Bound Definition and 501 Threads

I read this wiki and some of the references https://en.wikipedia.org/wiki/Bound_state But I can't really understand. For example the electron in hydrogen has specific energies and not general relations that the articles seem to claim. Thanks

Homework Statement i have a questions on the piece of lecture notes attached: 2. Homework Equations The Attempt at a Solution [/B] I agree 2) of proposition 2.12 holds once we have 1). I thought I understood the general idea of 1), however, my reasoning would hold for ##M_k## it does not...

Homework Statement A binary stellar system is made of one star with ##M_1=15{M}_\odot## and a second star with ##M_2=10{M}_\odot## revolving around circular orbits at a relative distance of ##d=0.001pc##. At some point ##M_1## explodes in a supernovae leaving a neutron star of mass...

I'm pretty frustrated with this exam question: Adrian is making some orange squash. He makes 13 litres of orange squash correct to the nearest litre. Each glass holds 250ml of orange squash correct to the nearest 10ml. Adrian has 48 glasses. Does he have enough orange squash to fill all 48...

Homework Statement A bound particle is in a superposition state: \psi(x)=a[\varphi_1(x)e^{-i\omega_1t}+\varphi_2(x)e^{-i\omega_2t}] Calculate <x> and show that the position oscillates. Homework Equations <x>=\int_{-\infty}^{\infty} \psi(x) x \psi^*(x) \mathrm{d}x The Attempt at a...

I am confused about the mass-energy equivalence relation as it applies to nuclei and nucleons. For nuclei, I read of a "mass defect." Naively, I supposed that since it is a collection of nucleons bound together, it has a negative binding energy and this is the reason for the term "mass defect."...

Hello Forum, Just checking my correct understanding of the following fundamental concepts: Stationary states: these are states represented by wavefunctions ##\Psi(x,y,z,t)## whose probability density function ##|\Psi(x,y,z,t)|^2 = |\Psi(x,y,z)|^2##, that is, the pdf is only a function of space...

It seems that the upper bound of the CHSH inequality is ##2\sqrt{2}## How is it analytically derived?

I'm working on an assignment where I'm required to numerically find the energy of a delta-potential's bound state. To do this, we've converted the time-independent schrödinger equation to an eigenvalue problem with E the eigen value, ψ the eigen vector and H a matrix as follows: with ##t =...

Homework Statement Trying to help a friend with a problem. We are supposed to solve the below using polar coordinates. The actual answer is supposed to be π/16. Solving the integral is not the issue, just converting it. 2. The attempt at a solution What I got sort of worked, but it is only...

Homework Statement Consider a cyclinder made of linear dielectric material with uniform dipole distribution. An externally applied field ##E_{ext}## is applied in the direction parallel to the axis of the cyclinder. What are the values of the bound chrages at the surface. Homework Equations...

Before I begin, I would like to say what I am about to ask would require some sort of top-top-bottom bound state for it to function. Which (to my knowledge) has not been experimentally or theoretically predicted. Also, in case if you are wondering- no, this is not a homework question. --- So...

Homework Statement With regards to a one dimensional conducting wire with a homogeneous charge density λ surrounded by a cylindrical glass dielectric of radius R, find: (a). The displacement vector inside the dielectric (b). The surface bound charges on the surface of the dielectric Sorry...

Homework Statement Bayes' risk is ##L^*=0## for a classification problem. ##g_n(x)## is a classification rule (plug-in) such that ##g_n=0## is ##\eta_n(x)\leq 1/2## and ##g_n=1##$ otherwise. The function ##\eta##is given by ##\eta(x)=\mathbb{E}(Y|X=x)##. Then ##\mathbb{P}(g_n(X)\neq Y)\leq...

Given that the Hamiltonian is H = P^2/(2m) + aδ(X − x(naught)) + bδ(X + x(naught), where x(naught) is a positive number. Find the conditions for bound states to exist and calculate their energies. Find the scattering matrix for arbitrary values of a and b. Can someone help me solve this please.

Homework Statement "Use the simple MO method to show that the He2+ ion is bound"Homework Equations The hamiltonian for this system is; H=-(ħ2/2m)∇2 -2ke2[∑3i=11/riB+∑3i=11/riB-2/RAB] And as far as I know, the total wave function sould be; ψ = φ1φ2φ3 ... φn , n is the number of electrons and...

Hi all - forgive me, I'd asked a series of questions in a previous post that was deemed to be circular, but I still didn't obtain a satisfactory answer to the question I was asking. In this post, I'm going to try to be very careful to use terms that are at least less 'misplaced', per se...

