Harmonic oscillator Definition and 743 Threads

This is from *Statistical Physics An Introductory Course* by *Daniel J.Amit* The text is calculating the energy of internal motions of a diatomic molecule. The internal energies of a diatomic molecule, i.e. the vibrational energy and the rotational energy is given by...

See attached photo please. So, I don't get how equations of motion derived. Why is it that x dot is partial derivative of H in term of p but p dot is negative partial derivative of H in term of x.

Homework Statement Show that the virial theorem holds for all harmonic-oscillator states. The identity given in problem 5-10 is helpful. Homework Equations Identity given: ∫ξ2H2n(ξ)e-ξ2dξ = 2nn!(n+1/2)√pi P.S the ξ in the exponent should be raised to the 2nd power. So it should look like ξ2...

Homework Statement Show that application of the lowering Operator A- to the n=3 harmonic oscillator wavefunction leads to the result predicted by Equation (5.6.22). Homework Equations Equation (5.6.22): A-Ψn = -iΨn-1√n The Attempt at a Solution I began by saying what the answer should end...

Homework Statement This is a question asked in a entrance examination[/B] A charged particle is in the ground state of a one-dimensional harmonic oscillator potential, generated by electrical means. If the power is suddenly switched off, so that the potential disappears, then, according to...

If you consider b^2/m > 4*k, you can get the solution by using classic method (b = damping constant, m = mass and k = spring constant) otherwise you have to use complex numbers. How have the references books proved the solution for this differential equation?

Homework Statement An undamped harmonic oscillator (b=0) is subject to an applied force Focos(wt). Show that if w=wo, there is no steady- state solution. Find a particular solution by starting with a solution for w=wo+#, and passing to the limit #->0, it will blow up. Try starting with a...

Hi Q1: I was reading about ultraviolet catastrophe and it was said that atoms were assumed to be harmonic oscillators of radiation. I believe that two harmonic oscillators could have the same frequency but different amplitudes so it would mean that two different atoms (i.e. two harmonic...

I started to ponder following problem. I have a driven, damped oscillator where the mass is free to vibrate in y-direction. If I put a wall or a ground near the mass, the mass touches it if the drive amplitude is larger than the distance to the ground. How does this change the normal dynamics. I...

Hi everyone, I have a great doubt in this problem: Let a mass m with spin 1/2, subject to the following central potencial V(r): V(r)=1/2mω2r2 Find the constants of motion and the CSCO to solve the Hamiltonian? This is my doubt, I can't find the CSCO in this potencial. Is a problem in general...

Homework Statement In Griffiths' book "Introduction to Quantum Mechanics", Section 2.3, Chapter 2, the Fig. 2.7 gives the plots of the wave function (##\psi_{n}##) and its modulus of the harmonics oscillator, see the Appendix. With the order (##n##) increasing, they become both higher. However...

Homework Statement Hi everybody! In my quantum mechanics introductory course we were given an exercise about the 3D quantum harmonic oscillator. We are supposed to write the state ##l=2##, ##m=2## with energy ##E=\frac{7}{2}\hbar \omega## as a linear combination of Cartesian states...

Homework Statement Consider a two-dimensional harmonic oscillator, described by the Hamiltonian ##\hat H_0 = \hbar \omega (\hat a_x \hat a_x ^{\dagger} + \hat a_y \hat a_y^{\dagger} + 1)## Calculate ##\hat H_0 \hat L | n_1, n_2 \rangle## and ##\hat L \hat H_0 |n_1, n_2 \rangle##. What does...

Homework Statement I am having issues with d) and would like to know if I did the a, b, and c correctly. I have tried to look online for explanation but with no success. A harmonic oscillator executes motion according to the equation x=(12.4cm)cos( (34.4 rad /s)t+ π/5 ) . a) Determine the...

Homework Statement The acceleration amplitude of a damped harmonic oscillator is given by $$A_{acc}(\omega) = \frac{QF_o}{m} \frac{\omega}{\omega _o} \sqrt{\it{R}(\omega)}$$ Show that as ##\lim_{\omega\to\infty}, A_{acc}(\omega) = \frac{F_o}{m}## Homework Equations $$\it{R}(\omega) =...

Homework Statement A simple harmonic oscillator has a potential energy V=1/2 kx^2. An additional potential term V = ax is added then, a) It is SHM with decreased frequency around a shifted equilibrium b) Motion is no longer SHM c)It is SHM with decreased frequency around a shifted equilibrium...

Homework Statement Find the first-order corrections to energy and the wavefunction, for a 1D harmonic oscillator which is linearly perturbed by ##H'=ax##. Homework Equations First-order correction to the energy is given by, ##E^{(1)}=\langle n|H'|n\rangle##, while first-order correction to the...

