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Questions tagged [approximations]

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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
54 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 ...
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3answers
87 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 ...
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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
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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
111 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 \...
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1answer
28 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 $...
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1answer
87 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
36 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....
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2answers
50 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
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1answer
64 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
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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 ...
4
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2answers
59 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 \...
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2answers
78 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 ...
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0answers
72 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
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1answer
133 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
55 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_{...
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1answer
69 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
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2answers
88 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 ...
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1answer
122 views

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
95 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
98 views

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
76 views

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
40 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
30 views

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
30 views

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
81 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 ...
2
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1answer
44 views

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 ...
2
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2answers
70 views

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 ...
3
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1answer
90 views

My textbook sets $\ln \left\{ 1 + \left[ \frac{L}{L_0} - 1 \right] \right\}$ equal to $\frac{L}{L_0} - 1 $. What's the justification for this?

Sat to study physics. I started to study this new chapter Heat and Thermodynamics. It included these steps: $$ \int_{L_0}^L \frac{dL}{L} = \int_{\theta_0}^\theta \alpha d \theta; \text{ or } \ln \...
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1answer
36 views

What is a quasistationary approximation

I was reading an article which states : The linear-stability analysis for this system can be performed in complete generality; but it will be best for purposes of this review to go directly to ...
2
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1answer
52 views

High temperature expansion in general

I'm referencing this thesis which should be open-access. In Appendix D.1 "High temperature expansion in general", the author writes the high temperature expansion in the following way: $$ \begin{...
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2answers
65 views

Theoretical definition and pratical mesurement of differential cross section

In Sakurai's book, the definition of differential cross section is: $$d\sigma/d\Omega= transition \;rate / probability\; flux $$ However this def doesn't contain any information about the thickness ...
2
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3answers
75 views

Oscillator with decaying restoring force

Suppose a system that is described by the equation of motion: $$ \ddot{x} = -k\cdot x\cdot \exp\left(-\frac{t^2}{2\sigma^2}\right). $$ (For example a spring with decaying stiffness.) I'd like to ...
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2answers
122 views

Is the geodesic equation independent of an initial condition?

The following argument is used to determine the unknown factors (e.g., $A(r)$ and $B(r)$) in the Schwarzschild metric. $$ \lim_{r \to ∞}A(r) = \lim_{r \to ∞}B(r) = 1 \space\space\space\space\space\...
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2answers
34 views

Why should the spherically asymmetric part of the effective potential be small in the central field approximation?

In the central field approximation, each electron is supposed to move in an effective or average potential contributed by its attractive interaction with the nucleus and repulsive interaction with the ...
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1answer
181 views

What do the small terms in the series expansion of relativistic energy mean?

Through the fabulous Feynman Lectures of Physics and the introduction of relativistic mass, Richard Feynman made a link between the increase in kinetic energy of a heated molecule of gas, and its ...
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2answers
100 views

Composite particles Dirac spinor approximation

In many scattering process evolving composite particles such as proton, the composite particles are treated as an elementary particles. For example in electron proton scattering to the proton is ...
0
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1answer
84 views

Taylor expansion of scalar fields [closed]

Starting of with electrodynamics I have to compute the taylor expansion around $\vec{r} = 0$ of $\psi (\vec{r}) = |\vec{r} - \vec{r_0}|^{\frac{3}{2}}$ where $\vec{r_0}$ is a constant vector up to ...
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0answers
41 views

Time-independent perturbation theory: why i'th order perturbations are orthogonal to base state?

I have been learning about time independent perturbation theory (non-degenerate for the moment), and am not satisfied about a particular point: the justification for setting $\langle n^i|n^0\rangle = ...
0
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2answers
91 views

Why free-fall acceleration is considered constant rather than increasing? [duplicate]

The force acting on a body of mass $m$ is $mg$, where $g$ is acceleration of free fall ! But why should there be a #uniform acceleration of free-fall in the first place? As per Newton's universal law ...
3
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0answers
74 views

Limitations of RPA (random phase approximation)

I'm interested in the possible limitations of the Random Phase Approximation (RPA). When is it expected to fail? As I understand it, RPA can be derived from the GW approximation, as can be seen here, ...
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1answer
36 views

Energy differentiation with cut-off function

I am a new learner of molecular dynamics (MD) simulations methods and has a simple question regarding handling of cutoff functions. In MD, pairwise energy between two atoms is assumed to be a function,...
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1answer
75 views

How to expand this equation? $H_{1}=\frac{e^{2}}{R}+\frac{e^{2}}{R+x_{1}+x_{2}}-\frac{e^{2}}{R+x_{1}}-\frac{e^{2}}{R+x_{2}}$ [closed]

$$H_{1}=\frac{e^{2}}{R}+\frac{e^{2}}{R+x_{1}+x_{2}}-\frac{e^{2}}{R+x_{1}}-\frac{e^{2}}{R+x_{2}}$$ in the approximation $ \left |x_{1}\right |,\left |x_{2}\right |\ll R $ we expand to obtain in lowest ...
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1answer
71 views

Models in physics [closed]

As I said in another question I am just a physics enthusiast so I am sorry for my very poor knowledge. What is meant by models in physics? what is their function and why physicists imply them? Are ...
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1answer
678 views

How can I Derive the Equation for Coefficient of Linear Thermal Expansion?

I know the relationship between change in temperature and change in length. When the ambient temperature around any substance is increased, its length increases. This is due to molecules gaining more ...
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2answers
117 views

Why does Coulomb's law hold only for two point charges? [closed]

What is the condition for validation of Coulomb's Law?
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2answers
52 views

Is the harmonic oscillator approximation valid in occasion of very powerful fields?

I noted that in physics, to study electromagnetic wave phenomena when there is a sinusoidal behaviour, often is used the approximation of harmonic oscillation. I tried to understand the basics of why ...
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0answers
41 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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3answers
201 views

Is the speed of light in vacuum $3\times 10^8\ \rm m/s$?

I saw another question which says the speed of light is "$3\times 10^8 \:\rm m/s$", and I know that the speed of light is $299,792,458\ \rm m/s$. My chemistry teacher taught me that $3.0$ means $3.0 \...