Harmonic oscillator Definition and 699 Threads

Homework Statement In the time interval (t + δt, t) the Hamiltonian H of some system varies in such a way that |H|ψi>| remains finite. Show that under these circumstances |ψi> is a continuous function of time. A harmonic oscillator with frequency ω is in its ground state when the stiffness of...

In the attached file, I have formulated a simple one dimensional harmonic oscillator and solved the model numerically. Such a model might represent a simple reaction coordinate along which a liquid drop actinide nucleus might split after absorbing a neutron. Clearly the complete model involves...

Pleae help me. a),b),c) was already solved. but question d) is not.

uncertainty relation. I think I'm on the right track. Currently, I'm at, E = (1/2m)*<p^2> + (1/2)*k*<x^2> and when applying the uncertainty relation, deltax = <x^2>^(1/2) deltap = <p^2>^(1/2) How do I go about connecting everything from here? Thanks!

Homework Statement A particle of mass m that is confined to a harmonic oscillator potential V(x) = \frac{1}{2} m \omega^2 x^2 is described by a wave packet having the probability density, |\Psi (x,t) |^2 = \left(\frac{m\omega}{\pi\hbar} \right )^{1/2}\textrm{exp}\left[-\frac{mw}{\hbar}(x -...

Homework Statement Verify that the ground state (n=0) wavefunction is an eigenstate of the harmonic oscillator Hamiltonian. Using the explicit wavefunction of the ground state to calculate the average potential energy <0|\hat{v}|0> and average kinetic energy <0|\hat{T}| 0> Homework...

1. Homework Statement Show that bohr's hypothesis (that a particle's angular momentum must be an integer multiple of h/2pi) when applied to the three dimensional harmonic oscillator, predicts energy levels E=lh/pi w with l = 1,2,3. Is there an experiment that would falsify this prediction...

The TISE can be written as -\frac{\hbar^{2}}{2m}\frac{d^{2}u}{dx^{2}} + \frac{1}{2}m\omega_{0}^{2}x^{2}u = Eu Now my lecture notes say that it is convenient to define scaled variables y = \sqrt{\frac{m\omega_{0}}{\hbar} x} and \alpha = \frac{2E}{\hbar\omega_{0}} Hence \frac{d}{dx} =...

It is well known that for an isolated system, the normal mode frequency of a N-body harmonic oscillator satisfies Det(T-\omega^{2}V)=0. How about a non-isolated, fixed temperature system? In solid state physics I have learned that in crystal the frequency does not change, but the amplitude of...

Homework Statement A particle in the ground state of the harmonic oscillator with classical frequency \omega, when the spring const quadruples (so \omega^{'}=2\omega) without initially changing the wave function. What is the probability that a measurement of the energy would still return the...

Homework Statement The pendulum of a grandfather clock has a period of 1s and makes excursions of 3cm either side of dead centre. Given that the bob weighs 0.2kg, around what value of n would you expect its non negligible quantum amplitudes to cluster? Homework Equations [/B] The...

Hi! Info: This is a rather elementary question about the creation a(+) and annihilation (a-) operators for the 1D H.O. The problem is to calculate the energy shift for a given state if the weak perturbation is proportional to x⁴. Using first order perturbation theory for the...

Hello, Attached are two problems I can not solve, thanks for the help. The Attempt at a Solution For the first question, I understand that I need insert A1coswt+A2sinwt into the homogenous equation , but don't know what's then .. But I'm pretty much lost on both of em :(

Homework Statement Ok so the question is, is the state u(x) = Bxe^[(x^2)/2] an energy eigenstate of the system with V(x) = 1/2*K*X^2 and what is the probability per unit length of this state.Homework Equations The Attempt at a Solution So the way i did this was, to find if the state is an...

A particle of mass m moves along the x-direction such that V(x)=½Kx^2. Is the state u(¥)=B¥exp(+¥2/2), where ¥ is Hx (H = constant), an energy eigenstate of the system?. What is probability per unit length for measuring the particle at position x=0 at t=t0>0?

Homework Statement Given that a^+|n\rangle=\sqrt{n+1}|n+1\rangle a|n\rangle=\sqrt{n}|n-1\rangle and that the other eigenstates |n> are given by |n\rangle=\frac{(a^+)^n}{\sqrt{n!}}|0\rangle where |0> is the lowest eigenstate. Define for each complex number z the coherent state...

Imagine a fictitious universe where springs want to stretch: the spring force is proportional to, and in the same direction as, displacement from equilibrium. We'll call these anti-springs. (a) Set up a differential equation modeling the motion of a damped anti-spring if the mass is m = 1 kg...

Homework Statement Given the Hamiltonian for the harmonic oscillator H=\frac{p^2}{2m}+\frac{1}{2}m\omega^2 x^2 , and [x,p]=i\hbar . Define the operators a=\frac{ip+m\omega x}{\sqrt{2m\hbar \omega}} and a^+=\frac{-ip+m\omega x}{\sqrt{2m\hbar \omega}} (1) show that [a,a^+]=1 and that...

My answer would be "yes," and here's my argument: If we let H = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + \frac 12 m \omega^2 x^2, it is a Hermitian operator with familiar normalized eigenfunctions \phi_n(x) (these are products of Hermite polynomials and gaussians) and associated...

Hi, I don't understand why the momentum probability distribution of the quantum mechanical oscillator has the same shape as the position probability distribution (with peaks at the extremes), I mean, I understand the mathematics but I don't understand the concept. This is my reasoning (which...

Homework Statement Show that G(q_2,q_1;t)=\mathcal{N}\frac{e^{iS_{lc}}}{\sqrt{\det A}} where \mathcal{N} is a normalization factor independent of q1, q2, t, and w. Using the known case of w=0, write a formula for G such that there is no unknown normalization factor. Homework Equations I...

Homework Statement What is the probability that a particle in the ground state of a simple harmonic oscillator 1-D potential will be found outside the region accessible classically Homework Equations ∫(between 1 and infinity) e^(-y^2 ) dy=0.08π^(1/2) I feel like it's quite a...

Homework Statement A particle is moving in a simple harmonic oscillator potential V(x)=1/2*K*x^2 for x\geq0, but with an infinite potential barrier at x=0 (the paddle ball potential). Calculate the allowed wave functions and corresponding energies.Homework Equations I am thinking that the...

For part of my lab write up on pendulum motion, my professor wanted us to find out why a pendulum was not a simple harmonic oscillator, and under what conditions it was. He also wanted to show this mathematically. So far what I have is that if there is no damping(friction?) and if the the...

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...