Homework Statement I have ##H'=ax^3+bx^4##, and wish to find the general perturbed wave-functions. Homework Equations First-order correction to the wave-function is given by, $$\psi_n^{(1)}=\Sigma_{m\neq n}\frac{\langle\psi_m^{(0)}|H'|\psi_n^{(0)}\rangle}{n-m}|\psi_m^{(0)}\rangle.$$ The...

Homework Statement Substitute \psi = Ne^{-ax^2} into the position-space energy eigenvalue equation and determine the value of the constant a that makes this function an eigenfunction. What is the corresponding energy eigenvalue? Homework Equations \frac{-\hbar^2}{2m}...

Hello, I'm studying the section 2.2 of "Introduction to Quantum Mechanics, 2nd edition" (Griffiths), and he shows this equation $$\frac{\partial^2\psi}{\partial x^2} = -k^2\psi , $$ where psi is a function only of x (this equation was derivated from the time-independent Schrödinger equation) and...

Homework Statement [/B] What is the shortest time required for a harmonic oscillator to move from ##x = A## to ##x = \frac{A}{2}##? Express your answer in terms of the period ##T##. Homework Equations [/B] ##x(t)=Acos(\omega t)=Acos(2\pi\frac{t}{T})## The Attempt at a Solution ##A=Acos(0)##...

Homework Statement I got an alpha particle (charge 2+) fixed at x=0 and an electron fixed at x=2. I then add a fluor ion (charge 1-) to the right of the electron and we note his position xeq. The question is to find the constant spring (k) relative to the harmonic oscillation made by the fluor...

A quantum mechanical oscillator with the Hamiltonian H1=p^2/2m +(m(w1)^2 x^2)/2 is initially prepared in its ground state (zero number of oscillatory quanta). Then the Hamiltonian changes abruptly (almost instantly): H1→H2=p^2/2m +(m(w2)^2 x^2)/2 What is the mean number of oscillatory quanta...

Homework Statement a mass is placed on a loose spring and connected to the ceiling. the spring is connected to the floor in t=0 the wire is cut a. find the equation of the motion b. solve the equation under the initial conditions due to the question Homework Equations ## \sum F=ma ## ##...

Homework Statement A damped harmonic oscillator consists of a block (m = 2.72 kg), a spring (k = 10.3 N/m), and a damping force (F = -bv). Initially, it oscillates with an amplitude of 28.5 cm; because of the damping, the amplitude falls to 0.721 of the initial value at the completion of 7...

The ladder operators of a simple harmonic oscillator which obey $$[H,a^{\dagger}]=\hbar\omega\ a^{\dagger}$$. --- I would like to see a proof of the relation $$\exp(-iHt)\exp(a^{\dagger})\exp(iHt)|0\rangle=\exp(a^{\dagger}e^{-i\omega t})|0\rangle\exp(i\omega t/2).$$ Thoughts?

Homework Statement Show the mean position and momentum of a particle in a QHO in the state ψγ to be: <x> = sqrt(2ħ/mω) Re(γ) <p> = sqrt (2ħmω) Im(γ) Homework Equations ##\psi_{\gamma} (x) = Dexp((-\frac{mw(x-<x>)^2}{2\hbar})+\frac{i<p>(x-<x>)}{ħ})##The Attempt at a Solution I put ψγ into...

How do you find the wave function Φα when given the Hamiltonian, and the equation: aΦα(x) = αΦα(x) Where I know the operator a = 1/21/2((x/(ħ/mω)1/2) + i(p/(mħω)1/2)) And the Hamiltonian, (p2/2m) + (mω2x2)/2 And α is a complex parameter. I obviously don't want someone to do this question...

Homework Statement How to calculate the matrix elements of the quantum harmonic oscillator Hamiltonian with perturbation to potential of -2cos(\pi x) The attempt at a solution H=H_o +H' so H=\frac{p^2}{2m}+\frac{1}{2} m \omega x^2-2cos(\pi x) I know how to find the matrix of the normal...

Homework Statement The separation between energies of an oxygen molecule is 2061 cm-1 (wavenumber). Treating the molecule as a simple harmonic oscillator whose fundamental frequency is related to its spring constant and reduced mass, calculate the spring constant for an O2 molecule. meff =...

Hello, I encountered a mass on a string problem in which the mass, moved from the equilibrium, gets a harmonic motion. The catch, however, is that the mass of the string is not neglected. On the lecture, the prof. wanted to calculate, for some reason, the complete kinetic energy of the system...

Hello everyone! For my quantum mechanics class I have to study the problem of two quantum oscillator coupled to each other and in particular to find the eigenstates and eigenergies for a subspace of the Fock space. I know that, in general, to solve this kind of problem I have to diagonalize the...

Homework Statement Consider the following potential, which is symmetric about the origin at ##x=0##: ##V(x) = \begin{cases} x^{2}+(x+\frac{d}{2}) &\text{for}\ x < -d/2\\ x^{2} &\text{for}\ -d/2 < x < d/2\\ x^{2}-(x-\frac{d}{2}) &\text{for}\ x > d/2 \end{cases}## Find the ground state energy...

