Questions tagged [approximations]

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13
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2answers
3k views

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 $$\...
0
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0answers
54 views

Perturbation theory with degenerate Zero Modes?

If we we know that the one dimensional Hamiltonian $\hat{H}=-\partial_{t}^2+V\left(\varphi\right)$ has two fold degenerate zero modes $\varphi_{1,2}\left(t\right)$ (i.e. has zero eigenvalue), if we ...
0
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1answer
206 views

Linearized gravity: When do we let the metric be $\eta_{\mu \nu} + h_{\mu \nu}$ and when does it reduce to $\eta_{\mu \nu}$?

I am following a standard text on GR. In the chapter on linearized gravity, the metric $g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}$ reduces to $\eta_{\mu \nu}$ when the metric act on tensor components ...
-1
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1answer
807 views

Non-relativistic approximation of kinetic energy-momentum relation

On Alonso Finn I found the following formula while studying the Compton effect, which should show that the relativistic relation between kinetic energy of electron $E_k$ and electron momentum $p_e$ ...
0
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1answer
506 views

Hamiltonian approximation of the coulomb interaction energy of two charged oscillators

I'm adding an excerpt from the book Introduction to Solid State Physics 7th edition by Charles Kittel. I don't see how they arrived at the approximation of the hamiltonian (2) by expanding it. If $...
1
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0answers
45 views

Is there a center of charge in a system in the same way as there is center of mass? [duplicate]

When we establish relation between mass and charges, we take parameters such as The law governing the property (Coulomb's for charge, Newton's for mass) The potential associated with them (...
0
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2answers
2k views

Significant Digits for different units [closed]

I read, From (iv), 12.3 has three significant figures. And from (v) we can infer that 12.30 has four significant figures. So let's say it's meters, then 12.30 m = 1230 cm = 12300 mm But 1230 and ...
1
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0answers
77 views

Validity of Ising model for mean field thoery

The Heisenberg model for the Hamiltonian of a ferromagnet is given by: $$H=-\frac{J}{2} \sum \vec{S}_i\cdot \vec{S}_j+\mu_B B \sum_i S^z_i$$ when performing mean field theory, to find $\chi$, we ...
0
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0answers
34 views

Avogadro number: Can not understand this notation and approximation, help please! [duplicate]

Here is the Avogadro number found everywhere on the internet:$6.022140857(74)×10^{23}$. Now what does the $(74)$ mean? Also this number is approximated in lots of books and schools as $6.023×10^{23}$. ...
0
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1answer
47 views

Lagrangian description of fluid - approximation

In the Lagrangian (i.e. not the Eulerian) description of fluids, the displacement field is given by $\vec{r} = \vec{r}\left(\vec{q},t\right)$ where $\vec{q}$ is the initial position of the particle. ...
3
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3answers
171 views

With calculations involving GR & gravity, when are Newtonian mechanics & Newtonian gravity sufficient and when are they not?

I understand that Newtonian mechanics is a good approximation of GR but at what extremities are the differences so great that GR must be used. I assume it to not be suitable at velocities nearing ...
0
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1answer
312 views

Deriving the Schwarzschild metric in the weak-field regime

I am trying to derive the weak-field Schwarzschild metric, but starting from the same form as Schwarzschild: $ds^2=-(1+2\Phi(r))dt^2+(1-2\Psi(r))dr^2 +r^2 d\Omega^2$ which has $R=-2\partial_r^2 \Phi(...
0
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1answer
40 views

Equivalence of 2 equations involving 4-momenta

In Thompson's Modern Particle Physics, in section 6 about electron-positron annihilation, it is stated (p. 152) that "If the final-state fermion mass is also neglected, (6.63) reduces to the ...
0
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1answer
89 views

Approximating equation of motion with polynomial [closed]

I have set up these equations for a projectile in a game: \begin{align} y(t) &= 100 (-(0.99)^t (v \sin(\theta) + 100 g) + v \sin(\theta) + g (-t) + 100 g)\\ x(t) &= -100 (0.99^t - 1) v \cos(\...
1
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0answers
109 views

How to approximate friction coefficients between different materials?

I want to add friction forces to my computer game. I know there are tables for friction, however I don't think that encoding big table of coefficients (of size $n^2$) is a good idea. I thought, that ...
9
votes
1answer
614 views

How to justify RPA (random phase approximation)?

