Harmonic oscillator Definition and 743 Threads

Homework Statement On June 10, 2000, the Millennium Bridge, a new footbridge over the River Thames in London, England, was opened to the public. However, after only two days, it had to be closed to traffic for safety reasons. On the opening day, in fact, so many people were crossing it at the...

Hi. As far as I know, the movement of a harmonic oscillator normally is not considered to be chaotic. Why not? Since the angular frequency can never be known to absolute precision, an error in the phase builds up. I can see that this build-up is only linear in time (if we assume the angular...

Homework Statement Homework Equations Find Time Period. Find the error in my solution. The Attempt at a Solution Where i am wrong ?

Homework Statement Consider a particle with mass m oscillates in a simple harmonic potential with frequency ω. The position, x, and momentum operator, p, of the particle can be expressed in terms of the annihilation and creation operator (a and a† respectively): x = (ħ/2mω)^0.5 * (a† + a) p =...

Homework Statement Spring with spring constant k=2000N/m has an object with mass 10kg attached to it. When it is pulled 0.1m away from the equilibrium state it starts oscillating and came to a stop. The coefficient of kinetic friction is 0.2 and the coefficient of static friction is 0.5. Find...

Homework Statement consider any damped harmonic oscillator equation m(d2t/dt2 +bdy/dt +ky=0 a. show that a constant multiple of any solution is another solution b. illustrate this fact using the equation (d2t/dt2 +3dy/dt +2y=0 c. how many solutions to the equation do you get uf you use this...

Hi, why there is only odd eigenfunctions for a 1/2 harmonic oscillator where V(x) does not equal infinity in the +ve x direction but for x<0 V(x) = infinity. I understand that the "ground state" wave function would be 0 as when x is 0 V(x) is infinity and therefore the wavefunction is 0, and...

Homework Statement Due to the radial symmetry of the Hamiltonian, H=-(ħ2/2m)∇2+k(x^2+y^2+z^2)/2 it should be possible to express stationary solutions to schrodinger's wave equation as eigenfunctions of the angular momentum operators L2 and Lz, where...

Homework Statement Part d) of the question below. Homework Equations We are told NOT to use the ladder technique to find the position operator as that's not covered until our Advanced Quantum Mechanics module next year (I don't even know this technique anyway). I emailed my tutor and he...

Homework Statement The wave function for the three dimensional oscillator can be written ##\Psi(\mathbf r) = Ce^{-\frac{1}{2}(r/r_0)^2}## where ##C## and ##r_0## are constants and ##r## the distance from the origen. Calculate a) The most probably value for ##r## b) The expected value of ##r##...

So this is something that troubled me a bit- in Shankar's PQM, there's an exercise that asks you to find the position expectation value for the harmonic oscillator in a state \psi such that \psi=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle) Where |n\rangle is the n^{th} energy eigenstate of...

For a 1D QHO we are given have function for ##t=0## and we are asked for expectation and variance of P at some time t. ##|\psi>=(1/\sqrt 2)(|n>+|n+1>)## Where n is an integer So my idea was to use Dirac operators ##\hat a## and ##\hat a^\dagger## and so I get the following solution ##<\hat...

Homework Statement The problem is attached Homework Equations f=2π/ω=2π√(m/k) The Attempt at a Solution My idea is that the mass doubles resulting in a √2 increase in the equation above. However, apparently the answer is (c). I have a strong feeling the book answer is wrong, but I wanted to...

Homework Statement A particle with a mass(m) of 0.500kg is attached to a horizontal spring with a force constant(k) of 50.0N/m. At the moment t=0, the particle has its maximum speed of 20m/s and its moving to the left. Find the minimum time interval required for the particle to move from...

When I work out $$b^+b$$, I get $$\widehat{b^+} \widehat{b} = \frac{1}{2} (ξ - \frac{d}{dξ})(ξ + \frac{d}{dξ}) = \frac{1}{2} (ξ^2 - \frac{d^2}{dξ^2}) = \frac{mωπx^2}{h} - \frac{h}{4mωπ} \frac{d^2}{dx^2}$$ So base on what I have about, (9) should be $$(9) = \frac{hω}{2π} (\frac{1}{2}...

Homework Statement I am trying to obtain the hermite polynomial from the schrødinger equation for a har monic oscillator. My attempt is shown below. Thank you! The derivation is based on this site: http://www.physicspages.com/2011/02/08/harmonic-oscillator-series-solution/ The Attempt at a...

Homework Statement In the exercise, we solved the 2D Harmonic Oscillator in kartesian (x,y) and polar (r,φ) coordinates. We found out that both have the same energy levels, but they look very different, when I plot them. What am I missing? The polar solution seems more like it. Homework...

