Harmonic oscillator Definition and 699 Threads

As far as I know we can express the position and momentum operators in terms of ladder operators in the following way $${\begin{aligned}{ {x}}&={\sqrt {{\frac {\hbar }{2}}{\frac {1}{m\omega }}}}(a^{\dagger }+a)\\{{p}}&=i{\sqrt {{\frac {\hbar }{2}}m\omega }}(a^{\dagger }-a)~.\end{aligned}}.$$...

In the paper below I've seen a new method to solve the quantum harmonic oscillator Introduction to the Spectrum of N=4 SYM and the Quantum Spectral Curve It is done using the concept of quasi momentum defined as $$p = - i \frac{d(\log \psi)}{dx}$$ See pg 7,8 Is this well know? is it discussed...

I posted yesterday but figured it out; however, a different issue I just detected with the same code arose: namely, why does the solution damp here for an undamped simple harmonic oscillator? I know the exact solution is ##\cos (5\sqrt 2 t)##. global delta alpha beta gamma OMEG delta =...

Dear PF community, I am back with a question :) The solutions for the quantum harmonic oscillator can be found by solving the Schrödinger's equation with: Hψ = -hbar/2m d²/dx² ψ + ½mω²x² ψ = Eψ Solving the differential equation with ψ=C exp(-αx²/2) gives: -hbar/2m (-α + α²x²)ψ + ½mω²x²ψ = Eψ...

The wavefunction is Ψ(x,t) ----> Ψ(λx,t) What are the effects on <T> (av Kinetic energy) and V (potential energy) in terms of λ? From ## \frac {h^2}{2m} \frac {\partial^2\psi(x,t)}{\partial x^2} + V(x,t)\psi(x,t)=E\psi(x,t) ## if we replace x by ## \lambda x ## then it becomes ## \frac...

Hey, I solved a problem about a double pendulum and got 2 euler-lagrange equations: 1) x''+y''+g/r*x=0 2) x''+y'' +g/r*y=0 (where x is actually a tetha and y=phi) the '' stand for the 2nd derivation after t, so you can see the basic harmonic oscillator equation with a term x'' or y'' that...

I'm reading through Lancaster & Blundell's Quantum Field Theory for the Gifted Amateur and have got to Chapter 17 on calculating propagataors. In their equation 17.23 they derive the expression for the free Feynman propagator for a scalar field to be...

As we see in this Phet simulator, this is only the real part of the wave function, the frequency decreases with the potential, so lose energy as moves away the center. we se this real-imaginary animation in Wikipedia, wave C,D,E,F. Because with less energy, the frequency of quantum wave...

I know that due to causality g(t-t')=0 for t<t' and I also know that for t>t', we should get g(t-t')=\frac{sin(\omega_0(t-t'))}{\omega_0} But I can't seem to get that to work out. Using the Cauchy integral formula above, I take one pole at -w_0 and get \frac{ie^{i\omega_0(t-t')}}{2\omega_0} and...

I tried by taking the derivative of the potential to find the critic points and the I took the second derivative to find which of those points are minimum points. I found that the point is ##x=- a##. I don't understand how to calculate the period, since I haven't seen anything about the harmonic...

I was reviewing the harmonic oscillator with Sakurai. Using the annihilation and the creation operators ##a## and ##a^{\dagger}##, and the number operator ##N = a^{\dagger}a##, with ##N |n \rangle = n | n \rangle##, he showed that ##a | n \rangle## is an eigenstate of ##N## with eigenvalue ##n -...

Homework Statement I'm trying to reconcile the answers to two questions regarding the average potential and kinetic energies in simple harmonic oscillator Question 1: The average potential energy of the vibrational motion in the ground state of a diatomic molecule is 12 meV. The average...

Greg Bernhardt submitted a new blog post An Accurate Simple Harmonic Oscillator Laboratory Continue reading the Original Blog Post.

Homework Statement The problem is from the Monbukagakusho exam.[/B] An object of mass M is hanging by a light spring of force constant k from the ceiling. A small ball of mass m which moves vertically upward collides with the object. After the collision, the object and the small ball stick...

Homework Statement Using the Schrödinger equation find the parameter \alpha of the Harmonic Oscillator solution \Psi(x)=A x e^{-\alpha x^2} Homework Equations -\frac{\hbar^2}{2m}\,\frac{\partial^2 \Psi(x)}{\partial x^2} + \frac{m \omega^2 x^2}{2}\Psi(x)=E\Psi(x) E=\hbar\omega(n+\frac{1}{2})...

When defining quantum fields as a sum of creation and annihilation operators for each momenta, we do it in analogy with the simple example of the harmonic oscillator in quantum mechanics. But why do we assume that the coefficients in the expansion can be interpreted in the same way as in the...