The Random Phase Approximation (RPA) is a technical method used in field theory to account for interactions when calculating correlation functions. It consists of only keeping a certain class of ...
1
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0answers
71 views

Taylor expansion of $Ei(x)$

I'm reading a note on regularization by Muruyama, link http://hitoshi.berkeley.edu/230A/regularization.pdf On the bottom of page 2, Muruyama Taylor expanded $$ -\frac{e^{m^2/\Lambda^2}}{4\pi} \...
0
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1answer
153 views

About linearisation of equations of motion rigid body

I am modeling a system of solid bodies. Consider $\theta \approx 0$ and $\chi \approx 0$. At a certain moment I get the following formula for the angular velocity: $$ \omega = \begin{bmatrix} 0 &-...
0
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2answers
64 views

If two differential values are multiplied, can the product be left ignored due to its small size?

We can leave square of $dt$ ($t$ is time) ignored. But I am confused whether what's a "small" change for quantity $A$ can always or not be regarded as "small" compared to the small change in quantity $...
1
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3answers
184 views

Why do we invent non-physical concepts (like e.g. point particles) to study physical phenomenons?

There is nothing exist like point particles in reality then why did we invented the notion of point particles and how does it relate to real world?
0
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3answers
1k views

Is my derivation of the potential energy formula $m*g*h$ correct?

I've just wondered where the formula $E_{pot} = mgh$ you learn at school comes from so I've tried to work it out - is my reasoning correct? The change in energy is given by $$\Delta E=\int_{e}^{e+h}G\...
26
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3answers
5k views

Why does a simple pendulum or a spring-mass system show simple harmonic motion only for small amplitudes?

I've been taught that in a simple pendulum, for small $x$, $\sin x \approx x$. We then derive the formula for the time period of the pendulum. But I still don't understand the Physics behind it. Also, ...
2
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2answers
1k views

Tidal force formula

My book explains what happens if we measure the difference in gravitational force by displacing a test mass by $\Delta r\ll r$. They give the following formula: $$ \Delta F=\frac{2GMm\Delta r}{r^3}. $$...
3
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1answer
191 views

Limit example in Zangwill “Modern Electrodynamics”

Zangwill shows that the potential of a finite line segment going from $-L$ to $L$ on the $z$-axis with constant line charge density $\lambda$ is: $$\phi(z,\rho) = \frac{\lambda}{4\pi\epsilon_0}\ln\...
23
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5answers
3k views

Far away from a charged conductor, the field is like a point charge. Where's the point located?

In the framework of classical electrodynamics, at distances much greater than a conductor's dimension, the field ought to approach that of a point charge located at the conductor. But where at? For ...
4
votes
2answers
400 views

Relation between perturbation theory and Taylor expansion in QM

So I am looking at non-degenerate perturbation theory. The idea is that the perturbing term in the Hamiltonian is small so you somehow expand the energies and wave functions in this small term and ...
1
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0answers
59 views

Why are second order PDEs so ubiquitous in physics? [duplicate]

Why are second order partial differential equations able to model such a wide variety of physical phenomena? Why do we never need to go to third or fourth order ?It seems just about everything can be ...
-2
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1answer
40 views

Special Relativity, Kinetics - Momentum and velocity [closed]

Suppose $ \bar{p} = \frac {v}{1-\frac 23 v^2} $; show that $ \bar{p} = {v} { (1+\frac 23 v^2+・・・)\approx v } $ when v is small.
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0answers
106 views

How does special relativity treat the limits of a rotation? [closed]

What happens when a traveller’s uniform motion follows an infinitesimally slight arc around a central inertial observer? Shouldn’t the traveller see the observer as time contracted rather than time ...
2
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1answer
122 views

Gravitational $n$-body problem with tidal forces

This year I'm working on modelling the gravitational $n$-body problem using Newton's law of gravity where I assume that for large enough distances, planetary bodies can be modelled as point masses. ...
0
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2answers
144 views

Can approximation formula for period of larger angles in simple pendulum be used to calculate period of small angles?

Can approximation formula for period of larger angles in simple pendulum be used to calculate period of small angles?
6
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1answer
227 views

How to solve the 10000th eigenvalue of the anharmonic oscillator?

Given a certain Hamiltonian, for example, $$ H = -\frac{1}{2}\frac{\partial^2}{\partial x^2 } + x^4 . $$ , what methods can we use to approximate the $n$th eigenvalue, for very large $n$? For ...
2
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0answers
140 views

Phase-shifting of instantaneous eigenstates in the adiabatic approximation

In my book Quantum Mechanics by B.H. Bransden and C.J. Joachain, there is a chapter on the adiabatic approximation. Here, the authors assume that the time-dependent Hamiltonian $\hat{H}(t)$ changes ...
0
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1answer
47 views

relativistic approximation made in 2 flavor neutrino oscillation derivation

Reading a 2 flavor neutrino oscillation derivation I saw the following approximation being made regarding energy and momentum: E-p ≃ m²/2E I can't see how this step is taken, and what is being ...
0
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1answer
37 views

Why do approximations work with diffraction?

