Questions tagged [approximations]
The approximations tag has no usage guidance.
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How is the gravitoelectromagnetism approximation of GR valid if it seems to yield unstable solutions?
In the gravitoelectromagnetism approximation of GR, we have equations analogous to Maxwell's equations with some sign changes. As pointed out in another post of mine, this leads to unstable run-away ...
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The multipole expansion of electrostatic potential and large distances
I'm reading Griffiths electrodynamics book and I'm currently studying the multipole expansion of electrostatic potential, and I have two questions if you don't mind:
Can I use the multipole expansion ...
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Kinetic friction and damped harmonic oscillators
I was taught in school that the magnitude of the kinetic frictional force does not depend on the speed.
Hence, the equation of motion of the harmonic oscillator in the presence of friction with the ...
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How small is $dt$ in this derivation of the kinetic energy of ideal gasses?
I was reading the derivation of the average translational kinetic energy of an ideal gas in Sears and Zemansky's University Physics. This derivation uses a cylinder with height
$|v_{x}|dt$ and base ...
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Is the drag force to raindrops at the terminal velocity quadratic or linear?
Air drag is approximated by the following equation
$$F=\frac12 \rho A v^2 C_D$$
Depending on the drag coefficient $C_D$, this force may be linear or quadratic with respect to velocity $v$. If $v$ is ...
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Equation of motion with air resistance
If I search for air resistance on a falling object, I find texts stating that it is proportional to the velocity.
$$F=mg-kv$$
Is this accurate or approximate?
Some texts say that when the speed is ...
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Can the cases of multiple fermions in any spherically potential be approximated by the Schrodinger Equation for a single fermion?
In this video https://www.youtube.com/watch?v=tq_y1qOmUBE&t=783s it's mentioned that the structor of atoms of multiple particles can be approximated using the Schrodinger Equation of the Hydrogen ...
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Double-slit interference pattern from double-slit diffraction pattern in the limit of zero slit-width
The intensity pattern of the double-slit experiment should change from the double-slit diffraction pattern $$I=4I_0{\rm sinc}^2\left(\frac{\pi b\sin\theta}{\lambda}\right)\cos^2\left(\frac{\pi(b+d)\...
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Approximation for two level system differential equations
I am currently reading the book "the quantum theory of light" link: http://rplab.ru/~as/2000%20-%20R.Loudon%20-%20The%20Quantum%20Theory%20of%20Light%20-%203rd%20ed%20Oxford%20Science%...
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What is the meaning/justification for the matching conditions in the WKB approximation?
Consider the time-independent Schrodinger equation in one dimension with a potential $V(x)$ at fixed energy $E$. In the WKB approximation, we obtain solutions in the classically allowed region (i.e. ...
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Is it possible to derive Navier-Stokes equations of fluid mechanics from the Standard Model?
We know that the Standard Model is a theory about almost everything (except gravity). So it should be the basis of fluid mechanics, which is a macroscopic theory from experiences. So is it possible ...
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Is the non-relativistic QM a limit of QFT at low velocity? [duplicate]
QFT works with creation and annihilation operators. What happens with them when the velocity of the studied object becomes low?
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Intuition behind gravitational potential
Gravitational potential at a point is equal to work done in bringing a unit mass from infinity to a particular point
That was the text book definition
$$V_{p} = -\frac{GM}{r}$$
If we calculate $V_p$ ...
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Trouble with the algebra in Srednicki book chapter 28
I'm studying chapter 28 in Srednicki (the renormalization group) and I'm having troubles figuring out how he derives eq. (28.15) (last summation above) from eqs. (28.7) and (28.9).
More specifically ...
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How does the net vertical force equation for a non-relativistic string derived?
In the following image from "A first course in string theory", we get the net vertical force of a string, dFv. While I understand the first equation, I don't understand how the second ...
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What scale are Maxwell's Equations valid?
Both Maxwell's equations and quantum mechanics are used to describe the behavior of electrons in circuits.
I am confused on the interlinking between the two and the dividing line between when you use ...
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Transitions in constant perturbations using time independent perturbation theory
The perturbed Hamiltonian is given as.
$$H=\begin{cases} H^{(0)}&\text{for }t\leq 0 \\ H^{(0)}+V(x)&\text{for }t>0\end{cases}.$$
Here $V(x)$ does not depend explicitly on time but it can ...
