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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Potential energy during vertical fall

Suppose I have a weightless spring connected perpendicularly to the ground, and it has on top of it some weightless surface. Now, I release some sticky object from height $h$ above the system of light ...
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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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The issue on existence of inverse operations of $a$ and $a^{\dagger}$

I have asked a question at math.stackexchange that have a physical meaning. My assumption: Suppose $a$ and $a^\dagger$ is Hermitian adjoint operators and $[a,a^\dagger]=1$. I want to prove that ...
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Does spatial coupling prohibit resonances due to an external source field?

The harmonic oscillator coupled to a sinodial external source $$\frac{\partial^2 x(t)}{\partial t^2}+\omega_0^2 x(t)=F_0\sin(\omega_\text{ext}\ t),$$ has the solution $$x(t)=x(0)\cos(\omega_0 t)+C ...
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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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Hyperbolic, parabolic, elliptical PDE related to under-, critical- and overdamped in harmonic osciallation

A damped harmonic oscillator has three cases for the damping: underdamped, critically damped and overdamped. With partial differential equations, I know the hyperbolic wave equation, the parabolic ...
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182 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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FWHM in resonance amplitude square derivation

Consider a linear harmonic oscillator subject to a periodic force: $$ \ddot x + 2 \beta \dot x + \omega _0 ^2x = f_0\cos \omega t$$ The solution tends to: $$A \cos (\omega t - \delta)$$ where: ...
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182 views

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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Can someone please derive $T=2\pi\sqrt{l/g}$ or prove it without using calculus?

I don't know much calculus, but I want to know that how one derives the formula to find time period $T$ of a simple pendulum.
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Why is simple harmonic motion called so?

Is the motion of a simple pendulum, a simple harmonic motion? It stops vibrating after sometime.
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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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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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885 views

Constant magnetic field applied to a quantum harmonic oscillator

I have a spinless particle of mass $m$ and charge $q$ which is an isotropic harmonic oscillator of frequency $\omega_0$, then I apply a constant magnetic field in the $z$ direction. We can show the ...
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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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316 views

Shift operator (integral calculus involving Hermite polynomials) [closed]

I didn't know whether to pose this question on Physics.stackexchange or Math.stackexchange. But since this is the last step of a development involving the eigenfunctions of an Harmonic oscillator and ...
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Partition function for quantum harmonic oscillator

Hi guys I'm currently trying to solve a mock exam for an exam in a few days and am a bit confused by the solutions they gave us for this exercise: Exercise: A solid is composed of N atoms which ...
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Why don't tuning forks have three prongs?

I was reading Why tuning forks have two prongs?. The top answer said the reason was to reduce oscillation through the hand holding the other prong. So if having 2 prongs will reduce oscillation loss, ...
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How to derive the period of spring pendulum?

So I wanted to find out how to (simply, if that's possible) derive the formula for a period of spring pendulum: $T=2\pi \sqrt{\frac{m}{k}}$. However, Google doesn't help me here as all I see is the ...
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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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Estimate the damping coefficient of my car

I was wondering how I can estimate the damping coefficient of my car by doing the hand bouncing the car body and watching the motion of the car? I just need a rough estimate of the damping ...
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461 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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Finding an equation for velocity and acceleration

I'm trying to derive an equation for the velocity and acceleration of an object undergoing simple harmonic motion. I have the equation for displacement: $x = A\sin (2 \pi ft)$ If I differentiate the ...
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Definition of “Quantizing”

Could anyone explain to me what "quantize" means in the following context? Quantize the 1-D harmonic oscillator for which $$H~=~{p^2\over 2m}+{1\over 2} m\omega^2 x^2.$$ I understand that the ...
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Noise spectrum of two systems and interacting Hamiltonian

I've been discovering recently the concept of noise spectrum, defined as: $$S_{xx}[\omega] = \int dt<x(t)x(0)>\text{e}^{-i\omega t}$$ Roughly the Fourrier transform of the two-point function. ...
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Zero point fluctuation of an harmonic oscillator

In a paper, I ran into the following definition of the zero point fluctuation of our favorite toy, the harmonic oscillator: $$x_{ZPF} = \sqrt{\frac{\hbar}{2m\Omega}} $$ where m is its mass and ...
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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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Showing that the probability density of a linear harmonic oscillator is periodic

The complete question I am trying to answer is the following: Show that the probability density of a linear harmonic oscillator in an arbitrary superposition state is periodic with period equal to ...
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319 views

