Questions tagged [approximations]
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What does Hooke's law have to do with molecular forces?
In The Feynman Lectures, in the chapter Characteristics of Force, In the section entitled Molecular forces, Feynman talks about the molecular forces, and then he states afterwards:
If the molecules ...
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Proof of adiabatic theorem on Wikipedia
I'm having trouble following the proof of the adiabatic theorem (apparently due to Messiah) on Wikipedia.
At one stage we have:
$U(t_1,t_0)=1+{1\over i}\int_{t_0}^{t_1}H(t)dt+{1\over i^2}\int_{t_0}^{...
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WKB and Virial Theorem contradiction in determining Bound States
Consider a potential that on the left of some point $x=x^*>0$ is infinite and on the right of that point it is of the form $$V(x)=-\alpha x^{-3}.$$
I tried to use the WKB method to determine the ...
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Is there a Lorentz invariant approximation to General Relativity?
Since General Relativity is the most accurate description of gravity is there any possible way to derive a Lorentz invariant theory from: $$R_{\mu\nu}-\frac{1}{2} g_{\mu\nu}R+\Lambda g_{\mu\nu}=kT_{\...
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Validity of the sudden/diabatic approximation
The Schrodinger equation is given by
$$i\hbar\ \frac{\partial}{\partial t}\ \mathcal{U}(t,t_{0})=H\ \mathcal{U}(t,t_{0}),$$
where $\mathcal{U}(t,t_{0})$ is the time evolution operator for evolution ...
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Evaluating low-temperature dependence of the BCS gap function
How does one go about evaluating the behavior of the BCS gap $ \Delta = \Delta(T) $ for $ T \to 0^+ $ under the weak coupling approximation $ \Delta/\hbar\omega_D \ll 1 $?
In Fetter & Walecka, ...
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Solution of QM tasks by using asymptotics
When we solve QM tasks by solving the Schrödinger equation, such as tasks about a particle in a Morse potential, a Poschl-Teller potential and many others, we usually find approximations (lets call ...
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What is meant by low-energy approximation?
I've seen the term 'low-energy approximation' several times and also 'low-energy Hamiltonian' but I didn't got the reasoning of these terms.
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Problem with Sudden Approximation in quantum mechanics
If the Hamiltonian of a system changes abruptly (over a very short time interval) from one form to another, we would expect the wave function not to change much, yet its expansion in terms of the ...
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The kinematic region for the operator product expansion
In Ch.18 of the textbook An Introduction to Quantum Field Theory by Peskin and Schroeder, on P.613 the operator product expansion (OPE) is introduced
$$\mathcal{O}_1(x)\mathcal{O}_2(0)\to \sum_n C_{...
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Approximating sums as integrals and divergent terms
I have the following sum (notice that the sum starts from 2, i.e. there's no divergence):
$$\sum_{i=2}^{N}C_i\dfrac{\exp{\left(-k| \mathbf{R}_i-\mathbf{R}_1| \right) }}{| \mathbf{R}_i-\mathbf{R}_1|}$$...
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Quasiclassical QM for centrally symmetric fields
Let's have quasiclassical QM for centrally symmetric field $V(r)$. The Schroedinger equation for radial part of wavefunction $R_{n\ell}$ after substitution $u_{n\ell} = rR_{n\ell}$ takes the form
$$\...
user8817
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Particles scattering on fluids: breakdown of the effective continuum description
When does the macroscopic continuum description of a medium like a fluid break down? Say I'm interested in a scattering process of some particles with momentum $p$ and energy $E$ off a fluid of ...
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Connection between a saddle point approximation and plain perturbation theory
I am currently studying functional integration in the context of classical and quantum equilibrium thermodynamics. However one thing puzzles me:
In the book "Phase Transitions and Renormalization ...
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Is there a guiding principle to determine when quantization is needed when modeling EM-matter interactions?
I'm an undergrad engineering student and I need some guidance to help me understand a few issues regarding classical electromagnetism (EM).
I'm interested in engineering applications where there is an ...
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Far field approximation for massive Klein-Gordon equation in 3+1D
For a massless scalar, one has the familiar Green's function
$$
G(t,r) = \frac{\delta(t - r)}{4\pi r}\,,
$$
and one may take the far-field approximation in a rather straight-forward way:
$$
\int d t d^...
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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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Why does mass increase when gravitational potential energy increases?
I saw a solved example in a book (Concepts of Physics by H.C. Verma, volume 2), where there is a body near surface of the earth, the problem is to calculate the increase in mass of the body when it is ...
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Appoximate solution for quantum scattering through Green's function: error and assumption
I've been trying to think of some approximate way of solving quantum scattering problem for a single electron obeying Schrodinger equation with a locally supported potential:
$$\nabla^2 \Psi( \mathbf{...
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Time averaging a Hamiltonian
There are a number of problems in quantum mechanics whose solution relies on time-averaging away parts of the Hamiltonian. In particular, two examples that come to mind:
The rotating wave ...
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Non-relativistic limit of complex scalar field Lagrangian
I am trying to derive the non-relativistic Lagrangian for a complex scalar field from taking the non-relativistic limit of the complex scalar field Lagrangian. I am following the steps in "QFT for ...
