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3
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3answers
331 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
20 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
30 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
533 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
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0answers
51 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
69 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
23 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
296 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
47 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
91 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
55 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
85 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
101 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
70 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
138 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
119 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
118 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 ...
3
votes
1answer
72 views

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, ...
2
votes
2answers
137 views

Is gravitational potential energy proportional or inversely proportional to distance?

We know that if an object has been lifted a distance $h$ from the ground then it has a potential energy change: $$\Delta U = mgh $$ so $h$ is proportional to $\Delta U$. However, we have also the ...
9
votes
3answers
368 views

What does Feynman mean when he says that $F=ma$ is not exact?

Chapter 12-2 in Feynman Lectures Vol. 1 states: In fact the law, $F=ma$ is not exactly true; if it were a definition we should have to say that it is always true; but it is not ... First, ...
0
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0answers
13 views

Modelling of nuclear motions (Classification) after invoking the BO approximation

I know that after invoking the Born-Oppenheimer approximation, the nuclei will move on the adiabatic potential provided by the electronic energy (also called potential energy surface (PES)). Such ...
1
vote
2answers
67 views

Taylor series: Epsilon not differentiated? [closed]

Why isn't epsilon differentiated with respect to time? (see my question on the right)
0
votes
0answers
66 views

Point source approximation

I have a 0.05 mm radius sperical source of Photons, and a 10 mm X 10 mm detector aligned to be orthogonal to their distance vector. Distance is D $\approx$ mm. I want to know how good the point ...
1
vote
0answers
158 views

Adiabatic approximation and time-dependent problems

I am an undergraduate physics student. I have a question in approximation methods for time-dependent problems in quantum mechanics. I read the proof of the adiabatic theorem but I didn't understand ...
1
vote
0answers
63 views

Can anyone outline the theory of plane wave Born approximation for direct nuclear reactions in detail?

Can anyone outline the theory of plane wave Born approximation for direct nuclear reactions in detail? Also What are the modification introduced in the distorted wave Born approximation? I was ...
2
votes
0answers
91 views

Limits of integration for the radial wave function of the Hydrogen atom in the WKB approximation

I am working a problem where we have to find the energy eigenvalues for the radial wave function of the hydrogen atom for $\ell=0$ using the WKB approximation. I am sure that I set up the integral ...
1
vote
0answers
76 views

Why cannot we apply perturbation theory in Born-Oppenheimer approximation

In Weinberg's Lectures on Quantum Mechanics, he mentions Unfortunately, we cannot simply use first-order perturbation theory, with $T_{nuc}$ taken as the perturbation and the state vectors ...
1
vote
1answer
123 views

Field from non-conducting plate?

For a non-conducting sheet, the electric field is given by: $$E = \frac{\sigma}{2\epsilon_0}$$ where $\sigma$ is the surface charge density. This equation holds well for a finite ...
3
votes
0answers
204 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 ...
1
vote
1answer
292 views

The Holstein-Primakoff Representation (approximation)

I have a question regarding the Holstein-Primakoff representation. In the HP-representation we define the spin operators in terms of bosonic creation and annihilation operators. $$ S_j^+ = \sqrt{2S ...
2
votes
1answer
181 views

Index of Refraction in Metal: Approximating Complex Perturbation

If you consider waves in a metal, you can write the index of refraction for the metal as, $$ n^2 = 1 - \frac{\omega_p^2}{\omega^2} $$ I am interested in what will happen if the index is perturbed by ...
1
vote
1answer
189 views

Adiabatic approximation

The adiabatic approximation for solid state systems is rather radical. I was wondering in which cases it breaks down. As it is based on the idea of the nuclii being much heavier than the electrons I ...
6
votes
3answers
427 views

Finding the energy eigenvalues of Hydrogen using WKB approach

I need help to find the energy eigen values of Hydrogen atom using WKB approach. So far I know, the radial equation is given by $$\frac{1}{r^2} \frac{\partial }{\partial r} \left( r^2 \frac{\partial ...
3
votes
1answer
159 views

Can a very small portion of an ellipse be a parabola?

We consider that when a body is projected from any height from the earth surface with a speed lesser than the orbital speed ( tangentially to the earth surface at that point.) it follows an elliptical ...
3
votes
2answers
68 views

Can I alternate between notes really fast and have it sound like a chord?

The question basically amounts to whether I can construct the illusion of superposition with adjacent sine waves of varying frequency. Context I'm trying to play music on a Tesla Coil (like OneTesla ...
0
votes
1answer
207 views

When to use ideal gas law in fluid mechanics?

The ideal gas law (aka the equation of state) is given by $$ p/\rho_N = k_BT, $$ where $\rho_N$ is number density. When am I allowed to use this to describe a fluid?
0
votes
0answers
44 views

What is the wave function outside the barrier region?

I've been trying to learn how to apply WKB for several days now. I asked a similar question already about trying to find the wave function inside the barrier region. Now I would like to understand how ...
1
vote
1answer
354 views

How to use the WKB approximation to find wave functions?

I'm trying to learn how to apply WKB. I asked a similar question already, but that question was related to finding the energies. Here, I would like to understand how to find the wave functions using ...
1
vote
1answer
635 views

How to apply the WKB approximation in this case?

I'm trying to learn how to apply the WKB approximation. Given the following problem: An electron, say, in the nuclear potential $$U(r)=\begin{cases} & -U_{0} \;\;\;\;\;\;\text{ if } r < ...
1
vote
1answer
83 views

Why does a difference in approach to projectile motion yields different results?

A body is projected vertically upwards from the surface of the earth with a velocity equal to half the escape velocity. If $R$ is the radius of the earth, then find the maximum height attained by the ...
2
votes
1answer
197 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 = ...
-3
votes
2answers
174 views

The nature of theoretical models

Mathematics is exact. It is a beautiful language that allows us to express quantities that aren't possible to be represented physically. We build theoretical models of physical systems that work out ...
1
vote
2answers
92 views

Discrete approximation of charge density

Given the electric potential $\Phi(r)$ and the Poisson's equation: $$ \nabla^2 \Phi(r) = - 4\pi \rho(r)$$ Consider the 2-dimensional case and let's say that I want to discretize this using a square ...
4
votes
1answer
407 views

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?
3
votes
1answer
179 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 ...
1
vote
0answers
125 views

Good book on deriving approximate solutions from first principles? [closed]

I have always been excited by examples in which a few simple assumptions and first principles are used to characterize a system. For example, I did an exercise in which Crawford estimates a lake to ...
1
vote
0answers
70 views

Any quadrupole approximation? Any example?

In atomic and molecular physics we quite often encounter with electric dipole approximation. The dipole approximation we do when the wave-length of the type of electromagnetic radiation which induces, ...
-3
votes
1answer
91 views

How can we depend on the mathematical axioms that break down at the nano-level? [duplicate]

Mathematics "makes sense" at our scope of view. For example, gravity seems to obey the fact that it accelerates at a rate of 9.8 meters a second^2. However, when an object drops, the force of gravity ...
5
votes
2answers
656 views

Small oscillations of the double pendulum

From the Lagrangian I've got the following equations of motion for the double pendulum in 2D. (The masses are different but the lengths of the two pendula are equal.) Let $m_2$ be the lowest-hanging ...
3
votes
1answer
211 views

Does effective theory have the same meaning in particle and condensed matter physics

I have a naive question about the meaning of effective theory in particle physics and condensed matter physics. In particle physics, from what I know, the effective theory comes from the Wilsonian ...