Harmonic Definition and 1000 Threads

Hi, in this article: http://dx.doi.org/10.1016/S0021-9991(03)00308-5 damped molecular dynamics is used as a minimization scheme. In formula No. 9 the author gives an estimator for the optimal damping frequency: Can someone explain how to find this estimate? best, derivator

Homework Statement A 2.12-m long rope has a mass of 0.116 kg. The tension is 62.9 N. An oscillator at one end sends a harmonic wave with an amplitude of 1.09 cm down the rope. The other end of the rope is terminated so all of the energy of the wave is absorbed and none is reflected. What is...

What's the relationship between DFT and harmonic amplitude? How do I find the harmonic amplitude using discrete Fourier transform? Here's what I have done so far. "harm.freq" is harmonic frequency here. I have done the DFT calculation and now what? Aftet I have performed DFT, how do I find the...

For a set of energy eigenstates |n\rangle then we have the energy eigenvalue equation \hat{H}|n\rangle = E_{n}|n\rangle. We also have a commutator equation [\hat{H}, \hat{a^\dagger}] = \hbar\omega\hat{a}^{\dagger} From this we have \hat{a}^{\dagger}\hat{H}|n\rangle =...

Homework Statement Let u be a harmonic function in the open disk K centered at the origin with radius a. and ∫_K[u(x,y)]^2 dxdy = M < ∞. Prove that |u(x,y)| \le \frac{1}{a-\sqrt{x^2+y^2}}\left( \frac{M}{\pi}\right)^{1/2} for all (x,y) in K. Homework Equations Mean value property for...

Could I get some hints on how to evaluate these question. The question asking to find where $f(re^{i\theta})$ is differentiable doesn't seem to involved, however would I use C-R equations, or would it just be for wherever $r \neq 0$. Although that is given in the domain, so I'm assuming they...

Homework Statement if a pendulum has a period of .36s on Earth, what would its period be on the moon Homework Equations T=2pi sqrt l/g The Attempt at a Solution How do u go about solving thAt without length?

Homework Statement A particle is in the ground state of a simple harmonic oscillator, potential → V(x)=\frac{1}{2}mω^{2}x^{2} Imagine that you are in the ground state |0⟩ of the 1DSHO, and you operate on it with the momentum operator p, in terms of the a and a† operators. What is the...

Homework Statement Mass = 2.4 kg spring constant = 400 N/m equilbrium length = 1.5 The two ends of the spring are fixed at point A, and at point B which is 1.9m away from A. The 2.4 kg mass is attached to the midpoint of the spring, the mass is slightly disturbed. What is the period of...

Homework Statement Find the eigenvalues of the following Hamiltonian. Ĥ = ħwâ^{†}â + \alpha(â + â^{†}) , \alpha \in |RHomework Equations â|\phi_{n}>=\sqrt{n}|\phi_{n-1}> â^{†}|\phi_{n}>=\sqrt{n+1}|\phi_{n+1}> The Attempt at a Solution By applying the Hamiltonian to a random state n I...

Homework Statement Prove that Hn converges given that: H_{n}=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n} The Attempt at a Solution First I supposed that the series converges to H...

Homework Statement The question is from Sakurai 2nd edition, problem 3.21. (See attachments) ******* EDIT: Oops! Forgot to attach file! It should be there now.. *******The Attempt at a Solution Part a, I feel like I can do without too much of a problem, just re-write L as L=xp and then...

A 10kg mass is suspended from a spring which has a constant K = 2.5kn/m. At time t=0, it has a downward velovcity of 0.5m/s as it passes through the position of static equilibrium. Determine the static spring deflection. I believe i first need to calculate the force which requires basic...

Homework Statement Two pendula of length 1.00m are set in motion at the same time. One pendula has a bob of mass 0.050kg and the other has a mass of 0.100kg. 1. What is the ratio of the periods of oscillation? 2. What is the period of oscillation if the initial angular displacement is...

Homework Statement Given (\mathcal{L} + k^2)y = \phi(x) with homogeneous boundary conditions y(0) = y(\ell) = 0 where \begin{align} y(x) & = \frac{2}{\ell}\sum_{n = 1}^{\infty} \frac{\sin(k_nx)}{k^2 - k_n^2},\\ \phi(x) & = \frac{2}{\ell}\sum_{n = 1}^{\infty}\sin(k_nx),\\ u_n(x) &=...

Homework Statement The generalized Green function is $$ G_g(x, x') = \sum_{n\neq m}\frac{u_n(x)u_n(x')}{k_m^2 - k_n^2}. $$ Show G_g satisfies the equation $$ (\mathcal{L} + k_m^2)G_g(x, x') = \delta(x - x') - u_m(x)u_m(x') $$ where \delta(x - x') = \frac{2}{\ell}\sum_{n =...

