Harmonic oscillator Definition and 743 Threads

Homework Statement The wave function \psi_0 (x) = A e^{- \dfrac{x^2}{2L^2}} represents the ground-state of a harmonic oscillator. (a) Show that \psi_1 (x) = L \dfrac{d}{dx} \psi_0 (x) is also a solution of Schrödinger's equation. (b) What is the energy of this new state? (c) From a look at...

Tuning forks are lightly damped SHO's. Consider a tuning fork who's natural frequency is f=392Hz. Angular frequency = w = 2(Pi)f = 2463 (rad/s) The damping of this tuning fork is such that, after 10 sec, it's amplitude is 10% of it's original amplitude. Here is my attempt to find the damping...

Homework Statement For a particle of mass m moving in the potential V(x) = \frac{1}{2}m\omega^2x^2 (i.e. a harmonic oscillator), it is often convenient to express the position and momentum operators in terms of the ladder operators a_{\pm}: x = \sqrt{\frac{\hbar}{2m\omega}}(a_+ + a_-) p =...

If I have mass on a spring that is oscillating in a linear motion, this system has a certain energy. Now if we imagine the system to be aligned along the vertical, why is the energy lower when gravity is turned on? I can calculate it and see that it is correct, but what is the "explanation" ...

Hi, I'm currently working through some exam papers from previous years before an upcoming module in Quantum Mechanics. Homework Statement See the attached image Homework Equations The Attempt at a Solution I'm a little stumped with this one, I'm assuming that I'm looking...

I need someone to please verify my work. Homework Statement A particle of mass m is suspended from the ceiling by a spring of constant k and initially relaxed length l_0. The particle is then let go from rest with the spring initially relaxed. Taking the z-axis as vertically oriented...

Homework Statement A body of mass 4[kgr] is moving along the x-axis while the following force is applied on it: F= -3(x-6) We know that at time t=0 the kinetic energy is K=2.16[J] and that its decreasing, that is, \frac{dK}{dt}<0 . The potential energy (with respect to the equilibrium...

I've looked at a few introductory treatments of the quantum harmonic oscillator and they all show how one arrives at the discrete energy values E_n = ( \frac {1}{2} + n ) hf \hspace {10 mm} n=0,1,2... usually by setting up and then solving the Schrodinger equation for the system...

Homework Statement How does change acceleration of relativistic linear harmonic oscillator with distance of equilibrium point in laboratory reference system? Homework Equations The Attempt at a Solution x=x_0sin(\omega t+\varphi) \upsilon=\omega x_0cos(\omega...

Homework Statement \omega_{x} = \omega \omega_{y} = \omega + \epsilon where 0 < \epsilon<<\omega Question: Find the path equation. Homework Equations I started with the 2D equations: x(t) = A_{x}cos(\omega_{x}t + \phi_{x}) y(t) = A_{y}cos(\omega_{y}t + \phi_{y}) The Attempt at a Solution...

A classical harmonic oscillator follows a smooth, sinusoidal path of oscillation. Since on a quantum level energy levels are discrete, does a quantum harmonic oscillator actually oscillate in the everyday sense?

Hello there, Can anyone help me, I am struggling with solving LHO in two dimension,but in the polar coordinates. I transfer laplacian into polar from decart coordinates, write Ψ=ΦR, and do Fourier separation method for solving differential equation. But I do not know how to solve...

Hi all, I am having trouble with a certain integral, which I got from Quantum Physics by Le Bellac: \int dxdp\;\delta\left( E - \frac{p^2}{2m} - \frac{1}{2}m\omega^2x^2 \right) f(E) The answer to this integral should be 2\pi / \omega\; f(E) . My attempts so far: This integral is basically a...

Hello. I am trying to use the following equation: a\left|\psi_n\right\rangle=\sqrt{n}\left|\psi_{n-1}\right\rangle (where a is the "ladder operator"). What happens when I substitute \psi_n with \psi_0?

Homework Statement Hi, I'm currently studying for a quantum mechanics exam but I am stuck on a line in my notes: Ha\left|\Psi\right\rangle =\hbar\omega\left(a^{t}a a + \frac{a}{2}\right)\left|\Psi\right\rangleHa\left|\Psi\right\rangle =\hbar\omega\left(\left(a a^{t} - 1\right)a +...

The Hamiltonian of the diatomic molecule is given by H = p1^2 / 2m + p2^2 / 2m + 1/2 k R^2, where R equals the distance between atoms. Using this result, given in standard texbooks, I keep geting C = 9/2 kT instead of 7/2 kT for heat capacity. I've traced down my problem to the potential energy...

