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Questions tagged [approximations]

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3
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2answers
129 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 ...
1
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0answers
26 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]+\...
3
votes
1answer
55 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 ...
1
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0answers
52 views

Approximation of $x\log x$ [migrated]

I was wondering if anyone knows a good approximation for the function $$f(x)=x\log x $$ when $x$ goes to infinity. In particular, I would like to get rid of the $\log x$ and so I need a polynomial ...
0
votes
1answer
23 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 ...
1
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0answers
20 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 ...
0
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0answers
11 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 ...
2
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4answers
93 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 $...
0
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1answer
40 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 ...
0
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1answer
27 views

When is the approximation of neglecting the surface energy most useful during estimating the total energy of a spherical system? [closed]

The bulk energy of a spherical system of radius R is proportional to its volume and the surface energy is proportional to its area. The surface energy per unit area, S, and bulk energy per unit volume,...
3
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1answer
180 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 ...
0
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0answers
40 views

Lorentz-like equation from Geodesic equation

In Hobson's General Relativity, it explains the following (p.173) Consider the limit of a weak gravitational field in a coordinate system in which $g _ { \mu \nu } = \eta _ { \mu \nu } + h _ { \mu \...
2
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1answer
83 views

Why does quantum mechanics become unnecessary at sufficiently high temperatures?

In my statistical mechanics intro class, we are taught that at sufficiently high temperatures, the quantum treatment of things becomes unnecessary. Why is this? Can this be shown using certain ...
0
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1answer
175 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 ...
3
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0answers
56 views

Why can infinite planes be approximated as Gaussian surfaces?

A little background: I'm an undergraduate studying Electrodynamics, currently in Chapter 8 of Griffiths. A question I came across (8.4 part a for those curious) asks for a calculation of the force ...
0
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1answer
57 views

Why is the law of Hooke valid only for small displacements?

What changes when displacement exceeds a certain limit?
0
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1answer
22 views

Positive and negative powers of small parameter in perturbation problem

I'd like to perturbatively handle an eigenvalue problem similar to this: $$ \lambda f = (\hat{H} + (1/\epsilon^2) \hat{V} + \epsilon {W}) f, $$ where $f$ is a function, $\lambda$ is an eigenvalue, $\...
0
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1answer
39 views

Perturbation theory for molecules, dipole approximation, chromophore

I am interesting in chromophore group and dipole approximation. For example, i have a molecule (acetone or any other ketone/enol) which is belongs to some symmetry group. Because of the symmetry ...
1
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0answers
60 views

Does uncertainty principle truly represent the “lower bound” of the information we can obtained from a pair of noncommunicable operator?

Background I: Suppose the commonly used non commuting operator $\hat p$ and $\hat x$. The uncertainty principle told us that $\sigma_p\sigma_x\geq \frac{\hbar}{2}$. In standard quantum mechanic ...
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0answers
97 views

Deriving the Pauli-Schrödinger equation from the Dirac equation

Since the Schrödinger Pauli equation describes a non-relativistic spin ½ particle. This equation must be an approximation of the Dirac equation in an electromagnetic field. I was trying to derive this ...
2
votes
1answer
95 views

Changing Summation to Integral

This is the text from Reif Statistical mechanics. In the screenshot he changes the summation to integral(Eq. 1.5.17) by saying that they are approximately continuous values. However, I don't see how. ...
0
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3answers
85 views

Inertial and non-inertial reference frames

My book states that: A reference frame attached to earth is not inertial because it is revolving around the sun and it is rotating about its own axis. Don't we need a specified observer's frame ...
0
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0answers
88 views

Assumptions and limitations for ideal gas

Assumption: In the assumption for ideal gas law, it is stated:"The time it takes to collide is negligible compared with the time between collisions." For, this assumption, can i just say there is only ...
3
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1answer
120 views

Non-relativistic limit of Hamiltonian for a free particle in general relativity

The Hamiltonian for a particle moving in a gravitational field can be taken as $$\mathcal{H} = \frac12 \sum_{\mu,\nu=0}^3g^{\mu\nu}(x)p_\mu p_\nu\tag{1}$$ as long as the parametrization is affine. ...
0
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1answer
38 views

Do pulleys have no effect on atwood machine?

I was given the following example for an atwood machine: We should calculate the accelaration of the masses. The given solution to this problem was the same as like there is just one pulley: $$a=\...
1
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2answers
87 views

Physical example of line charge

Electric field due to an infinite line charge, sheet of charge, point charge, etc are popular problems solved in most text on Gauss's law of electromagnetism. My question: does an (exact or ...
0
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2answers
27 views

How do you calculate Reynolds and Mach's numbers before solving the Navier-Stokes equations?

Apologies in advance if the question is trivial. I am accustomed to electromagnetics but an amateur on fluid dynamics. My understanding is that the Navier-Stokes equations are solved to determined ...
2
votes
1answer
59 views

Small oscillations in the given potential

The task is to find the period of small oscillations in the potential $$U=U_0\tan^2{\Big(\frac{x^2}{a^2}\Big)}.$$ I started with finding the stable equilibrium points: $\frac{dU}{dx}=0$ $2U_{0}\...
1
vote
2answers
60 views

Thick vs thin lens

What is the criteria to know whether a lens is thin or thick ? Suppose we have a lens of r1= 10 mm , r2= -10 mm and thickness is 5 mm. So, with this information what can we say about the lens whether ...
1
vote
0answers
83 views

Perturbation to the flat space metric

from the geodesic equation for non-relativistic case where $$v_i\ll c$$ $$\frac{dx^i}{dt}\ll1,{\rm for }\ c =1$$ $$\frac{dx^i}{d\tau}\ll\frac{dt}{d\tau}$$using this the geodesic equation for proper ...
1
vote
1answer
109 views