Homework Statement A damped harmonic oscillator is driven by an external force of the form $$F_{ext}=F_0sin(\omega t)$$ Show that the steady state solution is given by $$x(t)=A(\omega)sin(\omega t-\phi)$$ where $$ A(\omega)=\frac{F_0/m}{[(\omega_0^2-\omega^2)^2+4\gamma^2\omega^2]^{1/2}} $$ and...

Homework Statement In ##1+1##-dimensional spacetime, two objects, each with charge ##Q##, are fixed and separated by a distance ##d##. (a) A light object of mass ##m## and charge ##-q## is attached to one of the massive objects via a spring of spring constant ##k##. Quantise the motion of the...

Homework Statement An automobile with a mass of 1000 kg, including passengers, settles 1.0 cm closer to the road for every additional 100 kg of passengers. It is driven with a constant horizontal component of speed 20 km/h over a washboard road with sinusoidal bumps. The amplitude and...

Hi, For a harmonic oscillator in 3D the energy level becomes En = hw(n+3/2) (Note: h = h_bar and n = nx+ny+nz) If I then want the 1st excited state it could be (1,0,0), (0,1,0) and (0,0,1) for x, y and z. But what happens if for example y has a different value from the beginning? Like this...

In the series solution of simple harmonic oscillator,why do we have a factor of 1/2 in the trial solution?

Homework Statement A three-dimensional harmonic oscillator is in thermal equilibrium with a temperature reservoir at temperature T. The average total energy of oscillator is A. ½kT B. kT C. ³⁄₂kT D. 3kT E. 6kT Homework Equations Equipartition theorem The Attempt at a Solution So I know the...

Homework Statement An electron (S=1/2) is free in a spherical symmetric harmonic potential: V(r)=\frac{1}{2}kr^2 a) Find energies and degeneracy of ground state and first excited state. b) For these states find the l^2 and l_z basis. c) How does these states split in a \vec{L} \cdot \vec{S}...

hello :-) here is my problem...: 1. Homework Statement For a linear harmonic oscillator, \hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2} m \omega^2x^2 a) show that the expectation values for position, \bar{x}, and momentum \bar{p} oscillate around zero with angular frequency \omega. Hint...

Hi, I am trying to analyze the an harmonic oscillator using kinematics. first i calculate the force applied by the spring (f = (-x)*k) then i calculate the acceleration (a = f/m) then speed (v= v0 + v0t + 0.5*a*t^2) and finally update x (x = x0+vt) this is a simplfied loop of my program...

Electromagnetic wave behaves like a harmonic oscillator. Similarly a photon behaves like a quantum harmonic oscillator. http://www.physics.usu.edu/torre/3700_Spring_2015/What_is_a_photon.pdf ##dA/dt## and ##A## behaves like ##dx/dt## and ##x## at a harmonic oscillator. I suppose that...

Derivation of energy levels in a quantum harmonic oscillator, ##E=(n+1/2) \hbar\omega##, is long, but the result is very short. At least in comparision with infinite quantum box, this result is simple. I suspect that it can be derived avoiding Hermite polynomials, eigenvalues, etc. I understand...

Homework Statement An isotropic harmonic oscillator has the potential energy function U = 0.5 k (x²+y²+z²). (Isotropic means that the force constant is the same in all three coordinate directions.) (a) Show that for this potential, a solution to the three dimensional time-independent...

Homework Statement Hi everybody! I'm a bit stuck in this problem, hopefully someone can help me to make progress there: A mass point ##m## is under the influence of a central force ##\vec{F} = - k \cdot \vec{x}## with ##x > 0##. a) Determine the equation of motion ##r = r(\varphi)## for the...

Here is the new thread. Please justify your criticism of my statement.

Homework Statement I am supposed to calculate Lyapunov exponent of a damped, driven harmonic oscillator given by ## \ddot{x} + 2\beta \dot{x} + \omega_0^2 x = fcos(\omega t)## Lyapunov exponent is ## \lambda ## in the equation ## \delta x(t) = \delta x_0 e^{\lambda t} ## The attempt at a...

Homework Statement Show that the partition function for the harmonic oscillator with an additional force H = \hbar \omega a^{\dagger} a - F x_0 (a + a^{\dagger}) is given by \frac{e^{\beta \frac{F^2 x_{0}^2}{\hbar \omega}}}{1-e^{\beta \hbar \omega}} and calculate \left<x\right> = x_0...

Homework Statement Experimental data for the heat capacity of N2 as a function of temperature are provided. Estimate the frequency of vibration of the N2 molecule. Homework Equations Energy of harmonic oscillator = (n+1/2)ħω C=7/2kB Average molecular energy = C*T But this is an expression...