Questions tagged [approximations]

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pNRQCD at high pT

pNRQCD is an effective field theory for heavy Quarkonium, where the velocities are non-relativistic due to large mass. But is pQCD applicable when the Quarkonium is moving at high velocities? The ...
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29 views

Gravitational potential energy on the Earth's surface [duplicate]

We assume that gravitational potential energy at a height $h$ from the Earth's surface is $mgh$. Is that accurate or only approximately correct ? Here is my approach. On the surface of the Earth, $...
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93 views

Why can the equation of adiabatic process $P_1 V_1^\gamma=P_2 V_2^\gamma$ be written as $\Delta P/P=-\gamma \Delta V/V$?

Why can the equation of adiabatic process $P_1 V_1^\gamma=P_2 V_2^\gamma$ be written as $$\frac{\Delta P}{P}=-\gamma \frac{\Delta V}{V},$$ where $P_2-\Delta P=P_1$ and $V_2+\Delta V=V_1$ as in ...
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1answer
153 views

Two-source interference in the perpendicular direction and small-angle approximations

In the following setup, we have two point sources of light producing monochromatic, spherical light waves in-phase of wavelength $\lambda$, and a screen positioned in a plane prependicular to the line ...
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37 views

Working on Newton's Gravitational Law and cannot understand this approximation

Working through some questions in a text book and came across one which I did not quite understand how to find the solution. I looked up the solution in the manual and was unable to understand how ...
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2answers
123 views

Why can large objects at greater distance be treated as a point particle?

Why can large objects at greater distance be treated as a point particle? "The bodies of our solar system are so far apart compared with their diameters that they can be treated as particles to an ...
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1answer
61 views

Taylor approximation vs setting to zero

I was recently solving a problem where I had to find the velocity of a particle. The correct result is the following: \begin{equation}v^2 = v_0^2 + \frac{k\delta^2}{m}\left(1-e^{-\frac{\Delta^2}{\...
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1answer
61 views

Is it possible to approximate the motion under a V-shaped potential as harmonic motion?

Let's assume an $xy$ plane and let there be a force field defined by the potential $$V=F_0|x|$$ Though the potential is not differentiable still its a perfectly realisable system. If we solve the ...
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Elliptical Trajectory, or Parabolic?

Discuss whether this statement is correct: “In the absence of air resistance, the trajectory of a projectile thrown near the earth’s surface is an ellipse, not a parabola.” Is the above statement ...
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23 views

Mean field scaling introduce a factor of $1/N$ instead of $1/N^2$?

In this arxiv paper (pg2) they introduce the concept of mean field scaling, which keeps the overall charge constant. To do this they introduce a factor of $1/N$ in the force term: $$\dot V_i =\frac{1}{...
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1answer
52 views

Approximations in general

In analysis, a statement like $f(x) \ll g(x)$ (as $x\to x_0)$, has a very precise meaning: $$ \lim_{x\to x_0}\dfrac{f(x)}{g(x)}=0. $$ I was wondering, when physicists write $L_1 \ll L_2$, for, say, ...
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94 views

Why do we use differential equations in physics instead of $h$-difference ones?

Since we don't know whether space and time are discrete or continuous wouldn't it be a better idea to use $h$-difference equations where the derivative is $$f'(x) =\frac{f(x+h)-f(x)}{h},$$ since they ...
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1answer
296 views

Appoximate solution for quantum scattering through Green's function: error and assumption

I've been trying to think of some approximate way of solving quantum scattering problem for a single electron obeying Schrodinger equation with a locally supported potential: $$\nabla^2 \Psi( \mathbf{...
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1answer
661 views

Understanding multipole expansion in classical electrodynamics

I am trying to better understand what the multipole expansion means from a phyiscal point of view. Although mathematically, one may say it is just another form of a series expansion, in this case, the ...
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1answer
58 views

Linear approximation of time dilation. In what point is it?

I have watched a video about linear approximation and there was an example, exactly here: https://www.youtube.com/watch?v=BSAA0akmPEU&feature=youtu.be&list=PL590CCC2BC5AF3BC1&t=32m50s ...
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422 views

Hartree-Fock decoupling of Hubbard model

Hartree-Fock approximation requires wavefunctions be as separable as possible. I know the basic idea of Hartree-Fock but having some trouble in formalism of second quantization. I am trying to ...
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2answers
154 views

Electric field charged disc and L'Hôpital's rule

I have been looking at the electric field of a charged disk and have a question about the use of l'Hopital's rule for the limiting case of electric field at points along the axis $z\gg$ disc radius $R$...
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1answer
479 views

