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2
votes
3answers
19 views

Why do we assume differential coefficients of number of molecules?

In many portions of physics (like Maxwell's velocity distribution law) we assume statements like- Number of molecules having velocity between $c$ to $c+dc$ is $dn$. But number of molecules $n$ ...
-2
votes
1answer
55 views

Why are these two angles the same? [closed]

I can't find no similar or other relations between the two triangles, the two right triangles. However, if I look at it this way, the two triangles are indeed similar. But then this is a whole ...
1
vote
0answers
36 views

Approximate Electric Potential $V$ so that it is of the form $V(r) + V(\phi) + V(z)$

I'm trying to simulate the conductivity of a nanowire that is modeled by a cylindrical shell of infinite potential with benzene rings in the core of the wire. (This is based on a coiled-coil protein ...
1
vote
4answers
750 views

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 ...
0
votes
1answer
50 views

Explain how $\Delta v_{\perp}=v\Delta\theta$

In The Feynman lectures, under the chapter entitled Vectors, Feynman writes: My two intimately related questions are: 1) What does he mean by the magnitude of velocity? is he talking about the ...
2
votes
2answers
108 views

Why does GW-DFT give higher bandgaps in semiconductors

Usually the GW Density Functional Theory (DFT) gives larger band gaps in semiconductors compared to the LDA and GGA methods. This seems to be related to the screened potential in GW, but it is not ...
0
votes
0answers
18 views

Off-axial Field of Finite Solenoid

Regarding the computation of the off-axial field of a finite solenoid: The Radial and Z components of the off-axial magnetic field of a solenoid are given as: \begin{align} B_r &= \frac{\mu ...
1
vote
1answer
80 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 ...
1
vote
1answer
61 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 ...
0
votes
1answer
53 views

What is the typical strength of the electric field in a particle accelerator?

I am working on a research project involving the closed orbits of hydrogen in the presence of an external electric field and I am curious what a reasonable approximation for the electric field ...
4
votes
2answers
156 views

Is Torricelli's law “wrong” for big holes? - Tank draining problem

Consider a tank full of water with a constant cross-sectional area A1 placed vertically on the ground. Now someone drills a hole of an area A2 in the bottom of the tank, and the liquid starts escaping ...
3
votes
1answer
283 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 ...
0
votes
1answer
64 views

Why do some approximations give exact results?

The moment of inertia of a sphere of mass $M$ and radius $R$ can be calculated exactly (meaning, with certainty) using integrals. The formula we get is $\frac{2}{3}MR^2$. However, there's an other ...
1
vote
0answers
36 views

WKB boundary conditions [duplicate]

Technically, this is a math problem but I think it is better here than in the math stacks. Consider the differential equation \begin{equation} y'' = (x^4 - E)y \end{equation} The boundary ...
0
votes
2answers
107 views

How is the uniform gravitational field approximation $F_g\approx mg$ near Earth's surface derived from Newton's law $F_g=GMm/r^2$ of gravitation?

I am really bothered about how we can derive the equation of projectile motion. Suppose a point mass will move in the gravitational field of the Earth according to the equation $$\ddot R ...
1
vote
0answers
49 views

I have a problem with the variational method approximation in quantum mechanics. Is my issue valid, or am I misunderstanding something?

The variational method for approximating the ground state of a Hamiltonian $H$ by providing a lower bound is simple enough. If we construct any test wave function $|\bar{0}\rangle$ then the claim is ...
2
votes
1answer
67 views

Lagrangian for small oscillations

For a double pendulum we can consider 2 generalised coordinates $\theta_1$ (angle between first mass and vertical axis) and $\theta_2$ (angle between second mass and vertical axis). The Lagrangian to ...
1
vote
0answers
53 views

How fast do I have to dry myself for a hot shower to heat my body?

I am not a physicist. I would like to know how fast do I have to dry myself after taking a hot shower to get more heat from the shower, than lost because the water on my skin increasing heat exchange ...
0
votes
0answers
32 views

Weizsäcker–Williams approximation

I'm having some trouble understanding the Weizsäcker-Williams approximation What I think it is is the following: I have a charged particle at high speed, close to the speed of light, at this speed ...
1
vote
1answer
51 views

Born approximation and dipole approximation

I'm having some trouble really understanding when it's okay to use these approximations and why. I've been looking myself blind on equations, but I'm not even sure I understand it qualitatively. So I ...
1
vote
1answer
307 views

Hartree-Fock: Coulomb integral [closed]

Today I was wondering how to better understand the Coulomb integral in the Hartree-Fock approximation. Extracted from: Szabo & Ostlund, Modern Quantum Chemistry, p. 112 The Coulomb term has ...
0
votes
1answer
52 views

Where do we get the terms involving $\Phi$ in parentheses come from in the static weak field metric?