Given \((\mathcal{L} + k^2)y = \phi(x)\) with homogeneous boundary conditions \(y(0) = y(\ell) = 0\) where \begin{align} y(x) & = \frac{2}{\ell}\sum_{n = 1}^{\infty} \frac{\sin(k_nx)}{k^2 - k_n^2},\\ \phi(x) & = \frac{2}{\ell}\sum_{n = 1}^{\infty}\sin(k_nx),\\ u_n(x) &=...

I need to prove that $$H_n = \ln n + \gamma + \epsilon_n $$ Using that $$\lim_{n \to \infty} H_n - \ln n = \gamma $$ we conclude that $$\forall \, \epsilon > 0 \,\,\,\, \exists k \,\,\,\, $$ such that $$\,\,\, \forall k \geq n \,\,\, $$ the following holds $$|H_n - \ln n -\gamma | <...

Homework Statement Which of the following statements about the harmonic oscillator (HO) is true? a) The depth of the potential energy surface is related to bond strength. b) The vibrational frequency increases with increasing quantum numbers. c) The HO model does not account for bond...

Homework Statement We were asked to try to make a theoretical description of the following phenomenon: Imagine a 2D Bose Einstein condensate in equilibrium in an harmonical trap with frequency ω. Suddenly the trap is shifted over a distance a along the x-axis. The condensate is no longer...

Homework Statement For a lightly damped harmonic oscillator and driving frequencies close to the natural frequency \omega \approx \omega_{0}, show that the power absorbed is approximately proportional to \frac{\gamma^{2}/4}{\left(\omega_{0}-\omega\right)^{2}+\gamma^{2}/4} where \gamma is...

For infinite square well, ψ(x) square is the probability to find a particle inside the square well. For hamornic oscillator, is that meant the particle behave like a spring? Why do we put the potential as 1/2 k(wx)^2 ? Thanks

Homework Statement Numerically determine the period of oscillations for a harmonic oscillator using the Euler-Richardson algorithm. The equation of motion of the harmonic oscillator is described by the following: \frac{d^{2}}{dt^{2}} = - \omega^{2}_{0}x The initial conditions are x(t=0)=1...

Homework Statement Compute ##\left \langle x^2 \right\rangle## for the states ##\psi _0## and ##\psi _1## by explicit integration. Homework Equations ##\xi\equiv \sqrt{\frac{m \omega}{\hbar}}x## ##α \equiv (\frac{m \omega}{\pi \hbar})^{1/4}## ##\psi _0 = α e^{\frac{\xi ^2}{2}}##The Attempt at...

This is more of a conceptual question and I have not had the knowledge to solve it. We're given a modified quantum harmonic oscillator. Its hamiltonian is H=\frac{P^{2}}{2m}+V(x) where V(x)=\frac{1}{2}m\omega^{2}x^{2} for x\geq0 and V(x)=\infty otherwise. I'm asked to justify in...

Consider the harmonic oscillator equation (with m=1), x''+bx'+kx=0 where b≥0 and k>0. Identify the regions in the relevant portion of the bk-plane where the corresponding system has similar phase portraits. I'm not sure exactly where to start with this one. Any ideas?

Homework Statement The Hamiltonian for a particle in a harmonic potential is given by \hat{H}=\frac{\hat{p}^2}{2m}+\frac{Kx^2}{2}, where K is the spring constant. Start with the trial wave function \psi(x)=exp(\frac{-x^2}{2a^2}) and solve the energy eigenvalue equation...

Problem: Consider a harmonic oscillator of undamped frequency ω0 (= \sqrt{k/m}) and damping constant β (=b/(2m), where b is the coefficient of the viscous resistance force). a) Write the general solution for the motion of the position x(t) in terms of two arbitrary constants assuming an...

So, I've read conference proceedings and they appear to talk about counter-intuitive it was to create an infinite-energy state for the harmonic oscillator with a normalizable wave function (i.e. a linear combination of eigenstates). How exactly could those even exist in the first place?

Homework Statement Two-dimensional SHM: A particle undergoes simple harmonic motion in both the x and y directions simultaneously. Its x and y coordinates are given by x = asin(ωt) y = bcos(ωt) Show that the quantity x\dot{y}-y\dot{x} is also constant along the ellipse, where here the...

Homework Statement Consider a mass hanging from an ideal spring. Assume the mass is equal to 1 kg and the spring constant is 10 N/m. What is the characteristic frequency of this simple harmonic oscillator? Homework Equations No idea I think Hookes law F=-ky Some other relevant...

These are practice problems, not homework. Just wanting to check to see if my process and solutions are correct. 1. Given the following functions as solutions to a harmonic oscillator equation, find the frequency f correct to two significant figures: f(x) = e-3it f(x) = e-\frac{\pi}{2}it 2...

Homework Statement A one-dimensional harmonic oscillator is in the ground state. At t=0, the spring is cut. Find the wave-function with respect to space and time (ψ(x,t)). Note: At t=0 the spring constant (k) is reduced to zero. So, my question is mostly conceptual. Since the spring...