Dear all, I am aware that a weakly driven dipole can be modeled as a damped driven simple harmonic oscillator. If I have to model the dipole as being driven by a classical monochromatic electromagnetic wave, would the corresponding simple harmonic oscillator then be in coherent state ? In...

Hi! I'm trying to calculate the allowed energies of each state for 3D harmonic oscillator. En = (Nx+1/2)hwx + (Ny+1/2)hwy+ (Nz+1/2)hwz, Nx,Ny,Nz = 0,1,2,... Unfortunately I didn't find this topic in my textbook. Can somebody help me?

My question is given an orthonormal basis having the basis elements Ψ's ,matrix representation of an operator A will be [ΨiIAIΨj] where i denotes the corresponding row and j the corresponding coloumn. Similarly if given two dimensional harmonic oscillator potential operator .5kx2+.5ky2 where x...

Hello everyone, For weeks I have been struggling with this quantum mechanics homework involving writing a code to determine the energy spectrum and eigenvalues for the stationary Schrodinger equation for the harmonic oscillator. I can't find any resources anywhere. If anyone could help me get...

Homework Statement Find the eigenfunctions and eigenvalues of the isotropic bidimensional harmonic oscillator in polar coordinates. Homework Equations $$H=-\frac{\hbar}{2m}(\frac{\partial^2}{\partial r^2}+\frac{1}{r}\frac{\partial}{\partial r}+\frac{1}{r^2}\frac{\partial^2}{\partial...

1. The problem statement I want to write the angular momentum operator ##L## for a 2-dimensional harmonic oscillator, in terms of its ladder operators, ##a_x##, ##a_y##, ##a_x^\dagger## & ##a_y^\dagger##, and then prove that this commutes with its Hamiltonian. The Attempt at a Solution I get...

I have trouble with finding the eigenstates of a spherical pendulum (length $l$, mass $m$) under the small angle approximation. My intuition is that the final result should be some sort of combinations of a harmonic oscillator in $\theta$ and a free particle in $\phi$, but it's not obvious to...

How would you solve for the Amplitude(A) and Phase Constant(ø) of a spring undergoing simple harmonic motion given the following boundary conditions: (x1,t1)=(0.01, 0) (x2,t2)=(0.04, 5) f=13Hz x values are given in relation to the equilibrium point. Equation of Motion for a spring undergoing...

Homework Statement [/B] A particle of mass 'm' is initially in a ground state of 1- D Harmonic oscillator potential V(x) = (1/2) kx2 . If the spring constant of the oscillator is suddenly doubled, then the probability of finding the particle in ground state of new potential will be? (A)...

Hello, in every book and on every website (e.g. here http://farside.ph.utexas.edu/teaching/315/Waves/node13.html) i found for driven harmonic osciallation the same solution for phase angle:θ=atan(ωb/(k−mω^2)) where ω is driven freq., m is mass, k is spring constant. I agree with it =it follows...

1. I have been trying to plot wavefunctions of QHO for different states with potential energy function using excel. I followed Griffith's Quantum Mechanics, 2nd edition. I got the nature but they have same reference level. Basically I tried to draw fig2.7a (the first one) and got like the second...

Homework Statement Does the n = 2 state of a quantum harmonic oscillator violate the Heisenberg Uncertainty Principle? Homework Equations $$\sigma_x\sigma_p = \frac{\hbar}{2}$$ The Attempt at a Solution [/B] I worked out the solution for the second state of the harmonic oscillator...

I've been trying to form a proof using , using majorly dirac notation.There has been claims that its much better to use in QM. The question i wanted to generally show that the expected value is Zero for all odd energy levels.I believe i have solved the question but I am a bit Iffy about a step...

Hello, I have a question regarding Damped Harmonic Motion and I was wondering if anyone out there could help me out? Under normal conditions, gravity will not have an affect on a damped spring oscillator that goes up and down. Gravity will just change the offset, and the normal force equation...

Making use of the partition function, it is straight forward to show that the entropy of a single quantum harmonic oscillator is: $$\sigma_{1} = \frac{\hbar\omega/\tau}{\exp(\hbar\omega/\tau) - 1} - \log[1 - \exp(-\hbar\omega/\tau)]$$However, if we look at the partition function for a single...

I am confused about the difference between the two In Griffith's 2.3 The Harmonic Oscillator, he superimposes the quantum distribution and classical distribution and says What I understand for quantum case is that ##|\Psi_{100} (x)|^2## gives the probability we will measure the particle...

For the harmonic oscillator, I'm trying to study qualitative plots of the wave function from the one-dimensional time independent schrodinger equation: \frac{d^2 \psi(x)}{dx^2} = [V(x) - E] \psi(x) If you look at the attached image, you'll find a plot of the first energy eigenfunction for...