Homework Statement Consider a linear harmonic oscillator with the solution defined by the ladder operators a and a†. Use the number basis |n⟩ to do the following. a) Construct a linear combination of |0⟩ and |1⟩ to form a state |ψ⟩ such that ⟨ψ|X|ψ⟩ is as large as possible. b) Suppose that...

Hi i want to find propagator of inverse harmonic oscillator to find time dependent wave function, but I can't have any ideas about this. Is it possible to help me to find it Thanks

hy physics forum, I am doing an advanced qm course, but i still have some doubts about the correct formalism and so on.. so i have this problem of a one-dimensional harm. oscillator including 2 bosons. the hamiltonian would thus be ## \hat{H}=1/2 (\hat{p_1}^2+\hat{p_2}^2+\omega...

Homework Statement A linear harmonic oscillator with frequency ω = hbar / M is at time t = 0 in the state described by the wave-function: Ψ(x,0) = C( 1 + √2x) e-x2/2 Determine the values of energy which can be measured in this state. I'm not really sure where to start this question and was...

I studied this from Griffith Chapter 2, with the algebraic (raising and lowering operator) method, we reached the ground state by setting a_Ψ0 = 0 , then we got what the ground state is, and then plugged it in the Schrodinger equation to know the energy, and it turned out to be 0.5 ħω. My...

Homework Statement Consider the one-dimensional harmonic oscillator of frequency ω0: H0 = 1/2m p2 + m/2 ω02 x2 Let the oscillator be in its ground state at t = 0, and be subject to the perturbation Vˆ = 1/2 mω2xˆ2 cos( ωt )at t > 0. (a) Identify the single excited eigenstate of H0 for...

Homework Statement Show that for the one-dimensional linear harmonic oscillator the Hamiltonian is: [; H = \frac{1}{2}[P^2+\omega ^2 X^2]-\frac{1}{2}\omega \hbar ;] [; =\frac{1}{2}[P+i\omega X][P-i\omega X]+\frac{1}{2} \omega \hbar ;] where P, X are the momentum and position operators...

Homework Statement At t=0 the wave function of a two-dimensional isotropic harmonic oscilator is ψ(x,y,0)=A(4α^2 x^2+2αy+4α^2 xy-2) e^((-α^2 x^2)/2) e^((-α^2 y^2)/2) where A its the normalization constant In which instant. Wich values of total energy can we find and which probability...

Homework Statement For a quantum harmonic oscillator in an electric field, using ##\hat{V}=q\epsilon\hat{x}##, with the following trial state: $$|\psi\rangle=|0\rangle+b|1\rangle$$ Show that the energy can be written as $$E=\frac{\frac{\hbar...

Homework Statement The Hamiltonian is given by: H = \frac{1}{2} \sum_{i=1,2}[p_i^2 + q_i^2] We define the following operators: J = \frac{1}{2} (a_1^+ a_1 + a_2^+ a_2) J_1 = \frac{1}{2} (a_2^+ a_1 + a_1^+ a_2) J = \frac{i}{2} (a_2^+ a_1 - a_1^+ a_2) J = \frac{1}{2} (a_1^+ a_1 - a_2^+...

The ground state wave-function of a 1-D harmonic oscillator is $$ \psi(x) = \sqrt\frac{a}{\sqrt\pi} * exp(-\frac{a^2*x^2}{2}\frac{i\omega t}{2}). $$ a) find Average potential energy ? $$ \overline{V} = \frac{1}{2} \mu\omega^2\overline{x^2} $$ b) find Average kinetic energy ? $$ \overline{T} =...

Hello. I'm studying quantization of electromagnetic field (to see photon!) and on the way to reach harmonic oscillator Hamiltonian as a final stage, sudden transition that the Fourier components of vector potential A become quantum operators is observed. (See...

Homework Statement A particle in SHM is subject to a driving force F(t)= ma*e^(-jt). Initial position and speed equal 0. Find x(t). Homework Equations F = -kxdx = mvdv F(t) = F(0)*e^(iωt) x(t) = Acos (ωt +φ) The Attempt at a Solution I have no idea how to deal with the exponential term. I...

Homework Statement The problem asked me to derive an expression for the stationary wave function of the 3d harmonic oscillator which I have done. It then tells me a particle is in the stationary state $$\psi_{n_x,n_y,n_z}(x,y,z)=\psi_{100}(x,y,z)$$ and to express this in spherical coordinates...

I was wondering how to derive the sinusoidal equation for the simple armonic oscillator. But I am currently trying to understand this step in this webpage: I don't get where do P and Q come from and why it is summing pe^iwt + qe^-iwt. please I need some help. The rest of it pretty much makes...

Hi everyone I was wondering, why is vacuum energy related to the zero point energy of an harmonic oscillator? The hamiltonian of an harmonic oscillator is $H= \frac{p^{2}}{2m} + \frac{1}{2} \omega x^{2}$. Where does the harmonic potential term come from in the vacuum?