Coulomb's Law Question

The presentation of Coulomb's Law in various books occasionally has a note that the test charge, q2, must be small enough that it doesn"t alter the field of the first charge, q1. The same limitation ...
0
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0answers
37 views

QM limit of QFT in Schwartz [duplicate]

In Matthew Schwartz's QFT text, he derives the Schrodinger Equation in the low-energy limit. I got lost on one of the steps. First he mentions that $$ \Psi (x) = <x| \Psi>,\tag{2.83}$$ ...
0
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1answer
66 views

How to reach $U = mgh$ Using Newton's Law of Universal Gravitation? [duplicate]

today i was curious about the potential energy, so, i started studying the Newton's Law of Universal Gravitation which its equation is \begin{eqnarray} U= -\frac{GMm}{r}.\end{eqnarray} Well, since i ...
4
votes
3answers
103 views

Hooke's full unapproximated law

It is known that the Hooke's law relating the restoring force of a spring to the distance of retraction from the equilibrium position, is only an approximation. That is, the equation $F=-kx$ is only ...
13
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3answers
2k views

Why is a particle non-relativistic when its kinetic energy is small compared to its rest energy?

For example, nucleons in nucleus are in motion with kinetic energies of 10 MeV. Their rest energies are about 1000 MeV. Kinetic energy of nucleons is small compared to rest energy. They are hence ...
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0answers
47 views

How do we understand the results of $1/N$ or $\epsilon$ expansion beyond leading orders?

When we do $1/N$ expansions in, say, 2+1$D$ $O(N)$ models and try to extract all kinds of critical exponents from it, we get the following results for the scaling dimensions of various operators up to ...
6
votes
1answer
288 views

Rotating wave approximation and classical Rabi oscillations: why don't the fast oscillating terms seem negligible in the initial frame?

I am trying to understand better the rotating wave approximation (RWA). Consider an atom modeled as a two level system, interacting with a Laser. I have the dipole momentum operator $$\vec{D} = d \...
0
votes
1answer
34 views

Approximating the time it takes for a particle with a potential $-Ax^4$ to approach the origin [closed]

Here's the problem I'm solving: A particle of mass $m$ can only move along the $x$-axis and is subject to an interaction described by the potential energy function $U\left(x\right) = -Ax^4$, where $...
1
vote
1answer
266 views

How to derive a formula for the period of a simple pendulum? [duplicate]

The following formula is given in our lab manual: $$ T = 2 \pi \sqrt{\frac{L}{g}} \left( 1 + \frac{1}{4}\sin^2 \frac{\theta}{2} + \frac{9}{64}\sin^4 \frac{\theta}{2}+\cdots \right) $$ for the period ...
0
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3answers
43 views

Finding the equations of motion with observations

Let's say that we don't know the equations of motion. I will try to predict where my ball will fall when I shoot it with an angle $\alpha$ and and speed $v$ by finding the function that describe this....
-1
votes
2answers
51 views

What is the force of friction on the body? [duplicate]

This might seem like a stupid question, but my instructor could not give a straightforward answer. If a body is kept in contact with a completely vertical surface i.e. right angle to the ground, and ...
1
vote
1answer
97 views

Non-conservative forces in Lagrangian mechanics

In the Lagrangian formalism with a dissipative frictional force $F$, we can write $$\frac{d}{dt}\frac{\partial\mathcal{L}}{\partial\dot{q}_{k}}-\frac{\partial\mathcal{L}}{\partial q_{k}}=Q^{(nc)}_{k}...
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4answers
2k views

The reasoning behind doing series expansions and approximating functions in physics

It is usual in physics, that when we have a variable that is very small or very large we do a power series expansion of the function of that variable, and eliminate the high order terms, but my ...
4
votes
2answers
79 views

Slowly-varying envelope approximation: what does it imply?

I understand that the slowly-varying envelope approximation means that we can write an electromagnetic wave as $$ E(x,t)=V(x,t)e^{i(k_0x-\omega_0 t)},$$ where $$ \left \vert \frac{dV}{dx} \right \...
0
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2answers
81 views

How does $\sin\theta=\theta$ give a right answer even when it is an approximation?

Magnetic field at the center of circular current carrying loop is given by $$ B=\frac{\mu I}{2 R} $$ Where $\mu$ is the permeability of free space and $R$ is the radius of loop. In a question by ...
0
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0answers
78 views

Quantum numbers $n$ and $\ell$ relation in many electrons atom [duplicate]

Solving the Schrodinger equation for hydrogen atom we arrive to the conclusion that quantum numbers $n$ and $\ell$ have the relation $$\ell=0,1,...,n-1.$$ Now,since we can not solve the Schrodinger ...
4
votes
1answer
153 views

Coulomb's Law modified in general relativity?

It seems difficult to track down a clear explanation of this statement: So although the Coulomb law was discovered in a supporting frame, general relativity tells us that the field of such a charge ...
4
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0answers
68 views

The kinematic region for the operator product expansion

In Ch.18 of the textbook An Introduction to Quantum Field Theory by Peskin and Schroeder, on P.613 the operator product expansion (OPE) is introduced $$\mathcal{O}_1(x)\mathcal{O}_2(0)\to \sum_n C_{...
0
votes
1answer
113 views

Small Angle Approximation for Simple Pendulum

I am working on a simple pendulum problem. The $y$ direction is vertical and the $x$ direction is horizontal. Displacement in the $x$ direction is taken to be much less than the length of the string, ...
0
votes
2answers
98 views

Error of relativistic kinetic energy

I have recently begun working on the special relativity theory. I have then made the taylor series for the gamma factor to show that we get the classic formula for kinetic energy: $$E _ { k i n } = m ...