Questions tagged [approximations]

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Why can one neglect terms quadratic in derivatives of $h_{\mu\nu}$ in linearised gravity?

In the linear approximation, terms quadratic in the Christoffel symbols are all neglected in the Riemann Tensor. However, these are not quadratic in the $h_{\mu\nu}$ but quadratic in the derivatives ...
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1answer
67 views

1-Loop Approximation for Fermionic Effective Action

Given a partition function $$ Z = \int \mathscr{D}(\bar{\psi}, \psi) \, \mathrm{exp} \left( \, - S[\bar{\psi}, \psi] \right) $$ with fermionic (Grassmann) fields. I seek to calculate the effective ...
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1answer
29 views

Deriving expression for energy of a damped harmonic (1D) oscillator

The equation $$\ddot{x} + \gamma \dot{x} + \omega_0^2x=0 , \gamma \ll \omega_0$$ has the solution $$e^{-(\gamma t)/2}(\alpha sin\ \omega_0t + \beta cos\omega_0t) = Ae^{-(\gamma t)/2} sin(\omega_0t + \...
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38 views

Why are these parts of the equation $0$ in electrostatics and magnetostatics?

Two of Maxwells equations become something else in the static scenario but dont understand why. I know that in electrostatic scenario the charges are stationary and in magnetostatics the charges move ...
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1answer
96 views

WKB and Virial Theorem contradiction in determining Bound States

Consider a potential that on the left of some point $x=x^*>0$ is infinite and on the right of that point it is of the form $$V(x)=-\alpha x^{-3}.$$ I tried to use the WKB method to determine the ...
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72 views

Why does perturbation theory work for helium atoms?

I saw the following argument for calculating the energy levels of a helium atom. First, ignore the Coulomb interaction term between two electrons. For this simplified model, we have the same solutions ...
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1answer
56 views

Have we ever been able to fully model matter?

Chiral Anomaly clarified the following as: [do] we have a mathematical model that would probably account for everything we've observed about matter if we had the ability to do all of the calculations? ...
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1answer
73 views

Finding the form of an Infinitesimal Lorentz matrix

In the context of Lie groups, when looking for the form of the Lorentz generators we expand a general Lorentz matrix using some infinitesimal parameter $\epsilon$ such that $\Lambda = \mathbb{1} + \...
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2answers
127 views

$\log\left(\frac{p\cdot p’}{(E^2-p^2)/2}\right)=\log\left(\frac{-(p’-p)^2}{m^2}\right)$?

On page 182 of Peskin & Schroeder, equation (6.17) ends with the equality: $$2\log\left(\frac{p\cdot p’}{(E^2-p^2)/2}\right)=2\log\left(\frac{-(p’-p)^2}{m^2}\right),\tag{6.17}$$ where $p=E(1,\...
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How to define kinetic energy and potential energy from EM tensor in newtonian physics?

The question arises from here. People wants to define kinetic energy and potential energy from EM tensor. My question: How to define kinetic energy and potential energy from EM tensor in newtonian ...
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2answers
86 views

Can we always integrate numerically?

I dont know if its suitable here or on Math SE, Most of the times, when I watched online lectures most lecturers say that if we cant solve a integral exactly we can always numerically integrate it. (E....
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1answer
35 views

Hooke's law and atomic scale

Hooke's law is derived in this answer by Taylor expanding an energy potential with arbitrary functional form. This is dependent on the displacements involved being "small". By considering ...
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1answer
55 views

Faraday Rotation - $\Delta k$ Approximation

I'm currently studying the effects of Faraday rotation - specifically high frequency EM waves in a cold magnetised plasma, the external magnetic field is constant and uniform. I am considering EM ...
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Multipole expansion to approximate magnetic field of two currents with given distance and the field of a dipol with given momentum in 2D

I tried to approach this problem numerically, and someone pointed out that this was a classical multipole expansion problem. I dont know anything about multipole expansion, So i ask for help here. My ...
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2answers
73 views

Analytic solution to Kepler's Problem, exegesis

From 'Solving Kepler's Problem' by Colwell, the first analytic solution to Kepler's Problem used a theorem of Lagrange, and later Burmann, to invert Kepler's equation. When you look on the internet ...
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How to find the angle approximation of incidence for a simple pendulum undergoing SHM using Taylor series?

