Harmonic oscillator Definition and 743 Threads

Homework Statement Given a harmonic oscillator with mass m, and spring constant k, is subject to damping force F= cdx/dt and driven by an external force of the form F[ext]= FoSin(wt). A) Find the steady state solution. B) Find the amplitude and the phase. Homework Equations F=-kx the steady...

I seem to have two approaches that I've seen and understand, but I can't quite see how they relate. 1. Write a general time evolving state as a superposition of stationary states multiplied by their exp(-iEt/h) factors, and calculate <x>. We find that <x>=Acos(wt+b) as in classical physics (in...

Hello, I was just watching a youtube video deriving the equation for the Hamiltonian for the harmonic oscillator, and I am also following Griffiths explanation. I just got stuck at a part here, and was wondering if I could get some help understanding the next step (both the video and book...

Hello, I juste don't know how this was done it is on the solutionnary of a very long exercise and i am not getting this calculation 1. Homework Statement <1,0| ax+ay++ax+ay+axay++axay|0,1> = <1,0|1,0> Homework Equations 3. The Attempt at a Solution We have that |0,1> = ay+ |0,0> I don't...

Homework Statement [/B] The isotropic harmonic oscillator in 2 dimensions is described by the Hamiltonian $$\hat H_0 = \sum_i \left\{\frac{\hat{p_i}^2}{ 2m} + \frac{1}{2} m\omega^2 \hat{q_i}^2 \right\} ,$$ for ##i = 1, 2 ## and has energy eigenvalues ##E_n = (n + 1)\hbar \omega \equiv (n_1 +...

Homework Statement I must find the average number of energy levels of quantum harmonic oscillator at temperature T, and the answer is given as I must use Boltzmann distribution and the sum of geometric progression. For finding the average value I must use the equation <F>=trace(F*rho)...

Homework Statement A mass m sits on a horizontal frictionless surface and is attached to a wall by means of a spring having force constant k. The mass is now subjected to an additional force of the form. F(t) = Acosbt (a) Write the equation of motion for this mass.(b) What is the solution to...

1. After finding out that the wave function ##\Psi(z) \sim Ae^{\frac{-z^{2}}{2}}## in the limit of plus or minus infinity Griffiths separates the function into two parts ##\Psi(z)=h(z)e^{\frac{-z^{2}}{2}}## My question will be about a certain aspect of the function ##h(z)## After solving the...

I'm a bit confused wether or not there is a link between harmonic functions (solutions of the Laplace pde) and harmonic oscillating systems? What is the meaning of "harmonic" in these cases? Thanks!

My problem is described in the animation that I posted on Youtube: For the sake of convenience I am copying here the text that follows the animation: I have made this animation in order to present my little puzzle with the quantum harmonic oscillator. Think about a classical oscillator, a...

Homework Statement An oscillator with mass 0.5 kg, stiffness 100 N/m, and mechanical resistance 1.4 kg/s is driven by a sinusoidal force of amplitude 2 N. Plot the speed amplitude and the phase angle between the displacement and speed as a function of the driving frequency and find the...

when considering the quantum harmonic oscillator, you get that the wave function takes the form psi=ae^{-\frac{m\omega}{2\hbar}x^2} I have been trying to integrate \psi ^2 to find the constant a so that the wave function is normalised, and I know the trick with converting to polar coordinates...

Homework Statement Given the coherent state of the harmonic oscillator |z>=e^{-\frac{|z|^2}{2}}\sum_{n=0}^\infty\frac{z^{n}}{\sqrt{n!}}|n> compute the probability for finding n quanta in the sate |z> and the average excitation number <z|n|z>Homework Equations...

for harmonic oscillator, V(x) = 1/2*m*w^2*x^2. here, the spring can be stretch or compress. however, is if the spring can only stretch such that V(x) is infinity for x<0, then find energy level for this setup. I don't understand the part about spring only being able to stretch. what does that...

Homework Statement There is a harmonic oscillator with charge q and sudenly we turn on external electric field E, which direction is the same as oscillator's. We need to find probability, that particles energy calculated in electric field will be in m state. n=1, m=2 2. Homework Equations The...

Homework Statement One end of a light hacksaw blade is clamped in a vise with the long axis of the blade horizontal and with the sides vertical. A 0.665- kg mass is attached to the free end. When a steady sideways force of 20.5 N is applied to the mass it moves aside 13.3 cm from its...

Homework Statement mass "m" object is located near the origin of the coordinate system. External force was exerted on the object depending on the coordinate by the formula of Fx=-4*sin(3*pi*x). Find the short oscillation period, and the potential energy depending on the coordinate. Homework...