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The two ways to get Schrodinger equation from Klein-Gordon equation
We can take the Klein-Gordon equation describing the evolution of a complex scalar field. Taking the non-relativistic limit yields a classical wave equation that is identical in form to the ...
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When is classical electromagnetism applicable?
I'm currently studying intermediate level classical electromagnetism and I'm wondering, given that quantum mechanics is the suitable theory for describing the behaviors of microscopic systems, when is ...
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Electric field at a very distant point of an wire from generic point in space
I calculated the electric field at a generic point in the space $P(a,b,c)$ due to an wire with charge density $\lambda$, constant and positive, length $L$, with axis in $z$ direction and origin in the ...
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Notation for rule of thumb, without breaking dimensional homogeneity?
I'd like to know how to write rules of thumb in a concise way, without breaking dimensional homogeneity.
For example, if a runner has an average speed of ~10 km / h, an approximation of the covered ...
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WKB and Dirac Delta potential
The semi-classical approximation for using WKB in simple words says that in a region where the potential doesn't vary sharply compared to the wavelength of the wavefunction the momentum (or the ...
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Alternate Perturbation theory
Take $H=H^0+H'$
Start with $H^1=H^0+\frac{1}{N}H'$, $N$ is large
Approximating Eigenvalues and Eigenfunctions of $H^1$:
$E_n^1=E_n^0+\langle n_0|\frac{1}{N}H'|n_0\rangle$
$|n_1\rangle=|n_0\rangle + \...
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Understanding this quote by Feynman
This might be more of a question on semantics and interpretation and if this doesn't meet community guidelines, feel free to let me know and I'll delete it.
It doesn't matter how beautiful your ...
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Meaning of big $O$ notation with 2 values separated by a comma
I'm reading Classical Electrodynamics 3e by Jackson. In section 1.7 he performs a proof of the Poisson equation in the context of the electric potential. Near the end of the proof, he writes
$$
\...
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Expansion of $\phi (x)$ in the derivation of WKB approximation
In the derivation, we assume the eigenfunctions of $H$ to have the form $$\psi (x)=e^{i \phi(x)/\hbar}$$ where $\phi (x)$ is allowed to take any complex value.
But then suddenly we assume this ...
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When can you trust first-order QED?
In QED, we use the perturbation expansion. Usually, we use the first-order perturbation expansion, and sometimes the second. I want to know, physically, what conditions need to be satisfied for us to ...
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Is the continuum limit equivalent to the low-energy limit?
It is frequently stated that the continuum limit of a lattice model is equivalent to the low-energy limit, e.g. here, here and section IIB of this. I do not know how to show this for myself. Take for ...
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Approximation when $N$ is large in binomial distribution
Reif,pg 14. $n_1$ is the number of steps to the right in a 1D random walk. $N$ are the total number of steps
When $N$ is large, the binomial probability distribution $W\left(n_{1}\right)$ ,
$W\left(...
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Proper condition for a wave of fast varying phase, relative to its amplitude-polarisation?
In the context of special relativity (Minkowski spacetime), I define an electromagnetic wave of the following shape (I'm using units such that $c \equiv 1$ and metric signature $\eta = (1, -1, -1, -1)$...
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Approximation in Peskin's Chapter 6
In Peskin QFT's Chapter 6.4, IR divergence in 1st correction of interaction between electron and photon becomes
\begin{align}
F_1(q^2)\gamma^{\mu}+F_2(q^2)(p^{\mu}+p'^{\mu})
\end{align}
And
\begin{...
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How negligible is a term in the internal energy for the equipartion theorem in classical mechanics?
The equipartition theorem is a well-known result of classical statistical mechanics, and it states that if the Hamiltonian of a system can be written like this:
$$H=\sum_{j=1}^m {\alpha_j\ {x_j}^2}$$
...
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Approximating energy spectra of many-body Hamiltonian
This is related to the previous post here. Although there is no way to diagonalize this Hamiltonian to obtain the exact eigenvalues, is there a way to approximate the eigenvalues of such a Hamiltonian?...
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Electron Energy Approximation for Multielectron Atoms
Given the question: lithium ($Z = 3$) has two electrons in $1s$ and one in $2s$. Estimate the energy of the third electron when it is raised to $3d$.
