Harmonic oscillator Definition and 743 Threads

Edit: Problem solved please disregard this post Homework Statement A particle in the harmonic oscillator potential has the initial wave function \Psi(x, 0) = ∑(from n = 0 to infinity) Cnψn(x) where the ψ(x) are the (normalized) harmonic oscillator eigenfunctions and the coefficients are given...

Homework Statement I found this in Binney's text, pg 154 where he described the radial probability density ##P_{(r)} \propto r^2 u_L## Homework Equations The Attempt at a Solution Isn't the radial probability density simply the square of the normalized wavefunction...

I'm trying to plot the evolution of a simple harmonic oscillator using MATLAB but I'm getting non-sense result and I have no idea what's wrong! Here's my code: clear clc x(1)=0; v(1)=10; h=.001; k=100; m=.1; t=[0:h:10]; n=length(t); for i=2:n F(i-1)=-k*x(i-1)...

Homework Statement A particle of mass m moves in a 1-D Harmonic oscillator potential with frequency \omega. The second excited state is \psi_{2}(x) = C(2 \alpha^{2} x^{2} + \lambda) e^{-\frac{1}{2} a^{2} x^{2}} with energy eigenvalue E_{2} = \frac{5}{2} \hbar \omega. C and \lambda are...

Homework Statement For the n = 1 harmonic oscillator wave function, find the probability p that, in an experiment which measures position, the particle will be found within a distance d = (mk)-1/4√ħ/2 of the origin. (Hint: Assume that the value of the integral α = ∫0^1/2 x^2e^(-x2/2) dx is...

Homework Statement Consider a time-dependent harmonic oscillator with Hamiltonian \hat{H}(t)=\hat{H}_0+\hat{V}(t) \hat{H}_0=\hbar \omega \left( \hat{a}^{\dagger}\hat{a}+\frac{1}{2} \right) \hat{V}(t)=\lambda \left( e^{i\Omega t}\hat{a}^{\dagger}+e^{-i\Omega t}\hat{a} \right) (i)...

Homework Statement The pendulum of a grandfather clock activates an escapement mechanism every time it passes through the vertical. The escapement is under tension (provided by a hanging weight) and gives the pendulum a small impulse a distance l from the pivot. The energy transferred by...

Hello, my book explains detailed the proofs of these three formulas: y = Asin(ωt + φo) v = ωAcos(ωt + φo) a = -ω²Asin(ωt + φo) Where a is acceleration, v is velocity, ω is angular velocity, A is amplitude. The book uses the following figures: Figure a) -->...

I know I've seen this point about roots discussed somewhere but I can't for the life of me remember where. I'm hoping someone can point me the right direction. Here's the situation:- The standard derivation of the quantum HO starts with the classic Hamiltonian in the form H = p2 + q2...

Homework Statement Hey guys, So I have this equation for the entropy of a classical harmonic oscillator: \frac{S}{k}=N[\frac{Tf'(T)}{f(T)}-\log z]-\log (1-zf(T)) where z=e^{\frac{\mu}{kT}} is the fugacity, and f(T)=\frac{kT}{\hbar \omega}. I have to show that, "in the limit of...

Hello, I have been studying Introduction to Quantum Mechanics by Griffith and in a section he solves the Schrodinger equation for a harmonic oscillator potential using the power series method. First he rewrites the shroedinger equation in the form d^2ψ/dε^2 = (ε^2 - K)ψ , where ε=...

I am reading an article on the "energy surface" of a Hamiltonian. For a simple harmonic oscillator, I am assuming this "energy surface" has one (1) degree of freedom. For this case, the article states that the "dimensionality of phase space" = 2N = 2 and "dimensionality of the energy surface" =...

A harmonic oscillator with frequency ω is in its ground state when the stiffness of the spring is instantaneously reduced by a factor f2<1, so its natural frequency becomes f2ω. What is the probability that the oscillator is subsequently found to have energy 1.5(hbar)f2ω? Thanks

Homework Statement Consider a one-dimensional linear harmonic oscillator perturbed by a Gaussian perturbation H' = λe-ax2. Calculate the first-order correction to the groundstate energy and to the energy of the first excited state Homework Equations ψn(x) = \frac{α}{√π*2n*n!}1/2 *...

Shouldn't the integrating factor be ##exp(\frac{m\omega x}{\hbar})##? \frac{\partial <x|0>}{\partial x} + \frac{m\omega x}{\hbar} <x|0> = 0 This is in the form: \frac{\partial y}{\partial x} + P_{(x)} y = Q_{(x)} Where I.F. is ##exp (\int (P_{(x)} dx)##

Homework Statement We want to prepare a particle in state ##\psi ## under following conditions: 1. Let energy be ##E=\frac{5}{4}\hbar \omega ## 2. Probability, that we will measure energy greater than ##2\hbar \omega## is ##0## 3. ##<x>=0## Homework Equations The Attempt at a...

Homework Statement Potential energy of electron in harmonic potential can be described as ##V(x)=\frac{m\omega _0^2x^2}{2}-eEx##, where E is electric field that has no gradient. What are the energies of eigenstates of an electron in potential ##V(x)##? Also calculate ##<ex>##. HINT: Use...

Homework Statement One dimensional harmonic oscillator is at the beginning in state with wavefunction ##\psi (x,0)=Aexp(-\frac{(x-x_0)^2}{2a^2})exp(\frac{ip_0x}{\hbar })##. What is the expected value of full energy? Homework Equations ##<E>=<\psi ^{*}|H|\psi >=\sum \left | C_n \right |^2E_n##...

Homework Statement I need to show that for an eigen state of 1D harmonic oscillator the expectation values of the position X is Zero. Homework Equations Using a+=\frac{1}{\sqrt{2mhw}}(\hat{Px}+iwm\hat{x}) a-=\frac{1}{\sqrt{2mhw}}(\hat{Px}-iwm\hat{x}) The Attempt at a Solution...

Homework Statement The terminal speed of a freely falling object is v_t (assume a linear form of air resistance). When the object is suspended by a spring, the spring stretches by an amount a. Find the formula of the frequency of oscillation in terms of g, v_t, and a. Homework Equations the...

Homework Statement A harmonic oscillator oscillates with an amplitude A. In one period of oscillation, what is the distance traveled by the oscillator? Homework Equations I'm not sure which equation applies if any? The Attempt at a Solution My guess was 2A but the answer was 4A...

Homework Statement Given a quantum harmonic oscillator, calculate the following values: \left \langle n \right | a \left | n \right \rangle, \left \langle n \right | a^\dagger \left | n \right \rangle, \left \langle n \right | X \left | n \right \rangle, \left \langle n \right | P \left | n...

Homework Statement Parabolic harmonic oscillator potential well. A particle is trapped in the well, oscillating classically back and forth between x=b and x=-b. The potential jumps from Vo to zero at x=a and x=-a. The particle's energy is Vo/2. I need to find the potential function V(x) in...

An "attempt frequency" for a harmonic oscillator? Homework Statement What is the "attempt frequency" for a harmonic oscillator with bound potential as the particle goes from x = -c to x = +c? What is the rate of its movement from -c to +c? Homework Equations v...

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...

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

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 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 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 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...

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) &=...

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 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 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...

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...