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13

In the simplest form the saddle point method is used to approximate integrals of the form $I \equiv \int_{-\infty}^{\infty} dx\,e^{-f(x)}$ The idea is that the negative exponential function is so rapidly decreasing -- $e^{-10}$ is $10000$ times smaller $e^{-1}$ -- that we only need to look at the contribution from where $f(x)$ is at its minimum. Lets say ...


8

In the standard model (as in all traditional relativistic quantum field theory), particles are pointlike. All experimentally available facts about microphysics seem to be consistent with the standard model. This is the (fully sufficient) reason for believing that particles in Nature are pointlike. Pointlike is a technical term that refers to the fact that ...


8

You cannot use the second kinematical equation because it is valid only when the acceleration due to gravity, $g$ , is constant. This is incorrect for distances comparable to the radius of the earth, and velocities comparable to the escape velocity. The first correctly assumes a $\frac{1}{R^2}$ fall-off of the gravitational attraction on the body due to ...


7

No. There is nothing wrong with perturbation theory, or with theories with known, restricted accuracy. The point of theory is to explain the results of observation from as simple an initial theoretical standpoint as possible. Therefore: Since experiment always has a finite uncertainty, one can only ask that theory match the experimental value within its ...


5

In real life, the current can't jump instantaneously because there is always some finite inductance in a circuit. However, this is just a typical idealized textbook problem where the inductance is assumed identically zero, so the current can jump instantaneously according to the assumptions of the problem. Note the current also jumps in their solution for ...


5

BebopButUnsteady has explained the mathematics behind it and I'll provide you with some references I've found useful and quite like that get into the more technical mathematical details although they are still very readable. These deal more concretely with the complex analysis required and how to properly pick the correct contour so you don't get divergences ...


4

The modeling used in the GPS is based the static weak field metric $$ds^2 = -(1+2\Phi)dt^2 + (1-2\Phi)dS^2,$$ where $dS^2$ is the Euclidean metric and $\Phi$ is the gravitational potential of the Earth, though only the monopole and quadrupole terms are used. The proper time of a clock is therefore determined by $$d\tau = ...


4

Not exactly, since the total wave function is not factored as $\psi_j(q)\phi(R)$ (where $q$ are the electronic coordinates and $R$ the nuclear coordinates) but as $\psi_j(q;R)\phi(R)$. This means that whenever $R$ changes the electronic wave function instantaneously adapts to remain in the same quantum state. That necessarily implies that the electronic ...


4

There are many physical intuitions often presented in various texts on fluid dynamics. I won't mention those here. I will, however, mention that mathematically the passage from a particle point of view to a continuum point of view is still a largely un-resolved problem. (With suitable interpretation, this problem was already posed by Hilbert as his 6th of 23 ...


4

Yes, the continuity of the electric charge is an approximation that is valid whenever the relevant charges are much greater than the elementary charge (of the electron, or the proton). When we deal with numbers like $1,234,567\,e$, it doesn't really matter that it should have been $1,234,567.8\, e$: very large numbers may be approximated by a nearby integer ...


4

I) In this answer we discuss a systematic approach to linearization and stability analysis. Imagine that the physical system under consideration is described by an autonomous Lagrangian $L=L(q,\dot{q})$ of $n$ generalized coordinates $$\tag{1} q~=~(q^1, \ldots, q^n)~\in~ \mathbb{R}^n.$$ One of the first questions one would like to ask is, if a specific ...


4

A small portion of any smooth curve looks the same as a small piece of a parabola in the limit. Choose a coordinate system so that the tangential direction in the middle of the segment is along the $x$ axis and choose a translation for the middle of the segment to sit at $(0,0,0)$, the origin of the coordinates. Then $y,z$ on the curve (ellipse etc.) may be ...


3

Elementary particles don't really have a shape or a size, these are emergent qualities that stem from interactions between particles. In quantum physics a particle is represented by its quantum state, and if you want to describe that in space you get a wave function which tells us how much of the particle is present at any given point in space. Because there ...


