Questions tagged [approximations]

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134 views

Is Stokes equation a reduction of Navier-Stokes equations?

The following Stokes problem: $$\begin{cases}-\nu\Delta u+\nabla p=f&,\textrm{in }\Omega\\ \nabla\cdot u=0&, \textrm{in } \Omega\end{cases}$$ is a reduction of the Navier--Stokes equations? ...
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3answers
619 views

Index of Refraction in Metal: Approximating Complex Perturbation

If you consider waves in a metal, you can write the index of refraction for the metal as, $$ n^2 = 1 - \frac{\omega_p^2}{\omega^2} $$ I am interested in what will happen if the index is perturbed by ...
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1answer
58 views

Theoretical justification for the range of validity of the approximation $R(T)\approx R(T_0)[1+\alpha (T-T_0)]$

In the experiment for calibrating a platinum resistance thermometer, we are always approximating the resistance of the platinum thermometer by $$R(T) \approx R_0 (1+\alpha T),$$ taking the reference ...
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1answer
64 views

“Newtonian limit” property of special relativity

Books say that special relativity is indistinguishable from Newtonian mechanics when the speed of the primed frame ($v$) is small compared to the speed of light ($c$). This is what I mean by the "...
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1answer
197 views

First-Order Perturbation of Energy Eigenfunction

I have a homework questions where I'm struggling to understand the methodology to use. We derive first the energy functional for the energy eigenfunction equation (this is fine, I used some vector ...
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1answer
37 views

What does the notation $\mathcal{O}\left(\frac{1}{r^2}\right)$ mean? [duplicate]

I was reading a text about quantum scattering, and I faced a notation I don't understand. The equation is the following: $$ \nabla \psi_{\text{scattered}} = \frac{i k f(\theta) e^{ikr}}{r} \mathbf{\...
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0answers
73 views

The answer to this question is not close to the age of the universe, then why do we say it is?

I saw this question in our textbook A great physicist of the century (P.A.M. Dirac) loved playing with numerical values of Fundamental constants of nature. This led him to an interesting ...
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2answers
72 views

Why strong electric field leads to non-Ohmic behavior?

Homogenous conductors like silver or semiconductors like pure germanium or germanium containing impurities obey ohm's law within some range of electric field values. but if the field becomes too ...
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1answer
64 views

What exactly does Ohm's law say?

Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, R ...
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2answers
72 views

Boundary conditions in QM and statistical physics

I don't understand something about boundary conditions in problem that I discuss it below. in QM we solve the particle in Potential well and we obtain that we should have $k=\frac{n*pi}L$ that $n\in{...
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4answers
226 views

Is the statement that $U(x)$ is quadratic for simple harmonic motion equally strong as the statement that $F(x)$ is linear?

Is the statement "If the potential energy of a particle under oscillatory motion is directly proportional to the second power of displacement from the mean position, the particle performs a ...
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1answer
98 views

Is continuum mechanics a generalization or an approximation to point particle mechanics?

Newtonian Mechanics is usually presented as a theory of point particles (and forces). My impression of the status of continuum mechanics is that it is mostly taken as an approximate description for ...
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1answer
136 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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0answers
34 views

How to find the terms of Post-Newtonian approximation?

I am studying the Damour paper on the Post-Minkowskian approximation to the 2 bodies problem in General Relativity (paper) but can't really understand the explanation of the previous state of the art, ...
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2answers
923 views

When is the adiabatic approximation for solid state systems valid?

The adiabatic approximation for solid state systems is rather radical. I was wondering in which cases it breaks down. As it is based on the idea of the nuclii being much heavier than the electrons I ...
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3answers
21k views

How is the Saddle point approximation used in physics?

