My math book claims that the piecewise function f : [0,1] U (2,3] --> R defined by
f(x)=
x for 0<=x<=1
x-1 for 2<x<=3
is continuous. But it's undefined for 1<x<=2 so how can it be continuous? According to the definition of continuity, a function is at a point x0 if for a sequence x_n...
Homework Statement
Prove that the sequence {c_n} converges to c if and only if the sequence {c_n - c} converges to zero.
Homework Equations
The Attempt at a Solution
First I prove that the convergence of {c_n - c} to zero implies that {c_n} converges to c by (all limits take n to...
Hey!
I need to perform the following integration:
\int\limits_0^{\infty} J_1(k r)k dk
where J_1(x) is the cylindrical Bessel function of the first kind. This is an oscillatory function with amplitude decreasing as 1/\sqrt{x}. Due to the multiplication of k the integrand however, becomes...
Hey. Is the conductivity of a material related to the dielectric constant (permittivity) in some way? And what about the electrical susceptibility?
Thanks
Thanks for the answers.
I think the definition the book gives of equality between sets, is that for two sets A and B to be equal, A must be a subset of B and B must be a subset of A.
Hey
Im trying to study abstract algebra, set theory and group theory, on my own. I have trouble understanding how to construct mathematical proofs though. Some of the things the excercises tells me to prove, seems so intuitively clear and obvious that I don't know what's left to prove. For...
In a semiconductor the law of mass action states that the product of the electron concentration and the hole concentration is always equal to the square of the intrinsic carrier concentration (at a given temperature), i.e.:
n p = n_i^2
My book states that this law is valid for...
That makes sense to me... Thank you. :) And yes, I am talking about energy levels for multielectron atoms in the Russel-Saunder coupling scheme.
But, what I am basically trying to do is to calculate the energy of excited states of some atoms without the effect of spin-orbit. Now, let's say I...
Thanks for the reply! But, in an f-shell with 3 electrons for instance, even if we assume that the 3 electrons all have spin up we still get several atomic terms. How can that be, what degrees of freedom are left when the configuration is locked and the spins of the electron is also locked. The...
Can someone explain to me why different terms arising from a particular electron configuration have different energies? For example, for carbon in the configuration 1s^22s^22p^2 three terms arise, 3P 1D and 1S. These three terms have different energies but I don't understand why. The electron...
When calculating eigenstates in the hydrogen atom one finds plenty of eigenstates with angular dependence. The s orbitals are spherically symmetric, but an orbital like 2p is not, there is some angular dependence through the spherical harmonics. But why is there angular dependence? It is a...
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Could someone please help me find out how to express the product of two Legendre polynomials in terms of a sum of Legendre polynomials. I believe I have to use the recursion formula
(l+1)P_{l+1}(x)-(2l+1) x P_l(x) + l P_{l-1}(x)=0
but I am not sure how to do this. What is basically...
Light in a vacuum has momentum \hbar k but when the light enters a dielectric medium, fx. glass, the momentum of the light is increased to \hbar\sqrt{\epsilon} k, where \epsilon is the dielectric constant of the dielectric. Where does the extra momentum come from? How can this be in accordance...
Does the equation Sqrt(x) + 1 = 0 have a solution? I would say that it doesnt. But the equation x^2 + 1 = 0 doesn't have a solution either, unless you define the imaginary unit i as the solution to the equation.
So why don't one define some unit which is the solution to the equation Sqrt(x) +...
Hey...
Can someone recommend good software for creating 2D illustrations for articles and reports? I've been using SmartSketch for a while, but I don't really like it.
Thanks
What it the timescale of this radiation? I mean, if I jump into water at ~0 Celcius it will immediately feel extremely cold because heat is being rapidly removed from my body and transferred to the water. How will this feel in space? I am guessing that heat transfer by radiation is quite slow at...
What about buying a book on some of the subjects? For learning physics I would strongly recommend "Physics for Scientists and Engineers" by Serway and Jewett. This is a huge book that will get you started in many areas of physics including relativity, quantum mechanics, optics, electromagnetism...
What exactly is it that you don't understand? Are you having trouble understanding what exactly those matrix elements are and why they are called matrix elements or is it something else?
The direction of the torque vector is always perpendicular to the plane of rotation. Therefore it has basically only two directions, up or down. This direction tells you whether the rotation caused by the force is clockwise or counterclockwise. There is nothing more to the direction of the...
Is it possible to land a large plane on the water (in the ocean for example)? I am of course talking planes not meant for landing on the water. I am wondering whether the heavy weight of the water will cause such a fast braking that the plane would crash? Of course you could say that it depends...
