Harmonic Definition and 1000 Threads

Homework Statement Two points, each of charge Q, are fixed at either end of a frictionless rod of length 2R. Another point charge, of charge q (not Q) is free to move along the rod. Show that if charge q is displaced a small distance x (0<x<<R) from the centre of the rod, it will undergo...

I am a little confused with this subject. If you have a mass hanging from a spring, there is a specific equilibrium point, but what if you apply a force downwards on the mass, will this have an effect on the equilibrium position or will it remain the same? thanks!

Hello, I have two question regrading sound waves. The first one: The pressure P(x;t) at a point x at time t in a medium through which a harmonic wave is travelling can be described by: P(x,t) = Asin(wt -kx) If the equation describes a pressure wave traveling in air, with amplitude 2 Pa and...

Homework Statement So with pendulums in SHM, in my A level physics textbook (AQA Physics A), it shows a pendulum that has been displaced from equilibrium. It says that the restoring force is provided by the object's weight. Why isn't the restoring force provided by the tension in the string...

Homework Statement It's not a direct question, but it's an implied part of a larger question: can classical waves experience simple harmonic oscillator potentials, like a mass on a spring does? Homework Equations The Attempt at a Solution I'm thinking no, since I can't come up...

Well I was going through class lecture notes and my professor wrote this When x = A(the maximum value), v=0: E=1/2kA^2 When v = wA, x=0: E=1/2mw^2A^2 where w = omega, A = amplitude, k = spring constant, m = mass, v = velocity and apparently both equations are equal, i would like to...

Let a+,a- be the ladder operators of the harmonic oscillators. In my book I encountered the hamiltonian: H = hbarω(a+a-+½) + hbarω0(a++a-) Now the first term is just the regular harmonic oscillator and the second term can be rewritten with the transformation equations for x and p to the...

I have a couple of questions about what total harmonic distortion is, and what the measurement means. The definition I've read most places is: \frac{D}{S} × 100% , where S is the amplitude of the fundamental frequency, and D is the amplitude of the sum of all of the harmonics. A common...

Homework Statement A system is made of N 1D simple harmonic oscillators. Show that the number of states with total energy E is given by \Omega(E) = \frac{(M+N-1)!}{(M!)(N-1)!} Homework Equations Each particle has energy ε = \overline{h}\omega(n + \frac{1}{2}), n = 0, 1 Total energy is...

Homework Statement After four cycles the amplitude of a damped harmonic oscillator has dropped to 1/e of it's initial value. Find the ratio of the frequency of this oscillator to that of it's natural frequency (undamped value) Homework Equations x'' +(√k/m) = 0 x'' = d/dt(dx/dt)...

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 Consider the van del Pol equation [tex]\ddot{u}-ε(1-u^2)\dot{u}+u=0[\tex] Determine the limit cycle for ε=1 using the incremental harmonic balance method. Validate the result using numerical integration (e.g., Runge Kutta). Homework Equations It's incremental...

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 | < \epsilon H_n <...

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...

Homework Statement The generalization of the bohr rule to periodic motion more general than circular orbit states that: ∫p.dr = nh = 2∏nh(bar). the integral is a closed line integral and the "p" and "r" are vectors Using the generalized rule (the integral above), show that the spectrum for...

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##.

Hello, if I think of the harmonic oscillator action S = ∫L dt, L = 1/2 (dx/dt)^2 - 1/2 x^2, and then of the "scaling transformation" x -> x' = 1/a x (a>0, const), then x = a x', and in new coordinates x', S' has the same form as in x except for multiplication by the constant a, formally...