Homework Statement We have the lagragian L = \frac{m}{2} \dot{x}^2 - \frac{m \omega x^2}{2} + f(t) x(t) where f(t) = f_0 for 0 \le t \le T 0 otherwise. The only diagram that survives in the s -matrix expansion when calculating <0|S|0> is D = \int dt dt' f(t)f(t') <0|T x(t)x(t')|0>...

Hi, I am trying to find the wavefunction of a coherent state of the harmonic oscillator ( potential mw2x2/2 ) with eigenvalue of the lowering operator: b. I know you can do this is many ways, but I cannot figure out why this particular method does not work. It can be shown (and you can find...

Homework Statement Using the equations that are defined in the 'relevant equations' box, show that $$\langle n' | X | n \rangle = \left ( \frac{\hbar}{2m \omega} \right )^{1/2} [ \delta_{n', n+1} (n+1)^{1/2} + \delta_{n',n-1}n^{1/2}]$$ Homework Equations $$\psi_n(x) = \left ( \frac{m...

Hi, What is the physical meaning of zero probability of finding a particle in the square of the Quantum SHO wave function? the particle is supposed to oscillate about the equilibrium position, how would it go from an end point to the other end point without passing by certain points? Could the...

Hey, I'm stuck halfway through the solution it seems. I could use some tips on how to continue. 1. Homework Statement I have to determine a linear combination of the states |0\rangle, |1\rangle, of a one dimensional harmonic oscillator, so that the expectation value \langle x \rangle is a...

Homework Statement What is the normalized ground state energy for the 3-D Harmonic Oscillator Homework Equations V(r) = 1/2m(w^2)(r^2) The Attempt at a Solution I started with the wave fn in spherical coordinates, and have tried using sep of variables, but keep getting stuck when trying to...

I was wondering whether it's possible to write a code in C++ that would be able to solve the Harmonic Oscillator: \ddot{x} + \gamma x = F_{external}(t,x) With different F function inputs... I thought about creating a function with if clauses, so for different inputs by the user, the force F...

Homework Statement "Show that the Hermite polynomials generated in the Taylor series expansion e(2ξt - t2) = ∑(Hn(ξ)/n!)tn (starting from n=0 to ∞) are the same as generated in 7.58*." 2. Homework Equations *7.58 is an equation in the book "Introductory Quantum Mechanics" by...

Given the half harmonic potential: \begin{equation}V=\begin{cases}1/2\omega^2mx^2 & x > 0\\\infty & x < 0\end{cases}\end{equation}What will be the Hamiltonian of the half oscillator?I understand that for x>0 the Hamiltonian will be...

I was wondering why the average kinetic energy is calculated in this manner. They are dividing the kinetic energy function by the period (noted here as To) and integrating it with limits from 0→To. Why? (This is a tiny portion of a larger textbook example, not an actual assignment.) Sorry for...

Homework Statement The transition amplitude for the harmonic oscillator may be written as ##\langle x_2, t_2 | x_1, t_1 \rangle = N_{\omega}(T) \exp(i/\hbar S_{cl})##, where ##T=t_2-t_1## and ##S_{cl}## is the classical action. Let the wave function at ##t=0## be ##\psi(x,o) =...

Homework Statement A particle of mass m in the one-dimensional harmonic oscillator is in a state for which a measurement of the energy yields the values ##\hbar\omega/2## or ##3\hbar\omega/2## each with a probability of one-hald. The average value of the momentum ##\langle p_x\rangle## at...

For the harmonic oscillator in 1-D we get the 2nd time derivative of the x Heisenberg operator = -ω2 x. When that is integrated it gives xH (t) = Acos(ω t) +Bsin (ω t) where A and B are time independent operators. My question is why are the constants A and B incorporated into the terms as a...

I'm trying to figure out how to derive the equations for Energy from the differential equation corresponding to the (simple and damped) harmonic oscillator. Please note that I don't want to start with the expressions for kinetic and potential energy, I want to derive them. The references that I...

Hello, I'm was going through the simple harmonic oscillator, just as a recap, and I stumbled upon something which is causing me wonder. I'm solving the SHO with a shifted origin, and so I have the differential equation F=-k(x-x_0) \ddot{x}=-\frac{k}{m}x+\frac{kx_0}{m} Now, I get that I can...

Homework Statement Hello, I'm just curious as to whether I'm going about solving the following problem correctly... Problem Statement: A particle mass m and charge q is in the ground state of a one -dimensional harmonic oscillator, the oscillator frequency is ω_o. An electric field ε_o is...

Homework Statement Find the initial conditions for 2 interchangeable harmonic oscillators (undamped) so that they have the same amplitude of oscillation. Homework Equations x(t)=Xm*cos(wt+ϕ)The Attempt at a Solution The amplitude of the function is given by Xm so I would have thought that just...