Questions tagged [approximations]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
1
vote
0answers
108 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
598 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
vote
0answers
70 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
votes
1answer
147 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
votes
2answers
63 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
vote
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
votes
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
votes
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
votes
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
votes
1answer
190 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
votes
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
376 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
vote
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
votes
1answer
39 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.
1
vote
0answers
105 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
votes
1answer
117 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
votes
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
votes
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
votes
0answers
135 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
votes
1answer
46 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
votes
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
votes
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
vote
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
votes
1answer
256 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
vote
2answers
208 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
vote
2answers
328 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
vote
0answers
139 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
votes
1answer
162 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
votes
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
votes
0answers
96 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
votes
0answers
275 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
votes
4answers
277 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
votes
1answer
113 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
63 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
votes
1answer
112 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 ...
0
votes
1answer
343 views

How to derive the exponential Lorentz factor approximation?

In the book "Essential Relativity: Special, General and Cosmological" by W. Rindler, the author uses the approximation $$\gamma\approx10^n,$$ where $\gamma$ is the Lorentz factor and, $\frac{v}{c}=0....
0
votes
1answer
211 views

Velocity of small oscillations

We consider that the displacements about equilibrium are small, in small oscillation. Do we also tacitly assume that the velocity (rather, its magnitude) remains small as well? In Goldstein, it's ...
140
votes
8answers
16k views

Is $\pi^2 \approx g$ a coincidence?

In spite of their different dimensions, the numerical values of $\pi^2$ and $g$ in SI units are surprisingly similar, $$\frac{\pi^2}{g}\approx 1.00642$$ After some searching, I thought that this ...
2
votes
3answers
276 views

Moment of inertia related question [closed]

Sorry for the undetailed title, but when the moment of inertia is calculated in a solid cylinder, the volume of a "sheet" of the cylinder is calculated. I've only seen the volume as $$length*thickness*...
2
votes
2answers
146 views

Calculating size of opening of a Decanter in order to project 5 gallons of water per second 20 feet in the air vertically?

This question is attempting to determine the size of the opening of a Decanter of Endless Water for our gaming group. We are wanting to hook it up to some impromptu Jerry-rigged technology as either a ...
2
votes
1answer
339 views

Approximate formula for the volume of water at a given temperature

Sorry for asking this kind of question. Do you recongnize this formula? $$V(T) = 0.0000679T^3+0.0085043T^2- 0.0624T+999.87$$ when $V$ is volume of water in $\mathrm{ml}$, $T$ is temperature in ...
2
votes
2answers
87 views

In geometrical optics, how can we say that rays coming from a distant object are parallel to one another?

If two rays are not parallel in the start, how can they become parallel at the instant when they strike the lens of a telescope? If they don't become parallel, why do we consider them to be, in the ...
1
vote
1answer
48 views

An approximation question from “Nuclear Models” by Judah Eisenberg and Walter Greiner

I'm looking at page 311 of the book "Nuclear Theory" by Judah Eisenberg and Walter Greiner. Now for identity (49) which is: $$\omega_{1,2}^2=\frac{\omega_\alpha^2+\omega_\zeta^2-2bc\omega_\alpha\...
1
vote
2answers
923 views

Point object, Kinematics

I am very new to physics. This evening, I was reading about the concept of objects to be considered as point objects under some circumstances. And I was trying to think of circumstances under which ...
1
vote
0answers
547 views

WKB approximation to find energy levels of a step potential

Suppose the following potential: $$ V(x) = \begin{cases}V_0 & 0<x<\frac{a}{2} \\ 0 & \frac{a}{2}<x<a \\ \infty & \text{otherwise} \end{cases} $$ Also, assume that for every ...
3
votes
2answers
132 views

Difference between naive and Coriolis-force calculation

Consider the classical problem of dropping a coin from a tower at the equator of a planet without atmosphere and with spin $\Omega$: where in relation to a plumb-line will the coin land? When doing ...
2
votes
0answers
34 views

Adiabatic approximation with scaled time?

In this article (page 3) they have following expression for the coefficients in the adiabatic approximation: $$\dot a_k(t)=-a_k(t) \langle k| \dot k \rangle -\sum_{n\ne k} a_n(t) \frac{\langle k(t)|\...
12
votes
3answers
976 views

Validity of mean-field approximation

In mean-field approximation we replace the interaction term of the Hamiltonian by a term, which is quadratic in creation and annihilation operators. For example, in the case of the BCS theory, where $...
4
votes
2answers
363 views

Transversal wave speed derivation for small amplitudes

The above is a derivation for the wave speed equation in my physics textbook. However, I've read online that this equation is only true for waves with small amplitudes. I do not see where this ...