Homework Statement The problem requires to solve the integration to find ## G(t) ## after ##G(\omega)## is found via Fourier transform. We have G(\omega)= \frac{1}{2\pi}\frac{1}{\omega _{0}^2 - \omega ^2} Homework Equations As mentioned previously, the question asks to find ##G(t)## The...

Hey all, I want to preface this with I wasn't sure if I should put this in the physics thread since it is quantum mechanics or in chemistry thread since this is for my Physical Chemistry course. I will gladly move the thread if the community feels it is more appropriate in the physics thread. I...

Homework Statement A particle of mass m in the harmonic oscillator potential V(x) = (mω2x2)/2 is described at time t = 0 by the wavefunction χ(x, t = 0) = 1/[(2πσ2)1/4] exp[-x2/(4σ2)] What is <E> at time t? Homework EquationsThe Attempt at a Solution <T>+<V>= <E> I've found the expectation...

I have the following homework problem that I am having trouble with. Any guidance would be appreciated. Thank you in advance. Consider an object hanging on a spring, immersed in a cup of water. The water exerts a linear viscous force -bv on the object, where v is the speed of the object...

Hi people! Today I was doing some QFT homework and in one of them they ask me to calculate the Harmonic Oscillator propagator, which, as you may know is: W(q_2,t_2 ; q_1,t_1) = \sqrt{\frac{m\omega}{2\pi i \hbar \sin \omega (t_2-t_1)}} \times \exp \left(\frac{im\omega}{2\hbar \sin \omega...

This isn't homework. I'm reviewing physics after many years of neglect. Since a simple harmonic oscillator is a conservative system with no energy losses, then a driven undamped harmonic oscillator, once the transient solution has died out, can't be receiving any energy from the driving...

Homework Statement A harmonic oscillator of mass m and angular frequency ω experiences the potential: V(x) = 1/2mω^{2}x^{2} between -infinity < x < +infinity and solving the schrodinger equation for this potential yields the energy levels E_n = (n + 1/2)...

Homework Statement The spin 1/2 electrons are placed in a one-dimensional harmonic oscillator potential of angular frequency ω. If a measurement of $$S_z$$ of the system returns $$\hbar$$. What is the smallest possible energy of the system? Homework Equations...

Homework Statement The Hamiltonian for the 3-D harmonic oscillator in spherical polar coordinates is given in the question.The question then asks : using the trial wavefunction ##ψ=e^(-αr) ## show that Homework Equations ##<ψ|H|ψ>/<ψ|ψ> = (\hbarα)^2/2m + 3mω^2/2α^2## The following...

(I am referring to section 3.1 in Burkhardt's "Foundations of Quantum Physics", if you happen to have the book.) In that book it's pointed out that the apparent contradiction between the pdf's of the QM ground state solution to the harmoinc oscillator with its classical conterpart (at the...

Hello Forum, The 1D harmonic oscillator is an important model of a system that oscillates periodically and sinusoidally about its equilibrium position. The restoring force is linear. There is only one mode with one single frequency omega_0 (which is the resonant frequency). What about the...

Dear kind helpers, actually I am not 100% sure whether this is the right place to post, as it is not a homework in the sense of an exercise sheet. But I think it could be because it feels pretty basic and that I should be able to solve it. Though I really searched for a solution but could not...

if we have two non-interacting particles of mass M in a one-dimensional harmonic oscillator potential of frequency ω, with the wavefunction defined as: $$\Psi\left(x_1,x_2\right) = \psi_n\left(x_1\right) \psi_m\left(x_2\right)$$ where x_1 and x_2 are two particle co-ordinates. and ψ_n is the...

Homework Statement The e-functions for n=0,1,2 e-energies are given as psi_0 = 1/(pi^1/4 * x0^1/2)*e^(x^2/(2*x0^2) psi_1 =... psi_2 =... The factor x0 is instantaneously changed to y= x0/2. This means the initial wavefunction does not change. Find the expansions coefficients of the...

Definition/Summary An object (typically a "mass on a spring") which has a position (or the appropriate generalization of position) which varies sinusoidally in time. Equations x(t)=A\sin(\omega t)+B\cos(\omega t) \omega^2 =\frac{k}{m} Extended explanation According to...

Definition/Summary This is the quantum-mechanical version of the classical harmonic oscillator. Like the classical one, the quantum harmonic oscillator appears in several places, and it also appears in the quantization of fields. This article will discuss the one-dimensional version, but it...

I feel I understand what happens, and how to solve the equation of motion x(t) for a mass attached to a spring and released from rest horizontally on a smooth surface. We typically end up with x(t) = x_0 cos(ωt) as the solution, with x_0 as the amplitude of the oscillation. But I've...

