The term "harmonic oscillator" is used to describe any system with a "linear" restoring force that tends to return the system to a equilibrium state. There is both a classical harmonic oscillator and a quantum harmonic oscillator. Both are used to as toy problems that describe many physical systems.

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Equations of motion for a spherical pendulum in a non-inertial reference frame

Take a spherical pendulum with bob mass $m$, rod length $\ell$ and physical coordinates $\theta$, $\phi$ (spherical angles) and $h$ (the hinge height with respect to the coordinate origin). The rod is ...
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Quantum Mechanics Basics: product space

Consider a coupled harmonic oscillator with their position given by $x_1$ and $x_2$. Say the normal coordinates $x_{\pm}={1\over\sqrt{2}} (x_1\pm x_2)$, in which the harmonic oscillators decouple, ...
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From the local Hooke's law to the global one

My system consist of a cylinder with axis Z that can contract and dilate along this axis. It obeys microscopically Hooke's law of elasticity: ...
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Dynamics of a Vertical Mass-Spring Simple Harmonic Oscillator with Gravity

I am having some trouble obtaining the elastic potential energy and gravitational potential energy of a simple mass spring system. In this experiment, masses attached to a spring were dropped from a ...
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346 views

Writing equation for amplitude of driven harmonic oscillator in Lorentzian form

This harmonic oscillator is driven and damped, with the form: $$\ddot{x} + \lambda \dot{x} + \omega_0^2 x = A \cos(\omega_d t)$$ Now, I have used the ansatz (guess): $x(t) = B \cos(\omega_d t + ...
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Pendulum Wave Period

Recently I've seen various videos showing the pendulum wave effect. All of the videos which I have found have a pattern which repeats every $60\mathrm{s}$. I am trying to work out the relationship ...
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Metronome synchronisation applied to swings

The movement of several metronomes can be synchronised when a movable floor is utilised which couples the movement of the different metronomes. Is it possible to apply this sort of synchronisation to ...
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279 views

Simulating quantum network of harmonic oscillators

Let's say that I have a system of $n$ particles $p_1,\ldots,p_n\in\mathbb{R}^3$ (where $n$ here is on the order of 10,000). Furthermore, suppose we have a graph $G=(V,E)$ describing some network, ...
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What are some interesting coupled harmonic oscillators problems?

That I could create as a classical mechanics class project? Other than the classical examples that we see in textbooks.
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How to determine viscous dampening coefficient of spring?

I'm trying to determine the viscous dampening coefficient of a spring (c). Read about it on Wikipedia here. The two equations which I have are: f=-cv and ma+cv = -kx I know the spring constant ...
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39 views

Constant of motion

An exercise from Goldstein (9.31-3rd Ed) asks to show that for a one-dimensional harmonic oscillator $u(q,p,t)$ is a constant of motion where $$ u(q,p,t)=\ln(p+im\omega q)-i\omega t $$ and ...
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Measure energy state of quantum harmonic oscillator

When discussing the quantum mechanical harmonic oscillator we are talking about energy eigenstates. How would one actually measure in which state an harmonic oscillator is in? Could you weigh it and ...
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The Hamiltonian for clocks?

I am rather a theoretician and looking for a formalism to represent biological clocks by Hermitian operators. The simplest thought model I am looking for is a formal representation of a clock (for ...
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640 views

Effective mass in Spring-with-mass/mass system

Suppose you have a particle of mass $m$ fixed to a spring of mass $m_0$ that, in turn, is fixed to some wall. I'm trying to calculate the effective mass $m'$ that appears in the law of motion of the ...
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Energy Levels of 3D Isotropic Harmonic Oscillator (Nuclear Shell Model)

One simple way of detailing the very basic structure of the nuclear shell model involves placing the nucleons in a 3D isotropic oscillator. It's easy to show that the energy eigenvalues are $E = ...
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Using $\sin()$ or $\cos()$ for computing SHM?

In simple harmonic motion, you can use either the sin or cos form of the equation but my question is which one do you use when and why? I am having a tough time understanding this, so any help would ...
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Why Don't the Ladder Operators Commute?

