Infinite Definition and 1000 Threads

I can prove the twin prime counting function has this form: \pi_2(n)=f(n)+\pi(n)+\pi(n+2)-n-1, where \pi_2(n) is the twin prime counting function, f(n) is the number of twin composites less than or equal to n and \pi(n) is the prime counting function. At n=p_n, this becomes \pi_2(p_n) =...

I can prove the twin prime counting function has this form: \pi_2(n)=f(n)+\pi(n)+\pi(n+2)-n-1, where \pi_2(n) is the twin prime counting function, f(n) is the number of twin composites less than or equal to n and \pi(n) is the prime counting function. At n=p_n, this becomes \pi_2(p_n) =...

Hi, Does anyone know why k has to be real in an infinite system for bloch's theorem. I understand that the wavefunction becomes unphysical in an infinite system as it diverges. Why does that mean k has to be real? f(x)=u(x)exp(ikx)

Homework Statement An electron is bound in a square well of depthU0=6E1−IDW. What is the width of the well if its ground-state energy is 2.50 eV ? Homework Equations En = h2n2/8mL2 The Attempt at a Solution I used n = 1 so I get: 25eV*1.6*10-19 = h2/8*9.11*10-31*L2 I got L = .388 nm. It...

Let x+ and x- be the light cone coordinates x + t and x - t. The coordinate x- = log x+. The equations dx- = dx-/x+ and x+ = x+ dx- are like ds = dQ/T and dE = p dV. d- = d- dx+ D- = d- + d- dx+ In mechanics we have L = T - V, where L = L dt. This is a complex differential form just like x+...

Suppose we confine a spin 1/2 particle to an infinite annular region, in cylindrical coordinates, defined by the two cylinders r=a and r=b with a<b. How does such a region constrain possible spin and angular momentum? Thanks!

Homework Statement Homework Equations Provided in the questions I believe. Here's the triangle from question two. The Attempt at a Solution QUESTION SET 1 TOP OF PICTURE A.) I didn't know how to just "guess" what the constant should be so I actually worked it out. I found the constant...

Homework Statement Show that ##\sum_{n=1}^{\infty}\frac{1}{n^{4}}=\frac{\pi^{4}}{90}##. Homework Equations The Attempt at a Solution ##\frac{1}{n^{4}} = \frac{1}{1^{4}} + \frac{1}{2^{4}} + \frac{1}{3^{4}} + \dots##. Do I now factorise?

Homework Statement Consider a one-dimensional, non-relativistic particle of mass ##m## which can move in the three regions defined by points ##A##, ##B##, ##C##, and ##D##. The potential from ##A## to ##B## is zero; the potential from ##B## to ##C## is ##\frac{10}{m}\bigg(\frac{h}{\Delta...

Homework Statement The figure shows cross-sections through two large, parallel, nonconducting sheets with identical distributions of positive charge with surface charge density σ = 1.06 × 10-22 C/m2. What is the y component of the electric field at points (a) above the sheets, (b) between them...

I have read in the book 'A Brief History Of Time' that other bodies can't reach the speed of light because as its speed gets near the speed of light, it gets infinite mass and requires infinite energy to reach the speed of light. So light itself should be having infinite mass and would be...

I'm not sure where to put this question. It is by itself pretty basic, but it's a preamble to a Laplace Transform exercise, and I'll probably want to ask some follow up questions once the current query is resolved. 1. Homework Statement Unit stair-case function: f(t) = n, \ if \ \ n-1 \leq t...

Homework Statement "A dollar due to be paid to you at the end of n months, with the same interest rate as in Problem 13, is worth only (1.005)^{-n} dollars now (because that is what will amount to $1 after n months). How much must you deposit now in order to be able to withdraw $10 a month...

At the time of Big Bang the size of Universe equal the size of an atom.The Universe has expanded and the time from the Big Bang to the present is finite.Then at the present time the size of the Universe is finite or infinite?

Originally from the statistics forum but am told this is more of a calculus question. I flip 10 coins, if any of the coins land on tails, all of the coins split into 10 new coins and I flip them all again. I keep doing this until a round where every single coin lands on heads. Can I expect to...

Hi! I'm need some help with this question: Decide $h$ so that the linear system $Ax=b$ has infinite solutions. $$A=\pmatrix{ 5 & 6 & 7 \cr -7 & -4 & 1 \cr -4 & 4 & 16 \cr}$$ $$b=\pmatrix{ 6 \cr 30 \cr h \cr}$$ I solved a similar question before but with A being a 2x2 matrix (and B a 2x1) and...