We usualy consider that the distance between the slit and the screen is much larger than the slit width but why would that not matter? I mean sure if the wavelength was long enough yes, but visible ...
0
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1answer
89 views

On the ansatzes of potentials in quantum mechanics [closed]

Here are the list of quantum-mechanical potentials. My question is how did scientists physically model these potentials, what were the parameters they consider before mathematically constructing a ...
1
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0answers
54 views

Finding the lowest energies of electrons in a box with a delta separator with a tight-binding model

I am asked to find the two lowest energies of a 1D situation with two electrons in two adjacent $0$-potential wells of width $L$ with infinitely high barriers and a 'coupling potential' of $V(z)=\...
0
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1answer
266 views

Validity of the sudden/diabatic approximation

The Schrodinger equation is given by $$i\hbar\ \frac{\partial}{\partial t}\ \mathcal{U}(t,t_{0})=H\ \mathcal{U}(t,t_{0}),$$ where $\mathcal{U}(t,t_{0})$ is the time evolution operator for evolution ...
1
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2answers
212 views

Quantum Gravity In Particle Accelerators

It is my understanding that we have no fully working model for Quantum Gravity. However, I imagine one would need to take quantum gravity into account when making discoveries in, say, particle ...
1
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2answers
331 views

Obtaining Transmission Coefficient of Beam Upon a Linear Potential

I would like to determine the transmission coefficient $\mathcal{T}$ for a particle beam $$\Psi(x,t) = A_o e^{ikx}e^\frac{-iE_ot}{\hbar}$$ with energy $$E = \frac{\hbar^2k^2}{2m}$$ incident upon a ...
1
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0answers
142 views

Is the adiabatic theorem in Quantum mechanics valid in general for Non-Hermitian Hamiltonians?

Is the adiabatic theorem in Quantum mechanics valid in general for Non-Hermitian Hamiltonians? The proofs I have come across for adiabatic theorem all assume at some point in the proof that the ...
3
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1answer
170 views

Time averaging a Hamiltonian

There are a number of problems in quantum mechanics whose solution relies on time-averaging away parts of the Hamiltonian. In particular, two examples that come to mind: The rotating wave ...
13
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5answers
1k views

Near Earth vs Newtonian gravitational potential

Newton's Law of Universal Gravitation tells us that the potential energy of object in a gravitational field is $$U ~=~ -\frac{GMm}{r}.\tag{1}$$ The experimentally verified near-Earth gravitational ...
2
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0answers
100 views

Planck's theory question

Could you please tell me why Planck's theory ceases to be valid when the sizes of the bodies and/or their separation distances are comparable to, or smaller than, the wavelength. My professor told me ...
2
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0answers
278 views

Idealizations of an object as a point particle

Why is it that an object can be idealized as a point particle or 'particle like' to solve problems? What are the limits of such a tool?
2
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4answers
287 views

Velocity between two running animals

One animal $A$ can run $100$ km/h and another animal $B$ can run $85$ km/h. Suppose the slower animal $B$ starts running $25$ meters ahead of the faster animal $A$ in a direction. How can I ...
1
vote
4answers
3k views

Can there be tension in an inextensible string?

In physics, tension describes the pulling force exerted by each end of a string, cable, chain, or similar one-dimensional continuous object, or by each end of a rod, truss member, or similar three-...
0
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1answer
114 views

$m = m_0 + (½) m_0v^2/c^2$ vs $ m=m_0/\sqrt{1-v^2/c^2}$

Confused what equation I should use, $\displaystyle{m = m_o + \frac{1}{2} \frac{m_ov^2}{c^2}}$ or $\displaystyle{m=m_o\sqrt{1-\frac{v^2}{c^2}}}$, when solving for relativistic mass. When I plugged in $...
4
votes
0answers
64 views

gravitational force for a binary of point particles with GR term

i'm trying to simulate the two body problem with 2 equal masses and I want to account for general relativistic effects. I know that the difference in the gravitational force would be an additional ...
0
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1answer
113 views

How would I calculate the Reynolds number of a free falling body?

I know from fluid dynamics classes that I can calculate the Reynolds number of a fluid in a pipe using: $Re = \frac{\rho v d}{\mu}$ Where: $\rho$ is the fluid density, $v$ is the fluid speed, $d$ is ...