I have two problems with ladder operators. The first is that I feel they should somehow result in measurable things. The asymmetry of applying the plus operator versus the minus operator is very ...
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299 views

Why are overtones forbidden within the harmonic approximation?

In vibrational spectroscopy only transitions between neighboring vibrational states ($\Delta \nu = \pm 1$, $\nu$ being the vibrational quantum number) are allowed within the harmonic approximation. ...
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610 views

Hermite polynomials for expected value of harmonic oscillator

This was a problem on my final exam that has been really bugging me. Consider the quantum Harmonic oscillator prepared in an energy eigenstate, $\psi_n$(x). Calculate the expectation value of the ...
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Undamped oscillations. Why is the solution a linear combination of $\sin()$ and $\cos()$?

$ma = mg - cx$, where $x(0) = x_0 = 0$ is the position in which there is no tension in the rope. $dx/dt = v_0$ for $t = 0$; $v_0$ is a known constant. The discriminant of the characteristic ...
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193 views

Harmonic Oscillator (Quantum Mechanics)

Griffiths uses an algebraic "brute force" technique to solve the harmonic oscillator. I'm somewhat confused regarding a few parts. $$\frac{1}{2m}[p^2 + (m \omega x)^2] \psi = E \psi$$ $H = ...
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Infinite period in Simple Harmonic Motion

I'm studying the Simple Harmonic Motion, and I am hesitant about, how to get mass values for infinite period? When mass is 0. When mass is infinite. With $\tau=2\pi/\sqrt{k/m}$.
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Organs & Oscillations: An Analysis on the Temperature Dynamics of Solids

Does temperature have an influence on the frequency of an oscillating organ pipe?
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Complex energy eigenstates of the harmonic oscillator

Given the Hamiltonian for the the harmonic oscillator (HO) as $$ \hat H=\frac{\hat P^2}{2m}+\frac{m}{2}\omega^2\hat x^2\,, $$ the Schroedinger equation can be reduced to: $$ \left[ ...
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Why is linear independence of harmonic oscillator solutions important?

The equation of motion for the harmonic oscillator (mass on spring model) $$\frac{d^2x}{dt^2} + \omega_0^2 x = 0$$ with $\omega_0^2 = D/m$, $D$ and $m$ being the force constant of the spring and the ...
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Is it only the ground state of the quantum harmonic oscillator that has the minimum uncertainty product?

We know that the uncertainty product of general states is bounded by the inequality described by Heisenberg's uncertainty relation. And the ground state of the quantum harmonic oscillator falls under ...
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Harmonic oscillator

Let $|0\rangle,...$ be the states of the harmonic oscillator. Then a squeezed state was defined as $|\xi\rangle =S(\xi)|0\rangle $, where $S(\xi):=e^{\frac{1}{2}( \xi (a^{ \dagger ^2}-a^2))}$, where ...
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How To Use Ladder Operators?

I'm studying for a test in quantum mechanics and I'm having a hard time understanding how to use ladder operators. There are no examples in my text book, only definitions that I can't understand how ...
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146 views

Why are particles in harmonic motion in normal modes?

Why do we assume that in normal modes, particles oscillate in form cos (wt) ? How do we know that the general motion of particles can be expressed as a superposition of normal modes? In both French ...
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625 views

Probability for harmonic oscillator outside the classical region

I'm having some trouble finding an expression for the probability to find the particle outside the classical area in the harmonic oscillator. I have a wavefunction defined as: $\psi \left( x,\,t ...
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305 views

Expected number of quanta in harmonic oscillator states

I'm working my way through A Squeezed State Primer, filling in details along the way. Let $a$ and $a^\dagger$ be the usual annihilation and creation operators with $[a,a^\dagger]=1$ and ...
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Degeneracy of states in mixed infinite square well, harmonic oscillator

I'm trying to determine the degeneracy of states given by $g(\epsilon)=g_{0} \epsilon$ for a system that is trapped in a quite specific potential. In two dimensions, the particle has a potential as ...
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Harmonic oscillator - wavefunctions

I understand now how I can derive the lowest energy state $W_0 = \tfrac{1}{2}\hbar \omega$ of the quantum harmonic oscillator (HO) using the ladder operators. What is the easiest way to now derive ...
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Accessible microstates of harmonic oscillator in microcanonical enemble

While reading up on statistical physics, I am going through the calculation of the partition function of the harmonic oscillator in the microcanonical ensemble. The result for the partition function ...
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Why uncertainity is minimum for coherent states?