Since the $3d$ orbital is quite far away from the $...
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Function Values Surrounding Stationary Points
Taylor, in his widely read book "Classical Mechanics," writes on page 218 that
When $df/dx = 0$ at a point $x_0$, but we don't know which of the 3 possibilities obtains, we say that $x_0$ ...
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Does the stationary phase approximation equal the tree-level term?
Consider the scalar field transition amplitude
$$\tag{1} \mathcal{A} = \int_{\phi_i}^{\phi_f} D\phi e^{iS[\phi]/\hbar}. $$
Let $\phi_{cl}$ solve the classical equation $\frac{\delta S}{\delta\phi}=0$. ...
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How large does $N$ need to be for statistical mechanics to be a good approximation?
About how many components ($N$) does a system need for statistical mechanics to apply to that system?
I took stat mech and biophysics from the same professor in undergrad and I distinctly remember him ...
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"Classical" adiabatic approximation: please help me understand better
In this1 paper they use an adiabatic approximation to reduce two differential equations to one. Could you please recommend some alternative reading for this (semi) classical adiabatic approximation, ...
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Wood-Saxon Potential approximated to Harmonic-Oscillator Potential
Under what assumptions does a Wood-Saxon Potential for a nuclear bound state be approximated as a Harmonic Oscillator Potential?
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If there is no perfect black body then what's the point of learning about it? [closed]
My question is simple if there is no perfect black body then what's the point of learning about it?
Because definition of balck body goes like this
A body which can absorb and re emit all kinds of ...
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Converting between mass and binding energy in semi-empirical mass formula
The semi-empirical mass formula is given in terms of the binding energy as:
$$ B(Z,N) = aA - bA^{\frac{2}{3}} - s \frac{(N-Z)^{2}}{A} - d \frac{Z^2}{A^{\frac{1}{3}}} - \frac{\delta}{A^{\frac{1}{2}}}$$
...
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How to prove this relation (algebra: containing non-interacting Green function)?
There is a expression given in article arXiv:1505.01908
$$\bigg[n\big(\omega+\frac{\Omega}{2}\big)-n\big(\omega-\frac{\Omega}{2}\big)\bigg]g^R_{\omega-\frac{\Omega}{2}}g^A_{\omega+\frac{\Omega}{2}}
+n\...
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Questions about the approximation of the Maxwell-Boltzmann Distribution
My question is about calculating the number of quantum states $g$ within an infinitesimally small translational energy level range $dE$ and comparing that to how many true allowed energy levels exist ...
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Why is it called "rotating wave approximation"?
I am just wondering why it is called rotating wave approximation? Where does the rotating come from? According to wikipedia, it says
"Since in some sense the interaction picture can be thought ...
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Why do we multiply in the total wave function but add in the LCAO method?
If to particles with wave functions $\psi_a$ and $\psi_b$ are „combined“ their total wave function is given by:
$$\psi(r_1,r_2)= A[\psi_a(r_1)\psi_b(r_2) \pm \psi_b(r_1)\psi_a(r_2)]$$
(+ for bosons ...
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Vibrational spectrum of diatomic molecule from Schrödinger's equation
I'd like to find the vibrational spectrum of a diatomic molecule from its potential, which can be approximated as
$$V(R) = -V_0\Big[ \frac{1}{4}\Big(\frac{R_0}{R}\Big)^4 - \frac{1}{8}\Big(\frac{R_0}{R}...
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Perturbation theory and size of the perturbation
In quantum field theory, we usually perturb the free field by a little bit. What would be so bad about using a large perturbation to the free field?
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Justification for constant tension in transverse wave on a string
It is common in the derivation of the transverse wave equation on an ideal string to assume the tension along the rope is uniform in the limit that $$|\partial \psi / \partial x|\ll 1.$$ However, what ...
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Is the Newtonian gravitational potential $-\frac{GMm}{R}$ just an approximation?
Is $-\frac{GMm}{R}$ just an approximation? I believe that it is since we assume that one of the mass is at rest when deriving it.
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Dynamical Mean Field Theory (DMFT) does not take into account spacial correlations?
It is often said that the Dynamical Mean Field Theory (DMFT) does not take into account spacial correlations. What does this mean in layman terms?
Does that mean that we assume:
$$
\langle n_i n_j \...
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