I am a high school student in the process of writing his physics report. My investigation is quite simple and it is finding the relationship between time period and length of a string in a simple ...
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25 views

Different definitions of potential energy [duplicate]

I have been recently studying gravitation and till now I have been familiar with the concept of potential energy increasing with height.As mgh. But there it is also defined like $-\frac{Gm_1m_2}{r}$ ...
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1answer
30 views

Taylor expansion in semi-empirical mass formula derivation

I am trying to understand the derivation of the asymmetry term in the semi-empirical mass formula. I have found a useful derivation on Wikipedia (https://en.wikipedia.org/wiki/Semi-...
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2answers
90 views

Is the position kinematic equation an approximation?

Is the $\Delta x=v_0t+\frac{1}{2}at^2$ kinematic equation an approximation? I'm not asking with reference to relativity, but rather is it still an approximation within Newtonian Physics? I remember ...
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1answer
27 views

Approximation of Stable Orbits as Harmonic Oscillators

A textbook on classical mechanics I am currently reading considers the stable orbit (at $r_0$) of a body subject to the power law: $$\mathbf{F}(r)=-Kr^n\mathbf{\hat{r}},\quad n\in\mathbb{Z}$$ $$\...
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1answer
14 views

How do I derive the approximation of distance between two vectors (one being very small compared to the other)?

In solid states physics, there is an approximation that goes as follows: $$k_0|\vec{r}-\vec{R}|\approx k_0 \left(r-\frac{\vec{r}}{r} \cdot \vec{R}\right)$$ where $r$ is the norm of $\vec{r}$ and it is ...
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1answer
46 views

Analysis of the eigenvalues of the particle in a finite square well

The eigenstates of the particle in a 1D finite square well Hamiltonian: \begin{align} H = \frac{\hat{p}^2}{2m} + V(x) \end{align} \begin{align} V(x) = \begin{cases} -V_0 & \...
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1answer
27 views

Approximation of Rydberg Equation at Very Large $n$

I have been working the Rydberg equation with large $n$ to see how quantum systems when they become very large are functionally identical to classical systems. The problem I'm facing is trying to ...
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Graviton fluctuation suppressed in large $N$ matters

I have a question about semiclassical gravity approximation. For probing Hawking radiation, we usually treat gravitational theory as semiclassical assuming large $N$ matters. However, I do not know ...
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1answer
69 views

Linearized gravity: Derivatives of the metric perturbation

In linearized gravity, the metric is given by the Minkowski metric and a small perturbation, \begin{equation} g^{\mu\nu} = \eta^{\mu\nu}+h^{\mu\nu},\quad |h^{\mu\nu}|\ll 1. \end{equation} Plugging ...
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Amplitude scattering in Distorted-Wave Born Approximation

I am studying the scattering problem in electromagnetism from the book Devaney, Mathematical Foundations of imaging, tomography and wavefield inversion. In the treatment of non-homogeneous media, the ...
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38 views

The relation of two way to derive Hartree-Fock approximation

In Hartree-Fock approximation, we will solve the following self-consistent equations: $$\left[-\frac{\hbar^2}{2m}\nabla^2+v(r)\right]\psi_i(r)+\int\mathrm{d}^3r'V(r,r')n(r')\psi_i(r)-\sum_j\int\mathrm{...
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2answers
63 views

Approximations in deriving infinitesimal operator commutation relationship

I'm having trouble understanding some derivation from Sakurai's QM, chapter 1. To derive $[x,J(dx)]=dx$ he claimed that $dx|x+dx\rangle \simeq dx|x\rangle$ because the approximation is of second order ...
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1answer
51 views

Differential Equation & MacLaurin Series for Newton’s Second Law

I am currently working with a differential equation, where I think I need to take the derivative of $ma$ (corrected as per comment). I am trying to write $F = ma$ as a MacLaurin series and eventually ...
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1answer
43 views

Laws of Physics and Resolution of Measurement

Any 'physical' quantity is expressed as (generally) a Real Number. Real Numbers are abstract mathematical constructs. Laws of Physics are written as mathematical equations; where these real numbers ...
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How is it possible to differentiate or integrate with respect to discrete time or space?

As far as I have understood, the case is that there is nothing that argues that time or space is continuous, but at the same time we must assume this in order to be able to calculate derivatives or ...
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1answer
66 views

Physical interpretation of the nil ADM mass of gravitational waves

The ADM mass is defined for any asymptotic flat spacetime. Using cartesian coordinates: \begin{equation}\tag{1} E_{\text{ADM}} = -\: \frac{1}{16 \pi G} \, \lim_{r \, \rightarrow \, \infty}\oint_{\...
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Is it possible to make a tiny approximation in the equations of General Relativity so that they collapse to Newton's law of gravity? [duplicate]

GR and Newton give almost exactly the same result for the orbits of planets and the acceleration of falling bodies. Is it an incredible coincidence, or does GR have some tiny term (e.g., ict) that ...
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3answers
67 views

In what sense do we say that the Earth surface is almost inertial in Newtonian mechanics?