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Mean field theory Weiss Approximation for the Isling Model of a Protein
A model for protein in 2D can be formed by adding bonds of fixed length $l\sqrt{2}$ on a square lattice along the diagonal, ie $\hat{\mathbf{b}}_i=\frac{1}{\sqrt{2}}(\pm \hat{\mathbf{x}}\pm \mathbf{y})...
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Problem in Youngs double slit experiment
This is from Young Double slit experiment. But How to prove the the two $\theta$ are equal, I meant, how $\angle EAD= \angle PEC$? I see from the both triangle have $90^0$ but what about others?
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Self-consistent field approximation and uniform field approximation?
Can anyone give me explanation of self-consistent field approximation and uniform field approximation? I know self-consistent as when we write the Schrödinger equation as
$$[ -\frac{\hbar^2}{2m} \...
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Sub-series of a perturbation series and summation of infinite diagrams
In many-body perturbation theory like in, Altland and Simons: Condensed matter field theory, 2nd edition, we express a correlation function in terms in an perturbation series. My understanding is ...
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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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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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Has anyone ever actually considered a spherical cow? [closed]
Yes, I'm aware this question is somewhat whimsical.
I'm sure every physicist is familiar with "Consider a Spherical cow".
Has a cow (or any animal for that matter) ever been assumed spherical for ...
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Existence of solid mechanics problems that cannot be solved through Lax-Milgram approaches
Very often, solid mechanicians employ finite-element analyses to solve problems in linear solid mechanics. This approach is guaranteed to work because the Lax-Milgram theorem, along with some ...
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Hartree-Fock decoupling of Hubbard model
Hartree-Fock approximation requires wavefunctions be as separable as possible. I know the basic idea of Hartree-Fock but having some trouble in formalism of second quantization. I am trying to ...
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What is the meaning of the existence of a large $N$ limit in QFT?
Large $N$ limits are present in many different contexts: matrix models, gauge theories in various dimensions, conformal field theories (where $N$ is essentially the central charge).
We often hear ...
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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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Don’t understand how nonlinear resistors violate Ohm’s law
Ohm’s law states that the voltage across a resistor is directly proportional to the current through. This is given by the formula v=iR. But most textbooks say that this law is violated when the v vs i ...
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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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What's considered a small time step?
I was looking at the following identity that's often used in time evolution:
$$ (e^{xA/n}e^{xB/n})^n \approx (e^{x(A+B)/n})^n$$
This holds when $(\frac{1}{2}(x/n)^2[A,B])^n$ is small. I'm wondering ...
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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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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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Is it possible to get the SHO approximation of a pendulum without using energy conservation?
I tried to get the approximation for small angle of a simple pendulum using only $\sum \mathbf F = m\mathbf a$ and cartesian coordinates (that means only $x$'s and $y$'s, without $\theta$). After some ...
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What exactly is meant by infinity?
What exactly is meant by infinity when I see it in a physics equation (always something wrong?)?
And in experiment how many orders of magnitude can be treated as infinity (say, if infinity is ...
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Can a wavefunction be solved to any arbitrary precision, given enough computer time?
I learned that the wavefunction for the hydrogen atom can be solved analytically (we did the derivation in class), but that for more complicated atoms it is "impossible" to solve and that only ...
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Can there be tension in an inextensible string?
In physics, tension describes the pulling force exerted by each end of a string, cable, chain, or similar one-dimensional continuous object, or by each end of a rod, truss member, or similar three-...
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Derivation question of WKB method
Quantum Mechanics (2nd Edition) by Bransden and Joachain contains the following passage:
Substituting (8.176) into (8.171), we obtain for $S(x)$ the equation
$$-\frac{i\hbar}{2m}\frac{\mathrm{d}^...
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What's the difference between "numerical methods" & "mathematical analysis" as said by Feynman in his lectures?
While reading his lectures, I came to these lines:
On the basis of Newton's second law of motion,which gives the relation between the acceleration of any body & the force acting on it,any ...
user36790
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Classical formulation of mechanics applied to Quantum Mechanics
According to Ehrenfest's theorem, the expectation values of observables such as position ($x$), momentum ($p$), etc. behave not only in a deterministic way but in fact in a classical way. Therefore, ...
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Recover non-relativistic density of state
The density of state of a non-relativistic particle ($E = \hbar^2k^2/2m$) in 3D is:
$$\rho_{class}(E) = \dfrac{V}{4\pi^2}\left(\dfrac{2m}{\hbar^2}\right)^{3/2}E^{1/2}.$$
The density of state of an ...
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Faraday's law and Kirchhoff's law
Suppose we have a closed loop in a changing magnetic field. By Faraday's law this would induce an emf in the loop. However by the Kirchhoff's law the total emf around a closed loop is zero. It seems ...
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Why is Coulomb's law still relevant?
There is no scenario where we can use Coulomb's law. There is no static charge. Even if we consider the local charge density to be constant for a system of charges, the individual charges are still ...
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In which cases do KCL and KVL fail to apply in circuits?
For which circuits will KCL and KVL be applicable?
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Can artificial neural networks be effective theories?
Wikipedia describes "effective theories" as follows.
In science, an effective theory is a scientific theory which proposes to describe a certain set of observations, but explicitly without ...
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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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