Stirling's approximation for the entropy [closed]

I am trying to derive eq. (2.15) on page 34 of Birger Bergersen's and Michael Plischke's textbook on Equilibrium Statistical Mechanics second edition. We have equation $S(E,V,N)=k_B \log \Omega(E,V,N)...
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101 views

Linear electronic dispersion of graphene

The energy dispersion relation of graphene with the tight-binding approximation and interactions up to nearest neighbors is $$E(k_1,k_2)=\pm |t|\sqrt{3+2\cos(ak_1) + 4\cos\left(\frac{a}{2}k_1\right)\...
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1answer
244 views

Small Angle Approximations

I've seen a number of books take the small angle approximation of $\sin(a - b)=0$, and I'm confused because small angle approximation of $\sin(a)\approx a, \,\cos(a)\approx1$. Using trigonometric ...
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374 views

Expanding function about equilibrium?

This is Example 2.3.4 from Analytical Mechanics 7th Ed by Fowles & Cassiday. It uses the Morse function $V(x)$, given as $$V(x) = V_0[1 - e^{-(x-x_0)/\delta}]^2 - V_0.$$ The question is: "...
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1answer
127 views

Gravitational Field of a Photon compared to that of Massive Matter

[I'm aware relativistic mass is an outdated term, but I'm not sure what term to use in place of it] How is it that [as I've heard, perhaps incorrectly] photons can contribute to the stress-energy ...
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2answers
112 views

Violating Quantization of Charge?

Isn't it the violation of "the quantization of charge" when we use $dQ$ for calculation like integration or derivation?
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1answer
234 views

Why is it OK to use electrostatics in Bohr's Model if the electron is moving?

In the Bohr's Atomic model, we have assumed the centripetal force to be provided by the electrostatic force between the proton and electron and derived the radius, energy of orbit and the velocity of ...
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1answer
38 views

Entropy of two interacting spins with strong magnetic field

I want to calculate for two interarcting spins and an applied magnetic field the entropy. The values for the spins are $s_1= \pm 1$ and $s_2 = \pm 1$ and the Hamiltonian is given by \begin{align} H =...
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1answer
1k views

Why is that in the action principle, the Taylor's series is limited to the first order?

For the Hamilton's principle: $$\delta s =\int_{t_1}^{t_2}L(\mathbf {q+\delta q},\mathbf {\dot q+\delta \dot q},t) dt-\int_{t_1}^{t_2}L(\mathbf {q},\mathbf {\dot q},t) dt=0.\\$$ In the textbooks, ...
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236 views

In the case of tension on a massless string

The string can be thought of as made of little elements the size of elements can be made arbitrarily small. The forces acting on each element when the string is pulled are equal and oppositely ...
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110 views

Derive fluorescent scattering cross section as an approximation when $\omega \sim \omega_0$

The scattering cross section for an bounded electron with resonant frequency $\nu_0 =\frac{\omega_0}{2 \pi}$ under the effect of an EM with frequency $\nu=\frac{\omega}{2\pi}$ is $$\sigma_d=\bigg( \...
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505 views

What can be an example or a scenario where the continuum definition of fluid mechanics would not be valid?

The continuum definition of fluid mechanics may not be valid if a system contains too few molecules cause properties such as density, concentration and velocity are not well defined at a mathematical "...
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43 views

How was this approximation made? (Archimedes law and weights)

Excerpt from textbook: According to Archimedes' law the weight of a body of mass $m$ and density $\rho$ inside air is: $$G=mg\left(1 - \frac{\rho_v}{\rho}\right)$$ Where $\rho_v$ is the density ...
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2answers
418 views

Why are angular velocities of double pendulum small in small angle approximation? [duplicate]

In the lagrangian for double pendulum for small angles, the term $\dot{\theta}_1\dot{\theta}_2 \left [ 1-\frac{(\theta_1-\theta_2)^2}{2} \right ]$ is replaced with $\dot{\theta}_1\dot{\theta}_2$, ...
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0answers
81 views

Casimir energy expression for massive scalar field [closed]

While studying the Casimir effect, I had no problem getting the result for the energy of a massive scalar field: $$E(a,m)= -\frac{mc^2}{4} - \frac{\hbar c}{4 \pi a} \int_{2\mu}^{\infty} \frac{\sqrt{y^...
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415 views

Taylor expansion of a function whose arguments are matrices

I'm studying rotations expressed by a matrix $R$. $|\psi> \to \hat{U} (R) |\psi>$ When we assume infinitesimal rotations, we can write $R = E + \omega$ where $E$ is an identity matrix and $\...
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3answers
7k views

Why is acceleration due to gravity a constant? [duplicate]

I just learned of Newton's law of gravitation and that distance between two bodies is a factor in the gravitational force. My question is if that's true why is the Earth's gravitational acceleration a ...
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1answer
101 views

Multiplying significant figures and decimal places

One of my lecturers today said that multiplying a number leaves the number of significant figures to which that number is correct unchanged, however it doesn't leave the number of decimal places ...
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72 views

How to show transfer integral is a real number?

