Questions tagged [approximations]

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“Newtonian limit” property of special relativity

Books say that special relativity is indistinguishable from Newtonian mechanics when the speed of the primed frame ($v$) is small compared to the speed of light ($c$). This is what I mean by the "...
3
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1answer
138 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 ...
3
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2answers
141 views

Multiple Definition For Gravitational Potential Energy?

This may just be a simple Misconception Question, here goes: Definition for Gravitational Potential Energy: The work done by gravity to pull an object to the ground. $E=-(\frac{GMm}{r}-\frac{...
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1answer
1k views

WKB Quantization Condition - negative?

In deriving the quantization condition for a bound state in a potential with "no verticle walls" we start with the WKB connection formulas to find the wavefunction in the interior of the well ($x_1<...
3
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1answer
722 views

Understanding multipole expansion in classical electrodynamics

I am trying to better understand what the multipole expansion means from a phyiscal point of view. Although mathematically, one may say it is just another form of a series expansion, in this case, the ...
3
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2answers
131 views

Better approximations with two sticks

How approximation ought to be done always confuses me. Considering this problem in a textbook: Two massless sticks of length $2r$, each with a mass $m$ fixed at its middle, are hinged at an end. ...
3
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1answer
193 views

Limit example in Zangwill “Modern Electrodynamics”

Zangwill shows that the potential of a finite line segment going from $-L$ to $L$ on the $z$-axis with constant line charge density $\lambda$ is: $$\phi(z,\rho) = \frac{\lambda}{4\pi\epsilon_0}\ln\...
3
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1answer
179 views

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 ...
3
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1answer
745 views

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 ...
3
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1answer
448 views

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})...
3
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1answer
1k views

WKB approximation for multiple turning points

I'm working on a numerical program which approximates the eigenvalues of a Schrödinger equation by making use of the WKB approximation formulas. For example, if the Schrödinger equation is $$ y''(x) = ...
3
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1answer
469 views

Neglecting second order differentials

I am currently doing some Lorentz invariance exercises considering infinitesimal Lorentz transformations, and have been told to neglect second order differentials. It's not the first time I have come ...
3
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1answer
612 views

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} \...
3
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1answer
200 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 ...
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0answers
64 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 ...
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138 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, ...
3
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0answers
460 views

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 ...
3
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0answers
102 views

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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2answers
400 views

Temperature in statistical mechanics and differentiating entropy

In statistical mechanics, the entropy of an isolated system with energy $E$ (with fixed volume $V$ and chemical composition $N$) is defined as $S(E) = k \log \Omega$, where $\Omega$ is the number of ...
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110 views

Why does the time-independent perturbation theory become no longer useful when its order gets larger?

In Griffith's Introduction to Quantum Mechanics p. 256, after figuring out $$E_n^2=\sum_{m\neq n} \frac{|\langle\psi_m^0|H'|\psi_n^0\rangle|^2}{E_n^0-E_m^0}$$ he says We could go on to calculate ...
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0answers
572 views

What does it mean to expand a Hamiltonian using perturbation theory?

On UC Davis chemwiki website, the Hamiltonian for quadrupolar coupling in NMR is analyzed. (The details of this aren't important.) It is said in the analysis that: The expansion of the ...
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0answers
584 views

Born approximation to Lippmann-Schwinger integral equation

I am having the following problem understanding the Born approximation in the case of the Lippmann-Schwinger equation. This exercise is for something which is entitled "computational physics lab ...
3
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1answer
366 views

'Validity' of QED/QCD/Electroweak interaction

I am currently attending a course on Quantum Field Theory and I got into thinking how valid these theories are. As the theory attempts to describe reality only far above the Planck (length) scale, ...
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2answers
3k views

Massless string Paradox

If we introduce the notion of a massless string to denote the fact that net force on a massless string will always be $0$, since it is massless. How can these massless strings ever accelerate when ...
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3answers
1k views

Two bodies of finite size treated as two point masses in Newtonian gravity

When discussing gravitation between two bodies of finite size, for instance Earth around the Sun, we suppose the mass of Earth and the Sun to be perfectly localized at the center of each body. Is this ...
2
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1answer
142 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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4answers
115 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 $...
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3answers
599 views

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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3answers
447 views

Singularity in Newton's gravitational law [duplicate]

If $r=0$ in the well know equation $F= G\dfrac{m_1\cdot m_2}{r^2}$, it will not follow that the force will be infinite? May someone please clarify it to me?
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2answers
4k views

Why is the elemental volume of a sphere equal to $4 \pi r^2dr$?