Homework Statement I am considering the Klein Gordon Equation in a box with Dirichlet conditions (i.e., ##\hat{\phi}(x,t)|_{boundary} = 0 ##). 1-D functions that obey the Dirichlet condition on interval ##[0,L]## are of the form below (using the discrete Fourier sine transform) $$f(x) =...

Long time no see, PhysicsForums. Nevertheless, I have gotten myself into a statistical mechanics class where the prof is pretty brutal and while I can usually manage, this problem finally has me stumped. I'd like to be nudged in the right direction, not outright given the answer if possible. I...

Homework Statement Homework Equations Complex number solutions z= z0eαt Energy equations and Q (Quality Factor) The Attempt at a Solution For this question, I followed my book's "general solution" for dampened harmonic motions, where z= z0eαt, and then you can solve for α and eventually...

Homework Statement A particle is moving in a 1-dimensional harmonic osciallator with the hamiltion: ## H = \hbar \omega (a_+ a_- + \frac{1}{2})## at time ## t=0## the normalized wave function is given by ## \Psi(x,0) = \frac{1}{\sqrt{2}}(\psi_0(x) + i\psi_1(x)) ## Task: Calculate for ## t \geq...

Homework Statement A particle is moving in a one-dimensional harmonic oscillator, described by the Hamilton operator: H = \hbar \omega (a_+ a_- + \frac{1}{2}) at t = 0 we have \Psi(x,0) = \frac{1}{\sqrt{2}}(\psi_0(x)+i\psi_1(x)) Find the expectation value and variance of harmonic oscillator...

Homework Statement on page 51 (of my book, probably not current) section 2.3.2 equation 2.74 and 2.75 d2ψ / dξ2 ≈ ψξ2 Homework Equations This is an approximation of the Schrodinger equation with a variable introduced ξ = √(mω/h) The solution is given: ψ(ξ) = Ae-ξ2/2 +Beξ2/2The Attempt at...

Homework Statement An un-damped harmonic oscillator natural frequency ##\omega_0## is subjected to a driving force, $$F(t)=ame^{-bt}.$$ At time, ##t=0##, ##x=\dot{x}=0##. Find the equation of motion. Homework Equations ##F=m\ddot{x}## The Attempt at a Solution We have...

Homework Statement Consider an inertial laboratory frame S with coordinates (##\lambda##; ##x##). The Lagrangian for the relativistic harmonic oscillator in that frame is given by ##L =-mc\sqrt{\dot x^{\mu} \dot x_{\mu}} -\frac {1}{2} k(\Delta x)^2 \frac{\dot x^{0}}{c}## where ##x^0...

Suppose I have a 1-D harmonic oscilator with angular velocity ##\omega## and eigenstates ##|j>## and let the state at ##t=0## be given by ##|\Psi(0)>##. We write ##\Psi(t) = \hat{U}(t)\Psi(0)##. Write ##\hat{U}(t)## as sum over energy eigenstates. I've previously shown that ##\hat{H} = \sum_j...

$$m_1 \ddot{x} - m_1 g + \frac{k(d-l)}{d}x=0$$ $$m_2 \ddot{y} - m_2 \omega^2 y + \frac{k(d-l)}{d}y=0$$ It is two masses connected by a spring. ##d=\sqrt{x^2 + y^2}## and ##l## is the length of the relaxed spring (a constant). What is the strategy to solve such a system? I tried substituting...

Hi all, at the moment I am doing my Master Thesis and have the following problem. I am trying to measure Data and asign it a proper timestamp. My problem is, that the data is coming in bursts and the timestamps I assign with the software are wrong. The controller I am using for monitoring the...

I have some difficulties in viewing the literature on the topic. In textbooks on analytical mechnics the procedure given for Special relativistic motion is to write the kinetic term relativistically and attach the unchanged potential term. So, for a harmonic oscillator the Lagrangian is ##L =...

Homework Statement [/B] For the first excited state of a Q.H.O., what is the probability of finding the particle in -0.2 < x < 0.2 Homework Equations Wavefunction for first excited state: Ψ= (√2) y e-y2/2 where: The Attempt at a Solution To find the probability, I tried the integral of...

1. Homework Statement I've been using a recurrence relation from "Adv. in Physics"1966 Nr.57 Vol 15 . The relation is : where Rnl are radial harmonic oscillator wave functions of form: The problem is that I can't prove the relation above with the form of Rnl given by the author(above). I've...

I'm currently studying IR but my mind is having trouble tying everything together. While I see that vibrational frequency is determined really by just reduced mass, I can see from the equation that vib equation is the same throughout energy levels and so does energy (bc that basically depends...

Homework Statement I don’t have a specific problem to solve, and I’m not sure I would be able to correctly find one, but I need to know how to solve a harmonic Oscilator problem with Friction. I believe I should be starting with F = -kx -Ff, and that I will be given some information about the...

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...