From what I understand, inertial frames are the ones in which the momentum of every particle in the universe gets well accounted for. Like if there's any particle losing momentum, another particle ...
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43 views

Quantum factorization approximation for first order Coulomb energy

I'm working through "Advanced Quantum Mechanics" by Franz Schwabl, and he uses this G-correlation function to estimate the first order correction to the ground state energy in a Coulomb ...
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2answers
84 views

Why is Schrodinger equation taught while it does not describe an electron?

Strictly speaking, it is "wrong" because it does not describe spin-1/2 particle like an electrons. Why in every QM textbook is it taught, not as a historical equation, but as a current ...
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Finding second virial coefficient from a potential

Given the potential $\phi = \alpha/r^n$, I want to find the second virial coefficient of my system. My integration limits are from some cut-off length $D$ because the particles don't experience a self-...
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3answers
115 views

At which distance $r$ from a black hole does its gravity become Newtonian?

The force near a black hole (outside event horizon $r=3r_s/2$) onto a mass $m$ can be calculated by General Relativity: $$F=\frac{GMm}{r^2}\frac{1}{\sqrt{1-\frac{2GM}{c^2r}}}.$$ However, there must be ...
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723 views

Height is compared to earth radius?

If air resistance is neglected, the object is said to be in free fall. If the height through which the object falls is small compared to the Earth's radius, $g$ can be taken to be constant, equal to $...
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62 views

Lagrangian for nonlinear small oscillations

My original Lagrangian is this, but I want to obtain nonlinear terms considering small oscillations : $$ L = ma^2[\dot \theta^2(1+ 2\sin^2\theta) + \Omega^2\sin^2\theta + 2\Omega_0\cos\theta] . $$ ...
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71 views

Counting number of states for fermions

I have a system of $N$ fermions that can occupy $M$ single particle states, and $M$ is much larger than $N$, $M \gg N$. Since only a single or no fermion can be in a particular state, the number of ...
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35 views

Linearity of electromagnetism and gravity

If we have a very strong electromagnetic field, it stops being linear, Maxwell's equations stop working ($10^6$ Tesla or $10^9$ Newton/Coulomb); Why can't we say the same thing for gravity? Since the ...
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3answers
149 views

What does Schwarzschild refer to as Einstein's approximation regarding his exact solution?

In 1916, Schwarzschild published his $R$-metric solution that differs from the $r$-metric solution we are all familiar with. The relation between $R$ and $r$ is $R^3=r^3 + α^3$ with $r$ been the ...
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1answer
17 views

Shape of confining potential

So last time I posted a question on the same topic, but due to my bad english the question was not clear. This time I will try to be more specific and understandable. In literature I have observed ...
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3answers
96 views

Why are strings and springs often considered massless?

Ok, I know this type of question is already asked, but in every question I have seen, there is no answer to the question that I am asking right now, like people don't particular focus on the question ...
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1answer
98 views

Calculating Frequency of Oscillations About a Stable Equilibrium Point

Assume I have a particle $m$ moving in one dimension where function $U(x) = -Ax + Bx^2$ describes the potential energy. I am trying to figure out how I can calculate the frequency of small ...
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1answer
71 views

Could there be a new theory, which makes the same predictions as a known theory, but is less computationally hard?

Let me give an example at first: All of the calculations that are carried out in quantum chemistry rely on approximation methods to the Schrödinger equation. While these methods sometimes give quite ...
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2answers
32 views

Why specifically should a Foucault pendulum be long and massive?

For a pendulum to easily demonstrate the Foucault effect, it should have as long a cable as possible (this one is 52 feet) and a heavy symmetrical bob (this one is hollow brass, weighing about 240 ...
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8answers
3k views

What is the point of a voltage divider if you can't drive anything with it?

The voltage divider formula is only valid if there is no current drawn across the output voltage, so how could they be used practically? Since using the voltage for anything would require drawing ...
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1answer
116 views

Can you 'derive' mathematical approximations made from Taylor approximations from limiting cases in real life?

Here, the natural length of the string is $l_o$, and pulling the string up by $x$ increases its length by $ \sqrt{ l_{o}^{2} +x^2}$; thus, the increase in length can be approximated as $$ \delta l = \...

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