Homework Statement I've been reviewing some Taylor polynomial material, and looking over the results and examples here. https://math.dartmouth.edu/archive/m8w10/public_html/m8l02.pdf I'm referring to Example 3 on the page 12 (page numbering at top-left of each page). The question is asking...

Based on the paper by Visinelli (https://arxiv.org/abs/1605.06449), He stated in page 6 that the scalar spectral index as given by the Planck 2013 data (https://arxiv.org/abs/1303.5076) is, ##n_s = 0.9655 \pm 0.0062~~## (##68\%## C.L.) but when I looked into the Planck 2013 paper, I did not...

Homework Statement Homework Equations I have question for (a) section. The Attempt at a Solution I have two answer for the question but I can't figure out which one is right. (1)Since the partition function is to sum up all the state in the system, I write down the answer (2)In other...

Homework Statement A high-energy photon collides with a proton at rest. A neutral pi meson is produced according to the reaction ##\gamma + p \to p + \pi^{0}##. What is the minimum energy the photon must have for this reaction to occur? (The rest mass of a proton is ##938\ \text{MeV/c}^{2}##...

Hello, Before I begin, a lot of the math I try to describe in this post is stuff I worked out myself and have scribbled down. If something looks fishy or doesn't make sense, it could very well be totally wrong! Apologies for any confusions in advance. Consider a slab of linear isotropic...

Claim: Let A be a non-empty subset of R+ = {x ∈ R : x > 0} which is bounded above, and let B = {x2 : x ∈ A}. Then sup(B) = sup(A)2. a. Prove the claim. b. Does the claim still hold if we replace R+ with R? Explain briefly. So I have spent the past hours trying to prove this claim using the...

Homework Statement I am trying to show that the integrator is unstable by giving examples of bounded inputs which produce unbounded outputs (i.e. a bounded function whose integral is unbounded). Note: The integrator is a system which gives an output equal to the anti-derivative of its input...

Hi, I'm trying to understand the bound states of a periodic potential well in one dimension, as the title suggests. Suppose I have the following potential, V(x) = -A*(cos(w*x)-1). I'm trying to figure out what sort of bound energy eigenstates you'd expect for a potential like this. Specifically...

Hi. My question is why in photoelectric effect in coming photon interacts with bound electrons only? Thanks

Homework Statement Given the sequence ##\frac{1}{2}(x_n + \frac{2}{x_n})= x_{n+1}##, where ##x_1 =1##: Prove that ##x_n## is never less than ##\sqrt{2}##, then use this to prove that ##x_n - x_{n+1} \ge 0## and conclude ##\lim x_n = \sqrt{2}##. Homework EquationsThe Attempt at a Solution...

Homework Statement I'm currently working on a homework set for my intermediate QM class and for some reason I keep drawing a blank as to what to do on the first problem. I'm given three potentials, V(x), the first is of the form {A+Bexp(-Cx^2)}, the others I'll leave out. I'm asked to draw the...

Homework Statement Given ##v = {\left\{ {v}_{i} \right\}}_{i = 1}^{\infty}## and defining ## {v}_{n}^{\left( k \right)} = {v}_{n - k} ## (Shifting Operator). Prove that there exist ## \alpha > 0 ## such that $$ \sum_{k = - \infty}^{\infty} {2}^{- \left| k \right|} \left \langle {v}^{\left (...

Homework Statement A particle of mass ##m## is in a spherically symmetric potential ##V = -\alpha\delta(|r|-a)##. Find the minimum value of ##\alpha## such that there is at least one bound state. Homework Equations ##u = \frac{R}{r}## ##-\frac{\hbar^2}{2m} \frac{d^2u}{dr^2} + \left[V +...

If there are 3 positive charges of +1, +3, +5 coulombs equidistant from a negative charge of 1 coulomb what positive charge will this negative charge be attracted to ? Is the result different if the charges exist in a “bound” state (resulting in electrovalent compounds) where a positive charge...

I came to understand that the space between galaxies are expanding and not that they are speeding from each other and as space expands the gravitationally bound systems remain in their own relative position as the effect of gravity is more compared to the expansion of space. But now consider two...

In particular how does matter "clump" together to form stars and planets, and how do Galaxy/star systems form? For the latter question is the answer simply that near massive enough bodies, the spacetime curvature is significant enough that the geodesics within its vicinity are closed curves...

There is this summation, that I've been trying to solve, but am not able to do so. It is : $$\sum\limits_{i=k}^{n} \frac {1}{(n-i)! m^{i-1}}$$ I would be happy to find it's upper bound too. So what I did was intensely naive. I made the denominator the minimum by making ##(n-i)! = 1## and...

