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Questions tagged [approximations]

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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 \...
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1answer
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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 ...
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1answer
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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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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 ...
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1answer
51 views

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

What changes when displacement exceeds a certain limit?
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1answer
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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, $\...
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1answer
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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 ...
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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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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 ...
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1answer
80 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. ...
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3answers
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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 ...
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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 ...
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1answer
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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. ...
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1answer
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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=\...
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2answers
82 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 ...
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2answers
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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 ...
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1answer
51 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}\...
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2answers
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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 ...
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0answers
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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 ...
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1answer
78 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 ...
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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 $$ \Psi (x) = <x| \Psi>,\tag{2.83}$$ ...
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1answer
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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 ...
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3answers
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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 ...
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3answers
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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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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 ...
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1answer
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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 \...
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1answer
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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 $...
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1answer
182 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 ...
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3answers
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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....
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2answers
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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 ...
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1answer
80 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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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 ...
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2answers
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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 \...
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2answers
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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 ...
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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 ...
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1answer
144 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 ...
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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_{...
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1answer
90 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, ...
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2answers
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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 ...
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1answer
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Does a proton bend spacetime?

Protons have mass and as a result of einstein's field equation dictate that the spacetime is no longer flat. But yet I find in most Quantum Field Theory books the Minkowski flat spacetime metric is ...
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3answers
103 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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1answer
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Does Kirchhoff's Law always hold?

There's a bit of furore from this question on Youtube involving Dr. Walter Lewin and another Youtuber. With Dr. Lewin claiming Kirchhoff's Law doesn't always hold when magnetic fields are involved, ...
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3answers
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Why we often approximate a wave function of a particle to Gaussian wave function?

I was solving problem of two particle system. We were taking wave function generally $\psi$. Later we approximated this wavefunction of two-particle system to double Gaussian wave function. My ...
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0answers
52 views

Taylor expansion for a double well/perturbed infinite square well

I'm trying to estimate the ground state energy for a perturbed infinite square well directly. The potential is piecewise constant $$V(x)=\begin{cases}&\infty, \qquad x<-a/2\\ &0 \qquad -a/2&...
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1answer
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Expand the partition fct. of a simple harmonic oscillator

I come across a expansion of the partition fct. of a simple harmonic oscillator $q$ as: $$q=x^{-1}(1-\frac{x^2}{24}+...) \tag{1}$$ where $x=h\nu/kT$. It’s easy to get $$q=\frac{e^{-x/2}}{1-e^{-x}}=\...
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1answer
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Criteria on good inertial system approximation

I'm currently wrapping my head around Newton's First Law. I think I start to get a basic understanding on the meaning of this law in terms of "the existence of inertial system". Basically my ...
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2answers
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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
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How to interpret results of stationary phase approximation in GW case?

As time increases, the amplitude and frequency of the GW signal also increase. But after using the stationary phase approximation, the signal is proportional to ${1/f^{7/6}}$, where $f$ is the GW ...
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2answers
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Capacitor demo explanation

I know that for a charged capacitor as one separates the plates further apart the voltage increases while the capacitance decreases. But surely as the plates are pulled further and further apart the ...