Hi, I have to find the 'stationary position' of a particle of mass m and charge q which moves in an isotropic 3D harmonic oscillator with natural frequency \omega_{0}, in a region containing a uniform electric field \boldsymbol{E} = E_{0}\hat{x} and a uniform magnetic field \boldsymbol{B} =...

Homework Statement The energy eigenvalues of an s-dimensional harmonic oscillator is: \epsilon_j = (j+\frac{s}{2})\hbar\omega show that the jth energy level has multiplicity \frac{(j + s - 1)!}{j!(s - 1)!} Homework Equations partition function: Z = \Sigma e^{-(...

given the Hamiltonian H=p^{2}- \omega x^{2} we can see inmediatly that this Hamiltonian will NOT have a BOUND state due to a 'saddle point' on (0,0) , here 'omega' is the frecuency of the Harmonic oscillations the classical solutions are not PERIODIC Asinh( \omega t) +Bcosh( \omega t)...

Homework Statement A particle with mass m moves in the potential: V(x,y,z) = \frac{1}{2} k(x^{2}+y^{2}+z^{2}+ \lambda x y z) considering that lambda is low. a) Calculate the ground state energy accordingly to Pertubations Theory of the second order. b) Calculate the energies of...

Hey all, In the classical harmonic oscillator the force is given by Hooke's Law, F = -kx which gives us the potential energy function V(x)=(1/2)kx^2. Now I understand that the first derivative at the point of equilibrium must be zero since the slope at the point of equilibrium is zero. But what...

Homework Statement Can someone please give me some hints how to solve this problem. Show that expected value for the kinetic energy is the same as the expected value for the potential energy for a harmonic oscillator in gound state. Homework Equations how to start with it? The...

Homework Statement A simple harmonic oscillator has amplitude 0.49 m and period 3.7 sec. What is the maximum acceleration? Homework Equations a(max)=Aw^2 w=angular frequency Vmax=Aw w= angular frequency The Attempt at a Solution I attempted to divide the Amplitude (.49m) by...

Homework Statement An electron is confined by the potential of a linear harmonic oscillator V(x)=1/2kx2 and subjected to a constant electric field E, parallel to the x-axis. a) Determine the variation in the electron’s energy levels caused by the electric field E. b) Show that the second order...

Homework Statement What is the phase constant (from 0 to 2π rad) for the harmonic oscillator with the velocity function v(t) given in Fig. 15-30 if the position function x(t) has the form x = xmcos(ωt + φ)? The vertical axis scale is set by vs = 7.50 cm/s...

Homework Statement I need to find <x>, <x2>, <p>, and <p2> for a particle in the first state of a harmonic oscillator. Homework Equations The harmonic oscillator in the first state is described by \psi(x)=A\alpha1/2*x*e-\alpha*x2/2. I'm using the definition <Q>=(\int\psi1*Q*\psi)dx...

I'm looking at a question... The last part is this: find the expectation values of energy at t=0 The function that describes the particle of mass m is A.SUM[(1/sqrt2)^n].\varphi_n where I've found A to be 1/sqrt2. The energy eigenstates are \varphi_n with eigenvalue E_n=(n + 1/2)hw...

Quantum Harmonic Oscillator Operator Commution (solved) EDIT This was solved thanks to CompuChip! The entire post is also not very interesting as it was a basic mistake :P No need to waste time This is not homework (I am not currently in college :P), but it is a mathematical question I'm...

Ok, so I am trying understand how to derive the following version of the Schrodinger Equation for QHO: \frac{d^2u}{dz^2} + (2\epsilon-z^2)u=0 where \ 1. z=(\frac{m\omega}{hbar})^{1/2}x and \ 2. \epsilon= \frac{E}{hbar\omega} I've started with the TISE, used a potential of...

Homework Statement This is for my mechanics class. It seems like it should be easier than I'm making it. A single object of mass m is attached to the ends of two identical, very long springs of spring constant k. One spring is lined up on the x-axis; the other on the y-axis. Chpose your...

Homework Statement Consider a harmonic oscillator with mass=0.1kg, k=50N/m , h-bar=1.055x10-34 Let this oscillator have the same energy as a mass on a spring, with the same k and m, released from rest at a displacement of 5.00 cm from equilibrium. What is the quantum number n of the state of...

An experimenter has carefully prepared a particle of mass m in the first excited state of a one dimensional harmonic oscillator. Suddenly he coughs and knocks the center of the potential a small distance, a, to one side. It takes him a time T to recover and when he has done so he immediately...