Homework Statement The assignment is to find all values of k (in R) for which the system has 0 solutions, 1 solution and infinite solutions. If there are infinite solutions, find the amount of free variables. The system of linear equations: kx + (k+1)y + z = 0 kx + y + (k+1)z = 0 2kx + y + z =...

It is my understanding that fields store potential energy. That applies to both magnetic as well as electric fields. I know that the energy density also increases with the square of the norm of their vector value (at each coordinate). When I have an infinite current sheet, the math says[1] that...

I'm reading Griffiths' section on the infinite square well defined as having zero potential between 0 and a on the x-axis and being infinite everywhere else, and am confused about the following part when discussing the general solution inside the well. The bolded part is what confuses me, the...

Homework Statement Is state ψ(x) an energy eigenstate of the infinite square well? ψ(x) = aφ1(x) + bφ2(x) + cφ3(x) a,b, and c are constants Homework Equations Not sure... See attempt at solution. The Attempt at a Solution I have no idea how to solve, and my book does not address this type...

I just noticed in reading Griffiths that he places the base of the infinite square well at a zero potential while he places the base of the finite square well at a negative potential -V_0, where V_0 is a positive, real number; is there any reason for this? I just started learning about them/am...

Hello, I am taking some microwave engineering courses and was trying to explain the concept of reflection coefficients to my friend, but he asked me a question I am unable to answer... So we know that given a transmission line with characteristic impedance Z_0 terminated with a load impedance...

For time independent Schrodinger's equation in 3-D Where Enx,ny,nz=(nx/Lx2+ny/Ly2+nz/Lz2)(π2ħ2/2m and Ψnx,ny,nz=Asin(nxπx/Lx)sin(nyπy/Ly)sin(nzπz/Lz) How do I normalize A to get (2/L)^3/2? I don't think I understand how to normalize constants.

Homework Statement Let {b k } be a sequence of positive numbers. Assume that there exists a sequence {a k}, such that a k is greater than or equal to 0 for all k, a_k is decreasing, the limit of a_k is 0 and b_k = a_k - a _(k+1). Show that the sum from k=1 to infinity of b k exists and equals...

Write code to complete DoublePennies()'s base case. Sample output for below program: Number of pennies after 10 days: 1024 The if statement is what I am trying to complete, however this places it in an infinite loop #include <stdio.h> // Returns number of pennies if pennies are doubled...

I am trying to understand electric fields of conductors. Say that there is an infinite sheet of uniform positive charge; parallel to it lies a infinite, uncharged conducting sheet. What would the field look like between the sheets? Beyond the sheets? I would guess that the uniformly charged...

Hi, In my QFT course, the professor writes an infinite product like this: ∏n | k n0 > 0 ∫... My question is, what does the `|' in the subscript "n | k" representing? When I see `|', I think logical OR - obviously that is not it. Normally, if it's a sum over two indices, commas separate the...

Somewhere I saw that the sum of the infinite arithmetic series $$\sum_{n=1}^{\infty}n = \frac{-1}{12}$$ Why exactly is this? I thought infinite arithmetic series had no solution? Also... WHY is it negative? Seems counter-intuitive that the sum of all the NATURAL numbers is a decimal, a...

domainwhale submitted a new PF Insights post High Temperature Low Temperature Duality for the Ising Model on an Infinite Regular Tree Continue reading the Original PF Insights Post.

hi, i still don't understand why infinite thin-walled cylindrical shell or conducting rod use lambda rather than sigma ? lambda = C/m ,,, sigma = C/m^2 i mean when we look at conducting rod, the charges inside the conductor is zero, so the charges spread on the surface of conducting rod(have...

I have been wondering this for a while. I know that tires using static friction due to reasons that I forget and therefor maintain good traction with asphalt. A few years ago family member said that if there was too much friction on a road then a car wouldn't move, but I argued that the car...

Homework Statement Two infinitely long lines of uniform charge λ lay parallel on the xy plane (0, ±a) What is max E field in the xz plane. No values are given. Symbolic answer is expected. Homework Equations equation for an infinite line of charge E = λ / ( 2 π ε0 r) The Attempt at a...

I've been reading up a bit on semiconductor quantum wells, and came across a selection rule for an infinite quantum well that says that "Δn = n' - n = 0", where n' is the quantum well index of an excited electron state in the conduction band, and n is the index of the valence band state where...