I want to know how to show transfer integral is a real number. I am learning tight binding approach. Now let's consider the two-dimensional system whose unit cell has two atoms (for example, ...
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5answers
541 views

Why does a massless, frictionless piston move from high pressure to low pressure?

Consider an ideal gas kept in a rigid cylinder with a movable massless, frictionless piston at the top. Let the pressure inside the cylinder be $P$ at pressure exerted by the surrounding on the ...
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1answer
157 views

How to calculate the min Energy in SCF Hartree Fock Calculation?

I have just started writing a program using Hartree-Fock approximation. I have constructed my Hamiltonian (4 by 4 matrix, number of states=4) and found eigenvalues and eigenvectors(4 eigenvectors with ...
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1answer
156 views

What does it mean to Taylor expand free energy density “in gradients”?

I am self-studying Statistical Mechanics from J. Sethna's book Entropy, Order parameters, complexity. In one of the exercises (page 206), the Landau theory for the Ising Model is derived. Starting ...
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1answer
142 views

Gravitational field of a disc. Quadratic aproximation

I calculated the gravitational field above a disc positioned in the $x$,$y$ plane with its centre at $x=y=0$. The answer is: $$ 2G\frac{M}{R^2}\left(1-\frac{z}{\sqrt{R^2 + z^2}}\right) $$ with $M$=...
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291 views

Approximating the equivalent resistance of a large resistor network

Per the elegant solution of this Phys.SE question we can find the equivalent resistance between nodes $k$ and $l$ of a resistor network like this: $$R_{kl} = {{\rm det}A^{(kl)} \over {\rm det} A^{(l)}...
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2answers
197 views

How to obtain components of the metric tensor?

In coordinates given by $x^\mu = (ct,x,y,z)$ the line element is given $$ (ds)^2 = g_{00} (cdt)^2 + 2g_{0i}(cdt\;dx^i) + g_{ij}dx^idx^j, $$ where the $g_{\mu\nu}$ are the components of the metric ...
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249 views

A question on large-N limit?

Let's take $SU(N)$ for an example. The Lagrangian is $$\mathcal{L}=-\frac{1}{4g_{YM}^2}F_{\mu\nu}F^{\mu\nu}.$$ We can define the t'Hooft coupling as $$\lambda=g_{YM}^2N.$$ Then the large-$N$ limit ...
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139 views

What exactly are the approximations in Hartree-Fock?

I have read a book about Hartree-Fock but I am not sure of that simple question. Questions that should help me to understand are: Is an approximation to consider only one determinant? Is the product ...
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2answers
129 views

Better approximations with two sticks

How approximation ought to be done always confuses me. Considering this problem in a textbook: Two massless sticks of length $2r$, each with a mass $m$ fixed at its middle, are hinged at an end. ...
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1answer
839 views

Narrow Width Approximation

I understand how the NWA is used, setting the intermediate particles on-shell and allowing us to drop off-shell contributions, ultimately writing the NWA cross-section as a product of the production ...
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176 views

Does random phase approximation (RPA) response function obey Kramers-Kronig relations?

Consider the screened Coulomb interaction in electron liquid, which in the random phase approximation (RPA) takes the form $$ V(q,\omega)=\frac{v(q)}{1-v(q)\Pi(q,\omega)}, $$ where $v(q)$ is the ...
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101 views

What is the meaning of the existence of a large $N$ limit in QFT?

Large $N$ limits are present in many different contexts: matrix models, gauge theories in various dimensions, conformal field theories (where $N$ is essentially the central charge). We often hear ...
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124 views

Ohms law hold till what temp?

Is the Ohm's law verified to hold true at all temperatures? If not, then till what temperature does the Ohm's law hold? I think it is valid only till $0$ K and above. Am I right in my thinking?
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Why is the Newtonian expression for kinetic energy called the “first order” approximation of the relativistic expression?

In many texts, the non-relativistic (Newtonian) kinetic energy formula $$\text{KE}_\text{Newton} =\frac{1}{2}mv^2$$ is referred to as a first order approximation of the relativistic kinetic energy $$\...