I was doing this question on calculating the electric field at a certain point in a sphere (length $r$ away from the centre), where the charge density is given by an equation. When I checked the ...
2
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1answer
411 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
114 views

What are point objects?

I can't seem to get the idea of point mass into my head. Why are equations of physics applicable on only point masses and should be altered while dealing with object that has a collection of points? ...
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2answers
2k views

Why can we assume, when dealing with tension, that the mass of a rope is $0$ but still assume that there are forces on it?

I started studying tension and I can't understand the following concept: If I have two objects $A,B$ collegated by a rope, where $A$ applies a certain force $\vec{F}$ on it, I would have that the ...
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2answers
228 views

Derivation of velocities in the Coriolis force

In Fitzpatrick's Newtonian Dynamics book on the Coriolis force, he states \begin{align} v_{x'}&\simeq V_0\cos\theta-2\Omega t V_0\sin\lambda~\sin\theta \tag{433}\\ v_{y'}&\simeq-V_0\sin\...
2
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1answer
146 views

Are Quantum Physics and statistical theory always the same as semiclassical approximations?

Quantum Mechanics and Statistical physics is a bit hard , could we then study only the WKB approximation ? In the form: replace $ \sum_{n=0}^{\infty}exp(- \beta E_{n})=Z(\beta)\sim\iint dxdpexp(-p^{...
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2answers
73 views

Why strong electric field leads to non-Ohmic behavior?

Homogenous conductors like silver or semiconductors like pure germanium or germanium containing impurities obey ohm's law within some range of electric field values. but if the field becomes too ...
2
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5answers
149 views

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 ...
2
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1answer
121 views

Schwarzschild Black Holes

Page $190$ of “Tensors, Relativity, and Cosmology”: Consider a particle falling radially into a black hole with a radial velocity $u^1=dr/ds$. As the particle is falling radially, we have $u^2=u^3=...
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1answer
86 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 ...
2
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1answer
105 views

Kepler's Second Law: Why do we calculate the area of a triangle rather than the area of a sector?

Kepler's Second Law states that equal areas are swept in equal times. When calculating this area, why do we use the formula for the area of a triangle rather than the area of a sector?
2
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1answer
196 views

Why do materials obey Hooke's law? [duplicate]

Why do materials extend proportionally to the force exerted on them (Hooke's law)? I thought that when materials are compressed or extended under force, their atoms become closer or further apart; ...
2
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2answers
1k views

Tidal force formula

My book explains what happens if we measure the difference in gravitational force by displacing a test mass by $\Delta r\ll r$. They give the following formula: $$ \Delta F=\frac{2GMm\Delta r}{r^3}. $$...
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2answers
152 views

Calculating size of opening of a Decanter in order to project 5 gallons of water per second 20 feet in the air vertically?

This question is attempting to determine the size of the opening of a Decanter of Endless Water for our gaming group. We are wanting to hook it up to some impromptu Jerry-rigged technology as either a ...
2
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2answers
92 views

In geometrical optics, how can we say that rays coming from a distant object are parallel to one another?

If two rays are not parallel in the start, how can they become parallel at the instant when they strike the lens of a telescope? If they don't become parallel, why do we consider them to be, in the ...
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2answers
5k views

Why Earth is considered to be an inertial frame? [duplicate]

Earth rotates about its axis and also revolves around the Sun at the same time. So why Earth is considered as an inertial frame in Newtonian Physics. So technically, I'm effectively asking why the ...
2
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1answer
391 views

Finding the approximate solution for Schrodinger equation by using variational method [closed]

I need to find the approximate solution of nonlinear Schrodinger equation $$ i\hbar \partial_{t} \Psi + \frac{\hbar^{2}}{2m}\Delta \Psi - g |\Psi|^{2}\Psi - \frac{m\omega^2 (x^2 + y^2 + z^2)}{2}\Psi = ...
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1answer
62 views

Theoretical justification for the range of validity of the approximation $R(T)\approx R(T_0)[1+\alpha (T-T_0)]$

In the experiment for calibrating a platinum resistance thermometer, we are always approximating the resistance of the platinum thermometer by $$R(T) \approx R_0 (1+\alpha T),$$ taking the reference ...
2
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1answer
115 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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1answer
58 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 ...
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2answers
86 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 ...