I am trying to understand the saddle point approximation and apply it to a problem I have but the treatments I have seen online are all very mathematical and are not giving me a good qualitative ...
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5answers
143 views

Wave Equation derivation

I'm curious about part of the derivation of the wave equation as is done in all references that I've seen so far (I'm gonna reproduce only the part that's puzzling me). We apply Newton's second law ...
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1answer
47 views

Neglecting some wave functions by assuming that the angle between tension force and horizontal is small in the derivation of wave equation in $1D$

In the derivation of the wave equation in classical mechanics in one dimension in a string. It's assumed that the angle between the tension and the horizontal line is small. This is assumed to allow ...
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56 views

Taylor expansions giving different answers before and after applying Euler-Lagrange equation

I have the Lagrangian $$\alpha(\boldsymbol{\dot{r}} -\boldsymbol{v}(\boldsymbol{r}))^{2} + \beta \nabla \cdot \boldsymbol{v}(\boldsymbol{r}),\tag{1}$$ where $\boldsymbol{r}$ is the position and $\...
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52 views

Ignoring 2nd order terms in QM definitions of translation and momentum operators

In Sakurai's 'Modern Quantum Mechanics', he defines the infinitesimal translation operator as: $\mathcal{J}(d\mathbf{x}')=1-i\mathbf{K}\cdot d\mathbf{x}'$, and then he goes on to prove this ...
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2answers
65 views

Why do we don't get exact answers to our experiment but rather an approx value to it?

Why do we face a problem of not getting to the exact answer but to an approx to it, mostly everything we come through in practical physics is a approx but not the exact one.
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3answers
141 views

Wave speed derivation

The wave speed derivation approximates the wave as a circle. It uses that to know that $$a=\frac{v^2}{R}$$However, numerous functions can approximate the wave. A straight line, $x^2$, $x^3$, etc. If I ...
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0answers
51 views

High Temperature Expansions and Cumulants

In this paper the authors perform a high-temperature expansion of the correlation functions for a Heisenberg model on a lattice. Starting from $$\left<\mathbf{S}_i\cdot\mathbf{S}_j\right>_\beta ...
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0answers
29 views

What is the mathematical approximation for uniform gravitational field in Newtonian Mechanics? [duplicate]

When measuring the local gravitational field using a simple pendulum, one of the core assumptions is that the gravitational field is homogeneous. But the theory itself (Classical/Newtonian Mechanics) ...
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28 views

What is the static exchange approximation?

In this paper, on the 4th page (and throughout), they talk about studying electron-helium scattering in the "static-exchange approximation". I have scoured the literature and have not been able to ...
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2answers
67 views

What does the term $\mathcal O(\epsilon^2)$ mean?

In the highest upvoted answer to Where does the $i$ come from in the Schrödinger equation? the author writes the following equation: $$ U^\dagger U=(\mathbb I+\epsilon^* A^\dagger)(\mathbb I+\...
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2answers
700 views

Approximating an expression for a potential

In a problem which I was doing, I came across an expression for the potential $V$ of a system as follows $$V = k\left(\frac{1}{l - x} + \frac{1}{l + x}\right)\tag{1}\label{1}$$ where $k$ is a constant,...
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1answer
57 views

AC currents for self-inductance of a wire

I've been currently reading some stuff on self-inductance of a wire. There are different regimes where different approximations are used. For the AC currents the notes state that an approximation can ...
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2answers
151 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 ...
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2answers
120 views

Are the infinitesimal lengths in integrals bounded by the Planck length? [closed]

When we integrate something say work, $\int F\cdot ds $ then we will get work but what exactly is $ds$? how much is ds? Is it the Planck length? Are we just pretending there exists some infinitesimals ...
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1answer
63 views

Why is the Earth not an inertial frame of reference?

From many sources I have found the explanation that the Earth is not an inertial frame of reference because it rotates around its axis. However, nobody mentions the rotation about the Sun. What I ...
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1answer
207 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 ...
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2answers
36 views

Why is the velocity in the horizontal direction constant for 2D projectile motion?

