Questions tagged [approximations]

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Taylor expanding a vector function of many variables

How does one Taylor expand a vector function of many variables? The question arises in the context of deriving the geodesic deviation in Newtonian gravity, where we need to subtract as follows, and ...
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1answer
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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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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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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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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
62 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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Linearization of arbitrary equation [migrated]

If we define linearization of equation as here: http://academic.macewan.ca/physlabs/Linearization.pdf, is it possible to linearize any arbitrary equation? If so, is there an algorithm, I could code ...
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Approximation in series [migrated]

How to approximate $[1 - 1/(2x)]$ to $[1/(1 + (1/2x))]$ when $x$ very large. What power series should be used?. This problem is used to find the precession velocity with the Yukawa potential.
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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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36 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
62 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
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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
40 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
65 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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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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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
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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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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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4answers
715 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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1answer
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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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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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148 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
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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
94 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
63 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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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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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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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
115 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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Second derivative of energy as frequency of oscillations [closed]

Is there a way to algebraically see why when I take the second derivative of a potential energy in a point where it is minimal (force is zero), I generally get the frequency (squared) of the ...
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2answers
188 views

Problem in Derivation of Goldstein

In Goldstein, chapter three, third derivation, given as, Kepler's equation can be written as ${\rho} = e\sin({\omega}t + {\rho})$, Now I have to prove that the first approximation to ${\rho}$ is ${\...
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Metric Tensor in Weak Field Condition [duplicate]

Suppose that we are in the Weak Field Condition, that is: $$g_{\mu\nu}=\eta _{\mu\nu}+h_{\mu\nu}$$ where $\eta _{\mu\nu}$ is the metric tensor of flat space time (Minkowski spacetime) and $h_{\mu\nu}$ ...
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Approximation of an integral on a certain limit

In Peskin & Schroeder - Chapter 6 - the authors make the following approximation when $-q^2\rightarrow\infty$ $$\int_0^1 \!\!d\xi\, \frac{-q^2/2}{-q^2\xi(1-\xi)+m^2} \simeq \frac{1}{2}\int_0d\xi\, ...
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How is the Earth an approximate inertial frame? How can one validate this?

I have always been confused by the idea of Earth being an approximately inertial frame as the speed at which it rotates is so high. Everywhere vaguely it is referred through calculation it can be seen ...
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1answer
104 views

Newtonian Limit of Schwarzschild metric

The Schwarzschild metric describes the gravity of a spherically symmetric mass $M$ in spherical coordinates: $$ds^2 =-\left(1-\frac{2GM}{c^2r}\right)c^2 \, dt^2+\left(1-\frac{2GM}{c^2r}\right)^{-1}dr^...
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441 views

What's considered a small time step?

I was looking at the following identity that's often used in time evolution: $$ (e^{xA/n}e^{xB/n})^n \approx (e^{x(A+B)/n})^n$$ This holds when $(\frac{1}{2}(x/n)^2[A,B])^n$ is small. I'm wondering ...
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85 views

Approximating the energy levels of the anharmonic oscillator using WKB

I got stuck trying to solve this problem: Given the potential $$V(x) = \frac{m\omega^2x^2}{2}-\beta x^4,\ \beta>0$$ I need to evaluate the deviation of the energy levels from the harmonic ...
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What exactly are the field equations that hold in the “distributed-circuit” and the “lumped circuit”-models?

As far as I understood, there are 2 Models for electric circuits that aim to simplify Maxwells Equations (by reducing the number of degrees of freedom from infinite field-values to 2 variables (...
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2answers
626 views

Is this a Taylor approximation? How is it done? [closed]

I know what a Taylor series involves but you have to know the function; here $\mathcal{L}^*$ is just a function depending on $(v+w)^2$, any kind of function could be inside. How can the below ...
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1answer
21 views

Can we make uniformity in marking zero potential? Say it at earths surface or at infinity from earths surface?

The gravitational potential energy at infinity os supposed to be zero. Since body always moves towards lower potential, the gravitational potential is taken as negative so that gravitational potential ...
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1answer
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Why is there so much focus on highly symmetric configurations in electromagnetism?

I was told that in real life, there isn't always high symmetry in what one wants to work with. So why do we spend so much time dealing with highly symmetric problems that have elegant, straightforward ...
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68 views

Raising and Lowering Indices of a Perturbed Metric

I have seen in GR that if a metric is a perturbation of some base metric $g^{(B)}_{\mu \nu}$ such that $g_{\mu \nu} = g^{(B)}_{\mu \nu} + h_{\mu \nu},$ then $g^{\mu \nu} = g^{(B) \mu \nu} - h^{\mu \nu}...
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1answer
42 views

Index gymnastics in weak gravitational field

The metric in a weak gravitational field (TT gauge) is: $$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$$ with $$\eta_{\mu\nu}=\begin{pmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&...
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Why don't we consider the change in $g$ while determining the acceleration of a free falling object?

$$F=mg$$ Why is the acceleration constant? Shouldn't it change as it is thrown upward as the distance from the earth increases.I know the effects would be very negligible but is there any equation to ...
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Validity of weak gravitational field approximation (Schutz's First course in GR)

I'm studying GR with Schutz' First Course in General Relativity and I have some trouble. When field is weak enough, we can take such coordinate system that our metric is written as $$ g_{\alpha\beta} =...

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