When light propagates in a dielectric its wavevector is given by:
k = \frac{\omega n}{c}
where n is the refractive index. If light propagates in vacuum n=1 the momentum is \omega / c but if light propagates in a dielectric of for example n=1.33 the momentum is increased beyond the...
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When a photon is reflected off an interface doesn't it deliver some of it's momentum to the object that it hits due to radiation pressure? If so, shouldn't the reflected light have a longer wavelength (smaller wavenumber) than the incident light? Is this effect present, but just so small...
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I am a student from Denmark who would like to spend one year studying abroad. I study physics, primarily the fields of solid state physics, quantum mechanics and semiconductor physics. I will be begin studies in electronic and optoelectronic applications next year as well.
I would...
My question pretty much is the title of this post, but let me explain it a bit more. When in physics or mathematics, or some other discipline involving math, you run into an integral, a differential equation or some other expression, how do you know if it's solvable or not? I mean, how do I know...
What am I doing wrong here, I thought the order of integration and differentiation didn't matter in most cases:
\int_a^b \frac{d}{dx} f(x) dx = \frac{d}{dx} \int_a^b f(x) dx = \frac{d}{dx} (F(b) - F(a)) = 0
This is zero no matter what the expression of f(x) because F(a) and F(b) are...
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Could someone please explain to me what the exchange integral exactly represents? I understand the coulomb integral and the overlap integral, they have nice classical analogs. But there doesn't seem to exist a classical analog to exchange.
Thanks
Are you sure about that? I mean, the wavevector of a particle inside a barrier is complex so that we get exponentially decaying wavefunctions instead of oscillating ones, right? This means that the kinetic energy \hbar^2 k^2 / 2 m < 0 since k is complex.
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Can someone explain to me what the spin functions are? I understand that a spin up is described by a function which is often called \alpha and spin down is described by a similar function called \beta. But what are these functions? What do they look like and what parameters do they take...
There's not a general procedure for solving the 3D Schrödinger Equation (SE). I think you will get better responses if you describe for what kind of system you are trying to solve the 3D SE.
For systems with spherical symmetry (atoms, for example) you could rewrite the SE in spherical...
I don't really know where to post a question like this, so I hope this will be alright.
Does someone know of a site similar to www.unitednuclear.com, which accepts orders from non US countries?
To answer that question you would need to rewrite Div B into an integral using gauss' law. The integral you get is the magnetic flux through a closed surface, and this then implies that the magnetic flux through any closed surface is always equal to zero. In other words, there is always a...
The ASCII code for upper case letteres are 65 to 90 and lower case letters are 97 to 122, numbers are from 48 to 57. So you should search through the text string by using a loop and look for ASCII codes lying in these intervals and take action depending on which interval you are in.
But isn't the intensity given by \Psi \Psi^*? This gives an intensity equal to 1/r^2 for the wave described by a complex exponential function but an intensity equal to 1/r^2 cos^2(k r - w t) for the other one.
Okay, so if they both had time dependence -i \omega t so that
\Psi = \frac{1}{r} e^{i ( k r - \omega t)}
and
\Psi = \frac{1}{r} \cos( k r - \omega t)},
but they still don't have the same intensity, since the intensity of the second one is an oscillating function of r and t...
Hey!
Im quite confused about spherical waves. I mean, I understand that a spherical wave can be described by
\Psi = \frac{1}{r} e^{i r},
because the intensity of such a wave decreases as 1/r^2. The intensity of such a wave is given by I = 1/r^2 which makes sense to me. But a...
Wouldn't a discontinous wave function imply abrupt changes in probability density? How can that be physical? I mean, if there's a certain probability for a particle to be at location A there would be a discontinuity in the probability of being in a location A' immidiately adjacent to A.
I believe you are talking about the Doppler effect? This is not due to light traveling long distances, but due to the object emitting light moving away from you.
I believe that the Solid state physics book from the Manchester series might also be a great book. I have not read it myself, but I've read other books from the series and they have been great.
Like Neu said, the world of atoms and molecules is governed by quantum mechanics, and quantum mechanics is so much different from macroscopic quantities like basketballs. It is impossible to draw an analogy between the two.
In school atoms are sometimes visualized as planets (the nucleus)...
I believe that high frequency EM radiation penetrates much deeper into matter because it is much more energetic (E=hf). Wouldn't that explain why cell phones work better than FM radios? FM radios are operated at ~80-100 Mhz whereas cell phones are operated at ~1800 Mhz.