As the title says, why does the set of hydrogen bound states form an orthonormal basis? This is clearly not true in general since some potentials (such as the finite square well and reversed gaussian) only admit a finite number of bound states.

C1 1. Homework Statement : Using the ML inequality, I have to find an upper bound for the contour integral of \int e^2z - z^2 \, dz where the contour C = C1 + C2. C1 is the circular arc from point A(sqrt(3)/2, 1/2) to B(1/2, sqrt(3)/2) and C2 is the line segment from the origin to B...

error bounds for trapezoidal rule, midpoint rule, and Simpson's rule. can anyone please show me how to derive the formula?

Hi all, I am very confused about how one can find the upper bound for a Taylor series.. I know its general expression, which always tells me to find the (n+1)th derivative of a certain function and use the equation f(n+1)(c) (x-a)n+1/(n+1)! for c belongs to [a,x] However, there are...

Homework Statement A slab of material has parallel faces. One coincides with the xy plane (z = 0), while the other is given by z = zt . The material has a nonuniform polarization P = P(1 + αz)zˆ where P and α are constants. Fin the volume and surface densities of bound charges[/B] The Attempt...

I'm struggling about finding a way to find the upper bound of the error of Taylor polynomial approximation. I will explain better using a solved example I found... > $f: ]-3;+\infty[ \rightarrow \mathbb{R} $ $f(x)=ln(x+3) +1 $ >Find the upper bound of the error approximating the function in...

Find volume bound by $x_2 =\frac{y+1} {2} $ and $x_1=y^2$ $\int_{-{\frac{1}{2}}}^{1} (-{y}^{2 }+\frac{y}{2}+\frac{1}{2}) \,dx$ $ \begin{array}{l} {{\left[{\frac{{y}^{2}}{2}\mathrm{{+}}\frac{y}{4}\mathrm{{-}}\frac{{y}^{3}}{3}}\right]}_{\mathrm{{-}}{1}{\mathrm{/}}{2}}^{1}}\\...

Homework Statement Find a tight upper bound for the recurrence relation using a recursion tree argument Homework Equations T(n)=T(n/2)+T(n/3)+c The Attempt at a Solution I don't know how to do this problem because the tree doesn't have symmetry. One side of the tree can keep going because of...

Now I asked a question the other day about the gravitational binding energy of a torus, and someone responded that it cannot be gravitationally bound purely, but requires some opposing force. Okay, fine. But, qualitatively, can a toroidal planet be gravitationally bound if it has another force...

Hello, I am currently preparing myself for exams and I have a past exam question which I can't solve. This question concerns online learning and the following picture illustrates it: Is anyone able to help me out and propose a solution to this question?

As I understand the photoelectric effect as given by Einstein, an electron while attached to an atom is considered to be in a bound state and remains as such until a photon carrying a quantum of energy sufficient to overcome the work function of the electron frees the electron from its bound...

Homework Statement (Working with geometrised units) Consider the EFE ##G^{\alpha \beta }+\Lambda g^{\alpha \beta} = 8 \pi T^{\alpha \beta} ## work out (using weak-field considerations) an upper bound for the cosmological constant knowing that the radius of Pluto's orbit is 5.9 x 10^12 m...

Find $\text{glb}(A)$ if $A = \left\{(-1)^n \left(\frac{1}{4}-\frac{2}{n} \right): n \in \mathbb{N}\right\}$.$\displaystyle x_n = (-1)^n \left(\frac{1}{4}-\frac{2}{n} \right)$ then $ \displaystyle x_{2k} = \frac{1}{4}-\dfrac{1}{k} = \frac{k-4}{4k}$ and $ \displaystyle x_{2k+1} =...

Homework Statement Consider V(x)=-aV_{0}δ(x). Show that it admits a bound state of energy E=-ma^2V_{0}/2\hbar^{2}.Are there any other bound states?Hint:solve Schrodinger's equation outside the potential for E<0, and keep only the solution that has the right behavior at infinity and is...

I am using Spivak Calculus. I have a general doubt regarding the definition of least upper bound of sets. Let A be any set of real numbers and A is not a null set. Let S be the least upper bound of A. Then by definition "For every x belongs to A, x is lesser than or equal to S" Let M be an...

The supremum is defined as the "LEAST" upper bound. The word "least" makes me think, there is a "MOST" upper bound, or at least something bigger than a "least" upper bound. For a set of numbers, is there anything larger than a supremum? Supremum is analogous to a maximum, but I don't...