I am confused about the static weak field metric. As written in Hartle, it reads \begin{equation} ds^2 =-\left(1+\frac{2\Phi(x^i)}{c^2}\right)(cdt)^2 +\left(1-\frac{2\Phi(x^i)}{c^2}\right)(dx^2+dy^2 ...
1
vote
1answer
80 views

Energy levels in close-proximity of each other in time-independent degenerate perturbation theory

I've applied second order time-independent degenerate perturbation theory corrections to the energy with the method presented in Modern Quantum Mechanics by J.J. Sakurai. I shortly summaries this ...
1
vote
4answers
101 views

Approximations of the kind $x \ll y$ [closed]

I have an expression for a force due to charged particle given as $$F=\frac{kQq}{2L}\left(\frac{1}{\sqrt{R^2+(H+L)^2}}-\frac{1}{\sqrt{R^2+(H-L)^2}}\right) \tag{1}$$ where $R$, $L$ and $H$ are distance ...
2
votes
1answer
109 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 ...
1
vote
0answers
59 views

WKB formula and Langer correction [duplicate]

The general WKB approximation formula states that $$ \int_a^b\sqrt{E_k-V(x)} = (k+1)\pi \text{ with } x \in [a,b] $$ for a regular Schrödinger equation (without the $\hbar$ and such). However, in the ...
2
votes
2answers
83 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 ...
1
vote
1answer
69 views

Magnetization $\ M$ of a ferromagnet as a function of temperature $T$, nearby $T=0$

Using mean-field theory, the magnetization per spin, $M$, for a ferromagnet always obeys the equation: $M=\frac{g \mu_{\mathrm{B}}}{2}\mathrm{tanh} \left( \frac{2}{g \mu_{\mathrm{B}}} ...
0
votes
1answer
46 views

What do we get from the diagonalization of the $k\cdot p$ matrix?

In k.p theory, we expand the wave function around a known point ${\bf k}_0$ $$u_{\lambda}({\bf k})=\sum_{\nu} c_{\lambda,\nu}({\bf k})u_{\nu}({\bf k}_0).$$ If we now consider 8 bands (conduction, ...
1
vote
1answer
52 views

Post-Newtonian approximation for binary gravitating system

I have been studying gravitation waves radiated by a binary source. I have linearised Einstein's field equation and approximated the source to a Quadrupole moment to get the power radiated by the ...
12
votes
6answers
2k views

If the solar system is a non-inertial frame, why can Newton's Laws predict motion?

Since there is no object in the universe that doesn't move, and the solar system likely accelerates through space, how did Newton's Laws work so well? Didn't he assume that the sun is the ...
2
votes
0answers
45 views

How can the fictitious mass in the Car-Parrinello method reproduce the “real” dynamics?

In the Car-Parrinello method, to solve simultaneously the classical equations of motion for the atoms and the Kohn-Sham equations for the electrons, the following effective Lagrangian is used: $$ ...
7
votes
2answers
795 views

What will be the equation of motion of driven pendulum for amplitudes beyond the small angle approximation?

When finding the period of a pendulum beyond the small angle approximation, we have to use integration for small interval of $\theta$ and elliptical integration. I was trying to apply this situation ...
3
votes
3answers
614 views

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 ...
0
votes
1answer
41 views

Time Dependent Perturbation Theory Probabilities

(This is taken from Griffiths Quantum Mechanics): So suppose I have two states $\psi_{a}$ and $\psi_{b}$, and the particle starts out in the state $\psi_{a}$: $$ c_{a}(0)=1\qquad c_{b}(0)=0. $$ To ...
1
vote
0answers
52 views

Applying Statistical Mechanics to Formulate Corrosion (Rusting)

I wanted to try and take my current knowledge of statistical mechanics (first quarter undergraduate course completed, beginning researcher in far from equilibrium statistical mechanics, basic ...
3
votes
3answers
828 views

Why do we use the Coulomb potential for the hydrogen atom?

When solving the Schrodinger equation for the hydrogen atom, the Coulomb potential $V = \frac{e^2}{4 \pi \epsilon_0 r}$ is used. The Coulomb potential comes from classical electrodynamics, so why ...
2
votes
0answers
72 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 ...
2
votes
2answers
102 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}\\ ...
0
votes
0answers
31 views

Atmospheric refraction approximation

I am studying atmospheric refraction, reading ITU P.834 Effects of tropospheric refraction on radiowave propagation, and I have a question about an approximation. They say that refraction correction, ...
3
votes
1answer
447 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) = ...
1
vote
0answers
56 views

Does Planck's constant imply limits to computing *results*

... I don't mean quantum effects limiting hardware fabrication sizing. Such small scales have for some time exhibited issues. Rather, along the lines of imagining the smallest possible divisions of ...
3
votes
2answers
159 views

Mathematical approximation to physics

Why is it often said that any mathematical theory is just an approximate theory of the universe? Wouldn't there be accurate mathematical structures repressing the physical entities of the universe ...
1
vote
1answer
63 views

Electron electric field

As we know the fundamental unit of charge in our universe at the time of electrodynamics was an electron, and in any frame of reference, its radius is a finite number and assuming uniform charge ...
1
vote
1answer
110 views

How to reconcile these two approximations?

I'm working on what should be a simple problem (Taylor Classical Mechanics problem 6.23.), but I'm having a tough time reconciling the many ways it can be tackled. An aircraft whose speed is $v_0$ ...
3
votes
3answers
119 views

Solving differential equations without approximations?

In physics, many problems start with a mathematical relationship of the physical phenomenon at hand, and then, in many occasion, always only leave whatever in the first order to get a nice and ...
2
votes
1answer
105 views

Why do we consider the electric field of an infinite plane? [closed]

I never understood why one would calculate the electric field surrounding an infinite plane, if such thing does not exist. Is there physical motivation for using this model? Are the results applicable ...
2
votes
3answers
168 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?
3
votes
0answers
207 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 ...
2
votes
1answer
195 views

Proof of the conservation of the energy functional for the Gross-Pitaevskii equation?

From the Gross-Pitaevskii equation \begin{equation}i\hbar\frac{\partial\psi}{\partial t}=\left(-\frac{\hbar^2}{2m}\nabla^2+V+g|\psi|^2\right)\psi\end{equation} using the variational relation ...