For 2D projectile motion, why is the velocity in the horizontal direction constant? I learned that we neglect air resistance, but why? And isn't there horizontal acceleration at the beginning when ...
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0answers
22 views

Semiclassical limit $S \to\infty$ in spin model

In many literature, the limit $S \to \infty$ is considered as a semiclassical limit. My question is that when this approximation is valid? Since paticles, say electrons, have the fixed spin number $S=...
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1answer
160 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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3answers
70 views

Electric field generated by a small object

I've just begun to study electrostatics and I've read the coulomb law, it describes the electric field generated by a point charge as characterized by a spherical shape. I think it is due to the fact ...
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5answers
188 views

Which type of conductors don't follow Ohm's law? [closed]

Which type of conductors don't follow Ohm's law? I know that semiconductors and superconductors don't follow them, but why? And what about ideal conductors. What are they? Do they follow ohm's law? ...
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1answer
121 views

Schwarzschild Black Holes

Page $190$ of “Tensors, Relativity, and Cosmology”: Consider a particle falling radially into a black hole with a radial velocity $u^1=dr/ds$. As the particle is falling radially, we have $u^2=u^3=...
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0answers
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Validity of Random Phase Approximation in 2D/3D semimetals

In, for instance, this paper and this one the authors look at many-body effects in two- and three-dimensional semimetals, which have a low-energy quasiparticle dispersion relation of the form $\...
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2answers
139 views

Why cube roots or in general $(2n-1)$th roots are rarely seen in equations in physics?

I rarely saw any equation in physics which involved cube roots or odd roots.Even while solving problems I rarely saw any odd root or cube root. So why nature prefers even powers of physical ...
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0answers
32 views

Hartree-Fock approximation derivation

Some context: I'm having a hard time deriving the results of the Hartree-Fock approximation. Let $H$ have the form $$H = \sum_{i=1}^{n}\left[\frac{p_{i}^{2}}{2 m}+U\left(\vec{r}_{i}\right)\right]+\...
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1answer
118 views

problem with Sudden Approximation in quantum mechanics

If the Hamiltonian of a system changes abruptly (over a very short time interval) from one form to another, we would expect the wave function not to change much, yet its expansion in terms of the ...
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2answers
132 views

Series expansion of $N$-particle potential energy

I am trying to understand the so-called Taylor expansion (or series expansion) of potential energy of a system of $N$-particles. This expanded form is stated without derivation in some molecular ...
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1answer
24 views

Calculating Total energy of 2D Debye monoatomic solid

I am trying to find the total energy of a mono-atomic 2D Debye solid. I started with the density of states: $$D(\omega)=\frac{A\omega}{\pi c^2} $$ where A is the area, $\omega$ the frequency and c ...
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1answer
197 views

Second order relativistic corrections to Pauli equation from Dirac equation

I'm trying to derive the full and correct Hamiltonian for spin$\frac{1}{2}$ particles from Dirac equation up to second order in $v/c$. For a potential and magnetic field constant in time. In ...
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0answers
21 views

How does approximations to assumptions lead to approximations to solutions?

I was reading Physics SE 228313. It talks about using Born-von Karman condition to model metal lattices. Is there any mathematical justification that approximations to physical condition would give ...
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4answers
113 views

Approximation of multiplicity when Ideal gas low density is applied $\frac{M !}{(M-N)!} \approx M^{N}$

Our lecturer today mentioned how a piston's head being at equal pressure maximised the multiplicity of states. He mentioned the following: If I have a fixed number of particles $N_A$ on left and $...
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0answers
13 views

Paraxial approximation - time varying

I want to reproduce the results of a paper I am reading. In the paper, authors use paraxial approximation. My question is: in the potential V, I can put a time dependence of space? Or the paraxial ...
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1answer
50 views

How to ascertain that the Rayleigh-Ritz variational method gives the exact value of the ground state energy?

So the Rayleigh-Ritz variational method can be used to calculate the ground state energy of a quantum system. If $\phi(x)$ is a suitable (square integrable) and normalised function of the coordinates ...
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2answers
337 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 ...