3

The theory of fluids introduces material parameters in the stress tensor, which help model the substance. "The viscosity coefficient is the proportionality constant relating a velocity gradient in a fluid to the force required to maintain that gradient. The thermal conductivity is the proportionality constant relating the temperature gradient across a fluid ...


3

Whenever two quantum systems are united, the Hilbert space of the wavefunctions of the resulting system is always the tensor product of the Hilbert spaces of the wavefunctions of the source systems. This does not mean that the total wavefunction is always a product of two wavefunctions from the source spaces. In general case it looks like this: $$ \Psi = ...


3

first, I don't know exactly whta "sequence of infinitesimal Lorentz boosts" you're referring to. A boost of the flat Minkowski space gives you a Minkowski space back, so you can't get a curved space by any sequence of boosts that act globally on the spacetime. Also, the adjective "infinitesimal" could be pretty much inconsequential. Transformations in ...


3

Strictly speaking, light rays from distant stars are not perfectly parallel, but the typical angle (in radians) between them can be estimated as the diameter of the star divided by the distance from the star to the Earth; this value may equal $10^{-7}$ or less, so the rays are parallel to high accuracy.


3

"Newton's Laws" are - like most physics - a mathematical model that describes how the world - or the universe - works. All models are wrong, in that they don't describe the complete complexity of the physical world, but some models are useful, in that they let us make predictions. Newton's Laws, as a model, work well, unless you are dealing with things ...


3

Of course Newton's three laws of motion are correct, because they were verified several hundred of years ago and they continue working today, for such systems. Science is accumulative. What modern physics has done is to constraint the range of validity of those laws. Although some 18th century physicists believed that the laws were valid elsewhere, we know ...


3

1) The minus sign shows the fact, that the interaction between electrons and protons is attractive, not repulsive. 2) The coefficient $\frac{1}{2}$ is used in the cases, where you count the interaction between the same particles not to sum the same potential twice. I think you have slight mistake in the range of summation, let me try to explain your problem ...


3

Logarithmic series are a very broad topic. Generally speaking, many quantities in QCD can be expressed as power series of the form $$X(s) = \underbrace{\sum_n X_{0n}(\alpha_s\ln s)^n}_\text{LL terms} + \underbrace{\sum_n X_{1n}\alpha_s(\alpha_s\ln s)^n}_\text{NLL terms} + \cdots$$ The kinds of contributions that enter each set of terms depend entirely on ...


3

I think the issue here is that you need to keep a consistent level of approximation in your "small angle approximation." By small angles, we typically mean $\theta_1$ and $\theta_2$ are both of order $\epsilon$, where $\epsilon \ll 1$. Then the question is - to what order in $\epsilon$ do you want to write down the equations of motion? When you neglect ...


3

Mathematics is just a systematic way of stating facts about the world. It is only useful inasmuch as it is internally self-consistent. The latter fact means that there is nothing to "assume" about mathematics. It is a relationship between axioms and conclusions that enables one to succinctly summarize many observations. Something like Galileo's ...


3

Minimizing the energy with respect to $N$ will give a relationship between $\mu$, $N$, and the other parameters of the system. This will fix the chemical potential to something like $\mu=gN$ (this is the correct value in the limit $\omega\to0$). For sure the trial wave-function has a problem since $x^2/\omega^2$ is not dimensionless, but it seems quite ...


3

As it turns out, you can use it for the vast majority of the time. The ideal gas law is considered "ideal" because it assumes interactions between the gas molecules only occur due to collisions and no long-range forces are present. If the fluid you are modelling is non-polar and doesn't have things like van der Waals forces between molecules, then the ideal ...


3

It doesn't sound exactly like a chord, but this type of technique was widely used in the 8-bit and 16-bit computer game eras, when the number of sound channels available was limited. It has a very distinctive retro video game sound, but the ear is able to identify the chord that it's supposed to be. Here is a YouTube video explaining how to achieve the ...


2

In this case "perfectly parallel" means "more closely parallel than can be detected". How parallel is that? Consider the geometry. The biggest angular deviation possible is between a photon originating from the left side of the star impinging on the right side of the detector and vice versa. By the small angle approximation we can write this difference as ...



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