So over the weekend my physics prof has assigned an assignment where one of the questions is as follows and here is my thought process: A massless spring hangs from the ceiling with a small object attached to its lower end. The object is initially held at rest in a position yi such that the...

Homework Statement Prove harmonic series is divergent by comparing it with this series. ##\frac{1}{1}+\frac{1}{2}+(\frac{1}{4}+\frac{1}{4})+(...)## The Attempt at a Solution Clearly every term in harmonic series is equal or larger than the term in the second series ##n \geq 1##, hence like...

Hello, I was being taught AC in High School, It was good but the way they taught us DC, things like drift velocity, no of electrons per unit volume etc, it was easy to visualize electrons rushing in a conductor. I tried to visualise AC(which was not taught to us) and I came to a conclusion...

This challenge was suggested by jgens. The ##n##th harmonic number is defined by H_n = \sum_{k=1}^n \frac{1}{k} Show that ##H_n## is never an integer if ##n\geq 2##.

Homework Statement Derive the equilibrium state of a simple harmonic oscillation and show that the derivative of the maximum displacement is s^{'} = 2 \sqrt{E} Homework Equations F = -k x The Attempt at a Solution m a = -k s \rightarrow ms^{''}...

Homework Statement A spring is freely hanged on a ceiling. You attach a mass to the end of the spring and let the mass go. It falls down a distance of 49 cm and comes back to where it started. It contineous to oscillate in a simple harmonic motion going up and down - a total distance of 49...

Hi I am doing this completely out of self interest and it is not my homework to do this. I hope somebody can help me. Homework Statement In the book Biological Coherence and Response to External Stimuli Herbert Fröhlich wrote a chapter on Resonance Interaction. Where he considers the...

This thread will be dedicated for a trial to prove the following $$\sum_{k\geq 1} \frac{H^2_k}{k^2}=\frac{17}{4}\zeta(4)=\frac{17\pi^4}{360}$$ $$\mbox{where }\,\,H^2_k =\left( 1+\frac{1}{2}+\frac{1}{3}+\cdots \frac{1}{k}\right)^2$$ In this paper the authors give solutions to the sum and...

Prove the following $$\sum_{k\geq 1} \frac{H^2_k}{k^2}=\frac{17}{4}\zeta(4)=\frac{17\pi^4}{360}$$ $$\mbox{where }\,\,H^2_k =\left( 1+\frac{1}{2}+\frac{1}{3}+\cdots \frac{1}{k}\right)^2$$

Hello, Does the nth harmonic ALWAYS produce n loops, when referring to sound? If not, is there a general rule for this? Thanks.

Hello, When doing problems with SHM, my textbook says something like: An object in vertical shm is described by <insert some function>. Find the speed after X seconds. my question is, how do you know if the function is referring to the position of the object, or the velocity, or...

Homework Statement Please kindly help me. Actually I don't quite understand what the meaning of harmonic wave is and the mathematics that expresses it. h(x,y;t) = h sin(wt-kx+δ) h represents the position of the particle in a particular time? Or the wave motion? What is the physical...

The model of damped harmonic oscillator is given by the composite system with the hamiltonians ##H_S\equiv\hbar \omega_0 a^\dagger a##, ##H_R\equiv\sum_j\hbar\omega_jr_j^\dagger r_j##, and ##H_{SR}\equiv\sum_j\hbar(\kappa_j^*ar_j^\dagger+\kappa_ja^\dagger...

Homework Statement ## H=\frac{p^2}{2m}+\frac{1}{2}m\omega^2x^2## Show that ##[H,[H,x^2]]=(2\hbar\omega)^2x^2-\frac{4\hbar^2}{m}H## Homework Equations ##[x,p]=i\hbar## The Attempt at a Solution I get ##[H,x^2]=-\frac{i\hbar}{m}(px+xp)## what is easiest way to solve this problem?

I was reading through my Principles of Quantum Mechanics textbook and arrived at the section that discusses the quantum harmonic oscillator. In this discussion the equation ψ"-(y^2)ψ=0 presents itself and a solution is given as ψ=(y^m)*e^((-y^2)/2), similar to a gaussian function i assume. My...

The Hamiltonian is ##H=\hbar \omega (a^\dagger a+b^\dagger b)+\hbar\kappa(a^\dagger b+ab^\dagger)## with commutation relations ##[a,a^\dagger]=1 \hspace{1 mm} and \hspace{1 mm}[b,b^\dagger]=1##. I want to calculate the Heisenberg equations of motion for a and b. Beginning with ##\dot...

Homework Statement The velocity of an object in simple harmonic motion is given by v(t)= -(4.04m/s)sin(21.0t + 1.00π), where t is in seconds. What is the first time after t=0.00 s at which the velocity is -0.149m/s? Homework Equations N/A The Attempt at a Solution I thought this was...