Harmonic oscillator and Lorentz symmetry

There is a analog between harmonic oscillator $x=\frac{1}{\sqrt{2\omega}}(a+a^\dagger)$ and quantum field $\phi=\int dp^3\frac{1}{(2\pi)^3}\frac{1}{\sqrt{2\omega_p}}(a_p e^{ipx}+a^\dagger e^{-ipx})$, ...
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Expectation value of time-dependent Hamiltonian

I'm trying to solve a problem in QM with a forced quantum oscillator. In this problem I have a quantum oscillator, which is in the ground state initially. At $t=0$, the force $F(t)=F_0 \sin(\Omega t)$ ...
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Evolution operator for time-dependent Hamiltonian

When i studyed QM I'm only working with non time-dependent Hamiltonians. In this case unitary evolution operator has the form $$\hat{U}=e^{-\frac{i}{\hbar}Ht}$$ that follows from this equation $$ ...
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Spectrum of quantum fluctuations in a harmonic oscillator

If we have a harmonic oscillator and look at it on small scale the energy is quantized and we can calculate the different eigenstates. In general the energy eigenvalues are given by $$E_n = ...
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What is the formula for max kinetic and max potential energy of a spring?

What is the formula for max kinetic and max potential energy of a spring?
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Strange Behavior in Spring Computer Model

To learn more about oscillatory motion which I am learning about in my high school physics class, I have created a computer model of a damped spring where the damping force is proportional to ...
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Finding an efficient strategy for walking

Let's say you are already walking at a maximally efficient combination of pace and stride (or $\omega$ and $X_0$ I guess) but you need to reach your destination faster. Should you increase/decrease ...
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271 views

Understanding the concept of period of motion in simple harmonic motion formula

I have a spring system, whose position equation is $$x(t) = c_1cos(8 \sqrt{2}t) + c_2sin(8 \sqrt{2}t)$$ The textbook says it will have a period of motion of $\frac{2 \pi}{(8 \sqrt{2}t)}$. I ...
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Basis transformation between eigenstates of harmonic oscillators with different frequency

Given two harmonic oscillators with frequencies $\Omega$ and $\Omega'$, the eigenstates themselves are exactly known. Let's call them $\Psi_n$ and $\Psi'_n$. Is there a compact expression for the ...
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296 views

How do eigenstates of harmonic oscillators with different frequencies compare?

Suppose I have a harmonic oscillator with frequency $\Omega_1$ and another one with frequency $\Omega_2$. Is there a simple relationship between the eigenstates of the two? Especially, how would the ...
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351 views

Spring-mass physics homework question [closed]

I've been having trouble with my physics homework. The problem is: You may have measured the properties of a simple spring-mass system in the lab. Suppose you found ks = 0.9 N/m and m = 0.01 kg, ...
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994 views

Linear motion with variable acceleration

Consider the following problem I pull a mass m resting at x = 0 on a frictionless table connected to a spring with some k by an amount A and let it go. What will be its speed at x=0? I know how to ...
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295 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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Does every wavenumber of IR result in a different kind of vibration?

Does every wavenumber of IR result in a different kind of vibration? If that is true, what if a molecule absorb 2 different wavenumbers (which cause different rocking and symmetrical stretching for ...
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Frequency of a tuning fork in a vacuum

Consider this equation of a damped harmonic oscillator such that: $$ \ddot{x}+2\gamma\dot{x}+\omega^2_0=0 $$ with: $\gamma=\frac{b}{2m}$ and $\omega_0=\sqrt{\frac{k}{m}}$ Finally, we know that the ...
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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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How to think of the harmonic oscillator equation in terms of “acceleration = gradient”

This is related to another question I just asked where I learned that the equation of motion of a harmonic oscillator is expressed as: $$\ddot{x}+kx=0$$ What little physics I grasp centers on ...
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Question on Sakurai's treatment of the Harmonic Oscillator:

In Section 2.3 of the second edition of Modern Quantum Mechanics (which discusses the harmonic oscillator), Sakurai derives the relation $$Na\left|n\right> = (n-1)a\left|n\right>,$$ and states ...
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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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427 views

Symbol for dashpot/damper (in a harmonic oscillator)

In diagrams that contain the dashpot symbol, sometimes the mass is attached to the "interior" end of the dashpot, other times the mass is attached to the "base" end. For example, consider the ...
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Time period of torsion oscillation

For the oscillation of a torsion pendulum (a mechanical motion), the time period is given by $T=2\pi\sqrt{\frac{I}{C}}$ which is a result of the angular acceleration ...