This is not really homework, just a project I'm toying with in my sparetime. I'm doing some Path Integral Monte Carlo simulations, for now just for the 1D quantum harmonic oscillator. Anyways, currently I compare my results to the analytic mean energy of a 1D quantum harmonic oscillator, given...

Find the time dependence of the expectation value <x> in a quantum harmonic oscillator, where the potential is given by V=\frac{1}{2}kx^2 I'm assuming some wavefunction of the form \Psi(x,t)=\psi(x) e^{-iEt/\hbar} When I apply the position operator, I get: <x>=\int_{-\infty}^\infty...

Homework Statement P4-1. The Method of Frobenius: Sines and Cosines. The solutions to the differential equation y"+ y = 0 can be expressed in terms of our familiar sine and cosine: y(x) = Acos(x) + Bsin(x) . Use the Method of Frobenius to solve the above differential equation for the even...

I Have a brief idea about the equation and i have searched the web for its application to one dimensional harmonic oscillator but no use. Any Help Would Be Welcomed Especially about the latter:smile:

Homework Statement I have a simple harmonic oscillator system with the driving force a sinusoidal term. The question is to find the general solution and the amplitude of the steady state solution Homework Equations I found the steady state part of the solution. It is of the form...

I know that the HO hamiltonian in matrix form using the known eigenvalues is <i|H|j> = E^j * delta_ij = (j+1/2)hbar*omega*delta_ij, a diagonalized matrix. How do I set up the non-diagonalized matrix from the potential V=1/2kx^2?

Homework Statement Does a wavefunction have to be normalized before you can calculate the probability density? Homework Equations n/a The Attempt at a Solution Im thinking yes? so that your probability will be in between 0 and 1?

Hi. I just calculated the quantum mechanical harmonic oscillator with a driving dipole force V(x,t) = - x S \sin(\omega t + \phi) I used the Floquet-Formalism. Then I calculated the mean expectation value, in a Floquet-State, of the Hamiltonian over a full Period T indirectly by using the...

Homework Statement Calculate the ratio of the kinetic energy to the potential energy of a simple harmonic oscillator when its displacement is half its amplitude. Homework Equations KE=1/2mv2 = 1/2kA2sin2(wt) U=1/2kx2 = 1/2kA2cos2(wt) KEmax=1/2kA2 Umax=1/2KA2 The Attempt at a...

Homework Statement Consider a particle of mass m moving in a one-dimensional potential, V(x)=\infty for x\leq0 V(x)=\frac{1}{2}m{\omega^2}{x^2} for x>0 This potential describes an elastic spring (with spring constant K = m\omega^2) that can be extended but not compressed. By...

Hi, I'm desperately searching for some literature which discusses the harmonic oscillator, preferably simple, in terms of the path integral formulation. I am unfamiliar with dirac notation and want something as simple as possible which gives general results of the harmonic oscillator in terms of...

Homework Statement There is a mass attached to two springs on a table. Coefficients of static and sliding friction between the mass and table are equal with the value \mu. The particle is released at time t=0 with a positive displacement x0 from equilibrium. Given that 2kx0 > \mumg write...

Homework Statement You are working at a company that designs suspension systems. Some guy from the marketing department asks you to design a shock absorber that "bounces twice", meaning that after the initial bump, the spring should expand, compress, expand again and then gradually settle...

Homework Statement A quantum mechanical harmonic oscillator with resonance frequency ω is placed in an environment at temperature T. Its mean excitation energy (above the ground state energy) is 0.3ħω. Determine the temperature of this system in units of its Einstein-temperature ΘE = ħω/kB...

Homework Statement One possible solution for the wave function ψn for the simple harmonic oscillator is ψn = A (2*αx2 -1 ) e-αx2/2 where A is a constant. What is the value of the energy level En? Homework Equations The time independent Schrodinger wave equation d2ψ / dx2 =...

Hi everyone, I'm dealing with system identification for the first time in my life and am in desperate need of help :) The system is spring-mounted and I'm analyzing the vertical and torsional displacements. However, it seems like the vertical and torsional oscillations are coupled (shouldn't...

I've been looking at a coupled harmonic oscillator, and normal modes of this: http://en.wikipedia.org/wiki/Normal_mode#Example_.E2.80.94_normal_modes_of_coupled_oscillators At the bottom of this example it says: This corresponds to the masses moving in the opposite directions, while the...

Homework Statement Particle mass m is confined by a one dimensional simple harmonic oscillator potential V(x)=Cx2, where x is the displaecment from equilibrium and C is a constant By substitution into time-independant schrodingers with the potential show that \psi(x)=Axe-ax2 is a...