Questions tagged [approximations]

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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 ...
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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) - ...
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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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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 ...
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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 ...
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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 ...
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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 ...
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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 $$p=\dfrac1c\sqrt{2mE_k}$$ ...
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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_{\...
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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?
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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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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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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?
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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).
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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 ...
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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\...
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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]\...
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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 ...
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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 ...
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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 ...
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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?
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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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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. ...
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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 ...
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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. ...
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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?
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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 ...
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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\...
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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 ...
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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, ...
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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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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}|\...
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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 ...
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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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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 $\sin 1°=0.0174524064$ $\sin 2°=0.0348994967$ $\...
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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 ...
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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 ...
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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. ...
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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 ...
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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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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}...
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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) = ...
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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 ...
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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}{...
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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 ...
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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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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 ...
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2answers
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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 ...

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