While reading for quantum damped harmonic oscillator, I came across coherent states, and I asked my prof about them and he said me it is the state at which $\Delta x\Delta y$ is minimum. I didn't ...
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170 views

Coordinate representation of quantum ladder operator?

I can't seem to figure out how to derive the coordinate representation of the $a_+$ ladder operator in quantum mechanics. I know that $a_-$ is $\sqrt{\frac{1}{2mwh}} (mwx + i\dot{p}) $ in which where ...
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Sitting on the bob of a pendulum

Walter Lewin's best performance was the pendulum demonstration, and I copy the transcript now: Would the period come out to be the same or not? [students respond] Some of you think it's ...
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Non-Degeneracy of Eigenvalues of Number Operator for Simple Harmonic Oscillator [duplicate]

Possible Duplicate: Proof that the One-Dimensional Simple Harmonic Oscillator is Non-Degenerate? I'm trying to convince myself that the eigenvalues $n$ of the number operator ...
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Force to use in harmonic oscillation through the inside of a planet

I am to find an equation for the time it takes when one falls through a planet to the other side and returns to the starting point. I have seven different sets of values - mass of object falling, mass ...
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When are Maximum Velocity and Acceleration acheived in Simple Harmonic Motion?

Im trying to get my head around SMH out of curiosity because it seems simple yet I'm not getting the concept behind some ideas. For a SMH equation : $$ x=a \sin(\omega t+\phi) $$ Under what ...
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Using Fourier Transforms to Solve Systems with springs of high frequency

I'm trying to numerically solve the differential equations of motion in a system with multiple springs of very high frequency. Because the solution is often a combination of rapidly-oscillating sine ...
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How to calculate viscous damping coefficient?

The damping of a spring is calculated with: $$[\zeta] = \frac{[c]}{\sqrt{[m][k]}}$$ Where c is the 'viscous damping coefficient' of the spring, according to Wikipedia. m is the mass, k is the spring ...
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Harmonic Oscillator driven by a Dirac delta-like force

Consider that there is no damping for simplicity. As we know, a driving force of the form $\sin(\omega t)$ will make the oscillator at steady state vibrates at the external frequency $\omega$. What ...
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Harmonic Oscillator - Energy quantisation

The one-dimensional quantum HO can be solved in Schrodinger representation by getting Hermite Differential Equation $$ \frac{d^2y}{dx^2} - 2x \frac{dy}{dx} + \lambda y = 0 $$ with solutions $$ y(x) = ...
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304 views

Equations of motion for a pendulum in 3D?

I am trying to solve for the equations of motion to simulate a pendulum. I decided to use the spherical coordinates. The Lagrange equation is: where L = length of the rope ϕ= angle of the ...
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154 views

Harmonic Oscillator Expectation Value

In Calculating the expectation value of the quantum harmonic oscillator, I've come across a problem for finding $\left \langle x \right \rangle$ for the coherent state $\left| \alpha \right \rangle$ ...
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One body harmonic oscillator states expressed in terms of creation operators

I am reading trough chapter one of Moshinsky's "The harmonic Oscillator in Modern Physics". However i am having some trouble with the mathematics in section 8 of chapter 1. I will sketch a summary of ...
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Simple harmonic oscillator system and changes in its total energy

Suppose I have a body of mass $M$ connected to a spring (which is connected to a vertical wall) with a stiffness coefficient of $k$ on some frictionless surface. The body oscillates from point $C$ to ...
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Using eigenvalues to determine the stability/behaviour of the system

first time I've been on physics.se but have used the math and cs before... Anyway, here's my question: If we have a damped pendulum described by the equation $$y'' + ay' + b = 0 , a>0$$ Using the ...
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Different hamiltonians for quantum harmonic oscillator?

The Hamiltonian for a classical simple harmonic oscillator is $$ H = \frac{p^2}{2m} + \frac{1}{2}m\omega^2x^2$$ With the usual choice of the ladder operators $$a = ...