Homework Statement A two-dimensional isotropic harmonic oscillator of mass μ has an energy of 2hω. It experiments a perturbation V = xy. What are its energies and eigenkets to first order? Homework Equations The energy operator / Hamiltonian: H = -h²/2μ(Px² + Py²) + μω(x² + y²) The...

I am working through Leonard Susskinds 'the theoretical minimum' and one of the exercises is to show that H=ω/2(p^2+q^2). The given equations are H=1/2mq(dot)^2 + k/2q^2, mq(dot)=p and ω^2=k/m. q is a generalisation of the space variable x, and (dot) is the time derivative if this helps...

Homework Statement Find the expectation values of x and p for the state \vert \alpha \rangle = e^{-\frac{1}{2}\vert\alpha\vert^2}exp(\alpha a^{\dagger})\vert 0 \rangle, where ##a## is the destruction operator. Homework Equations Destruction and creation operators ##a=Ax+Bp##...

Homework Statement For a particle of charge ##q## in a potential ##\frac{1}{2}m\omega^2x^2##, the wavefunction of ground state is given as ##\phi_0 = \left( \frac{m\omega }{\pi \hbar} \right)^{\frac{1}{4}} exp \left( -\frac{m\omega}{2\hbar} x^2 \right)##. Now an external electric field ##E##...

Homework Statement Homework Equations The Attempt at a Solution I find this task very hard to understand. First of all, when adding more mass, wouldn't that change the acceleration, according to F=ma? And in that case the velocity should also change when adding more mass, shouldn't it? That...

Homework Statement In a port the tide and the low tide change with harmonic motion. at the high tide the water level is 12 meters and at the low tide it is 2 meters. between the tide and the low tide there are 6 hours. A ship needs 8 meters of water depth. how long can it stay in the port...

Homework Statement A mass of 0.1 kg has 2 springs of length 20 cm attached to each side, like in the drawing. they are loose. one has a constant of 50 [N/m] and the other 30. The system is between 2 walls 10 cm distant from each spring. At the second stage the springs are tied each to the...

Homework Statement Particle originally sits in ground state about x=0. Equilibrium is suddenly shifted to x=s. Find probability of particle being in new first excited state. Homework Equations The Attempt at a Solution Shifted wavefunctions are for ground state: ##\phi'_0 =...

Hello readers, Given the potential V(x) = - 1/ sqrt(1+x^2) I have found numerically 12 negative energy solutions Now I want to try to solve for these using matrix mechanics I know the matrix form of the harmonic oscillator operators X_ho, P_ho. I believe I need to perform the...

The energy changes correspond to infrared, h_bar * w. Which particles are actually oscillating? The neutrons or the electrons? Is it the electrons that fill up the stationary states, electronic configuration, or is it the nucleons that fill up the states?

A simple harmonic oscillator has total energy E= ½ K A^2 Where A is the amplitude of oscillation.  E= KE+PE a) Determine the kinetic and potential energies when the displacement is one half the amplitude. b) For what value of the displacement does the kinetic energy equal the potential...

Homework Statement 2N fermions of mass m are confined by the potential U(x)=1/2(k)(x2) (harmonic oscillator) What is the ground state energy of the system? Homework Equations V(x)=1/2m(ω2)(x2) The Attempt at a Solution I know the ground state energy of a simple harmonic...

Homework Statement Part (a): Derive Ehrenfest's Theorem. What is a good quantum number? Part (b): Write down the energy eigenvalues and sketch energy diagram showing first 6 levels. Part (c): What's the symmetry of the new system and what happens to energy levels? Find a new good quantum...

In the course of solving the simple harmonic oscillator, one reaches a fork in the road. x(t) = A1Sin(wt) + A2Cos(wt) At this point, you exploit a trig identity and arrive at one of two solutions x(t) = B1Sin(wt+phi1) or x(t) = B2Cos(wt+phi2) Both of these are correct solutions...

Given the equations for the harmonic oscillator $\frac{dy}{dz}=z, \frac{dz}{dt}= -y$if the system is approximated by the symplectic Euler method, then it gives$z_{n+1}= z_{n}-hy_{n}, \\ y_{n+1}= y_{n}+hz_{n+1}$which shows that the circle $y^2_{n} + z^2_{n} = 1$ is mapped into an ellipse...

Hi folks! Apparently \Psi(x) = Ax^ne^{-m \omega x^2 / 2 \hbar} is an approximate solution to the harmonic oscillator in one dimension -\frac{\hbar ^2}{2m} \frac{d^2\psi}{dx^2} + \frac{1}{2}m \omega ^2 x^2 \psi = E \psi for sufficiently large values of |x|. I thought this...