Homework Statement I think this is a square well potential problem. The question asks me to sketch the ground-state probability density, for the following situation: A quasielectron moves in a 'quantum dot' device. The potential V(x) = 0 for 0 ≤ x < L, and is infinite otherwise. Homework...

Hi everyone, I need help for preparing a Hamiltonian matrix. What will be the elements of the hamiltonian matrix of the following Schrodinger equation (for two electrons in a 1D infinite well): -\frac{ħ^{2}}{2m}(\frac{d^{2}ψ(x_1,x_2)}{dx_1^{2}}+\frac{d^{2}ψ(x_1,x_2)}{dx_2^{2}}) +...

In my book, applied analysis by john hunter it gives me a strange way of stating an infinite sum that I'm still trying to understand because in my calculus books it was never described this way. It says: We can use the definition of the convergence of a sequence to define the sum of an...

When I pick a random number on a number line made out of integers, starting from zero and expanding infinite to the right, what can I say about the position of this random number ? To the right the amount of numbers is infinite. To the left is an amount, a number, so that is finite, but it has...

Homework Statement Say x is an infinitesimal number on the hyperreal line, is this expression finite, infinite or infinitesimal Homework Equations (sqrt(4+x)-2)/x The Attempt at a Solution [/B] My approach so far has been that sqrt(4+x) is (2+y) where y is another infinitesimal and y<x...

I've been reading a bit about the very intriguing summation \displaystyle \sum_{n=0}^{\infty} {n} and it seems \frac{-1}{12} is the result but apparently with a lot of subtleties and caveats. It is those that I am trying to understand now. At first reading it appeared totally incongruous to...

Homework Statement Wave equation: ytt=yxx Initial conditions: Y(x,0) =f(x) = x (0 ≤ x < 1) 2.5(5-x) (1 ≤ x < 3) 0 (Otherwise) and yt(x,0) = 0 Boundary condition: y(0,t) =0 Semi infinite domain: 0 ≤ x < infinity Homework Equations d'Alembert solution...

I've read somewhere that hypothetical tachyons always travel faster than c, speed up when they lose energy, and slow down when they gain energy. And that it takes an infinite amount of energy to slow tachyons down to c. How did they derive all these concepts?

When I was taught Gauss's law. My teacher used a cylindrical Gaussian surface to find the electric field above an infinite uniformly charged plate. What I have trouble understanding is why the plate has to be infinite in order for the arguments to work...

Griffiths' Electrodynamics says that the electric field of a uniformly charged infinite plane, surface charge density sigma, is sigma/2e0. The field of an infinite sheet of charge is said to be sigma/e0, twice that of the plane. What is the supposed difference between the sheet and the plane...

Hello everyone. I am currently having trouble actually defining what qualifies as an infinite discontinuity. I have read several sources that state that both of the one sided limits must approach infinity (positive, negative or both). My problem is what happens when only one of the one sided...

In a lecture I heard that if we suspend two objects of different masses (and the system is accelerating) on both sides of a pulley of no resistance with a mass-less string then the tension on both sides of the string is same - this is fine till now. To explain that the tension is same, it was...

Homework Statement We have an infinite slab of conducting material, parallel to the xy plane, between z = −a and z = +a, with magnetic susceptibility χm. It carries a free current with volume current density J = J0z/a in the x direction (positive for z > 0, negative for z < 0). The integrated...

Hello, a dubt arose while doing some exercise. If I have a charge q at a distance d from the above-mentioned plane, i can find the solution to the laplace equations (thanks to the uniqueness theorems) finding a collection of image charges that satisfies the boundary conditions. These conditions...

İn a finite set, can we take limit to $\infty$ ? Also, can you give an example related to infinite subset of $\Bbb{N}$ ?

The quantum states ##\psi(x)## of the infinite square well of width ##a## are given by ##\psi(x) = \sqrt{\frac{2}{a}}\sin\Big(\frac{n \pi x}{a}\Big),\ n= 1,2,3, \dots## Now, I understand ##n \neq 0##, as otherwise ##\psi(x)## is non-normalisable. But, can't we get additional states for...

Homework Statement hello this question is discussed in 2009 but it is closed now If you invest £1000 on the first day of each year, and interest is paid at 5% on your balance at the end of each year, how much money do you have after 25 years? Homework Equations ## S_N=\sum_{n=0}^{N-1} Ar^n##...