mostly when dealing with turning points and patching wavefunctions) rely on the fact that the WKB approximation is a ...
13
votes
2answers
484 views
Calculating the total time elapsed until two pendulums “stop colliding” gives a divergent result [closed]
Setup
Consider the following situation: Two (small) balls hang by two identical ideal threads, such that in their initial states the threads are perfectly vertical. The two balls are moved by small, ...
2
votes
0answers
25 views
Error estimation for field theories
I am looking for resources on error estimation for field theories, both the error due to perturbation theory and measurement error. In other words, consider a field theory of a field $\phi$, with some ...
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0answers
43 views
Assumption in isolating ground state of the interacting theory in Peskin & Schroesder
In QFT (specifically Peskin & Schroesder), when developing an expression for the ground state of the interacting theory $|\Omega\rangle$, the following expression
$$e^{-iHt}|0\rangle=e^{-iE_0t}|\...
0
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1answer
35 views
Derivation of Commutators for Galillean Transformations in Ballentine
I am trying to follow along the derivation of the commutator relations for the generators of the Galilei Group in Ballentine.
He states that the product of two infinitesimal generators and their ...
0
votes
2answers
39 views
Change in areal element
I am reading Griffith's Introduction to Electrodynamics., On example 1.7 while calculating surface integral of $x = 2$ for a cube of side 2., the book states $da = dy \cdot dz$ I don't get this, what ...
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0answers
40 views
What's the difference between a post-Minkowskian approximation and a post-Newtonian one?
I'm studying the book Gravity by Poisson & Will. Specifically, I'm interested in the post-Newtonian and post-Minkowskian approximations showed in chapters 6-10. The problem I'm having is ...
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4answers
189 views
Problem in $\sin\theta \approx \theta$ approximation [closed]
This approximation is used oftenly in physics:
$$\sin\theta \approx \theta$$
This approximation is valid for small value of $\theta \leq10°$):
But $\sin 1°=0.0174524064$
$\sin 2°=0.0348994967$
$\...
0
votes
1answer
88 views
Riemann normal coordinates and the interpretation of the curvature scalar
I have read two demonstrations regarding the interpretation of the Ricci scalar:
https://arxiv.org/abs/gr-qc/0401099 (page 13)
And
https://arxiv.org/abs/0908.1395 (Page 258)
Both expositions are ...
0
votes
2answers
65 views
Range of Earth's gravitational field
We know that the acceleration due to gravity acting on a body situated h meter away from the surface of the earth is given by,
$$g' = (1 - 2h/r)g,$$ where $\,r$ = Radius of the earth ($R$) + $h$. Now ...
1
vote
1answer
46 views
What is the difference between $U(r)=mgr$ and $U(r)=-G\frac{M_{1}M_{2}}{r}$? [duplicate]
As I know gravitational potential energy of an object relative to $U=0$ point is defined as $U(h)=mgh$ and this came from work-energy theorem.
But In my book, there is another definition for it.
...
2
votes
1answer
124 views
How is a function approximated by Fourier analysis?
Quantum mechanics and QFT use extensively Fourier analysis.
When trying to approximate a periodic function by Fourier series (say a rectangular wave), it is possible to increase the number of terms ...
5
votes
1answer
101 views
When can a single particle be treated classically?
When can a single particle be treated classically?
I feel like after 15 years of doing physics I ought to know the answer to this question, but I don't. Here are two variants that will help describe ...
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0answers
29 views
Derivation of the Force Law predicted by General Relativity [duplicate]
I was reading a book where Precession of Mercury's perihelion was described using general relativity. There I found the following equation:
$$
F \approx \frac{G M_m M_s}{r^2}\left(1+\frac{\alpha}{r^2}...
3
votes
2answers
372 views
What does it mean when we say 'The difference between two quantities is of first order'?
This question is about the explanation below Eq.(6.19) of Modern Quantum Mechanics by Sakurai Nepolitano (2nd edition)
Let ${\bf j}(dx)$ be an operator that translates a point $x$ to $x+dx$.
jf(x) = ...
0
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0answers
43 views
How do I linearize the following expression?
My experiment involves dropping a magnet through different radii while everything in the equation is constant. I am trying to figure out if there is a correlation between the radius and terminal ...
0
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1answer
37 views
How does a multiple timescale analysis work?
I often see papers which analytically solve a set of ODEs taking into account the different timescales over which each of the variables change. For example, one might have a set of ODEs
$$\frac{dA}{...
0
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1answer
44 views
How to obtain the time period from the Lagrangian equation for a simple pendulum?
Solving the Lagrangian equation for a simple pendulum we get the following equation:
$$\ddot{\theta} + \frac{g \theta}{l} = 0,$$ (when $\theta$ is small enough).
We already know that time period of a ...
2
votes
1answer
55 views
Geometrical optics problem and broader questions about the correct use of $\approx$ in physical calculations
My textbook, Fundamentals of Photonics, Third Edition, by Saleh and Teich, gives the following:
This seems to be mathematically incorrect to me?
Firstly, the author stated that $\phi = \psi - \...
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0answers
17 views
Validity of the Poisson equation in high-frequency device simulations
Many device simulators use the quasi-stationary Poisson equation
\begin{equation}
\nabla\cdot(\epsilon(\vec{r})\nabla\varphi(\vec{r})) = -\rho(\vec{r})
\end{equation}
for calculation of the ...
-1
votes
2answers
69 views
Is charge quantization correctly interpreted in classical electrodynamics?
As I tried to answer the SE question Electric fields in continuous charge distribution, I faced many ambiguities regarding this matter. In classical electrodynamics, it is claimed that the electric ...
