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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Help explain how direction change relates to acceleration [duplicate]

I was doing some simple harmonic motion problems and I came across this picture describing the position, velocity and acceleration of a linear oscillator. At the moment in time when v is 0 the ...
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Questions related to resonance/standing-waves and sound

I understand resonance for a simple harmonic oscillator but not for more complex systems like standing waves. How can I be in resonance with the normal mode in an organ pipe? I understand that the ...
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How can one calculate the phase difference between two quantum harmonic oscillator (Hermite-Gauss) states?

The analytic solutions of a quantum harmonic oscillator are given by Hermite-Gauss states, which differ in the order $n$ of the Hermite polynomials. If two such states are plotted, there will be a ...
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What is the relationship between Q-Factor and viscosity for a Newtonian fluid based damper?

I am working with a rotary damper (paddle hanging off a shaft attached to the bottom of a rotary table and into a vat of fluid). I know the viscosity of the current Newtonian fluid, I know the flow is ...
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Why isn't a pendulum clock considered a perpetual machine?

Why do people say that a perpetual machine is one that runs for eternity? Wouldn't a machine that runs taking no energy from outside but sometimes needs to be restarted, such as pendulum clock, be ...
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Energy of damped harmonic oscillator begins to increase with very large Q in numerical integration

I have numerically integrated the (reduced) homogeneous equation of a damped harmonic oscillator in order to see how the error propagates. $$\frac{d^2 X}{d\phi^2} + ...
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Angular momentum for 3D harmonic oscillator in two different bases

I know that the energy eigenstates of the 3D quantum harmonic oscillator can be characterized by three quantum numbers: $$ | n_1,n_2,n_3\rangle$$ or, if solved in the spherical coordinate system: ...
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WKB approximation in two dimensions

Does anybody know how to implement the WKB approximation for the two-dimensional Schrodinger equation with a harmonic oscillator potential: $\frac{1}{2}\Biggl[-\biggl(\frac{\partial^2}{\partial ...
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187 views

Concrete example of a two-dimensional harmonic oscillator

I am a student of mathematics and some time ago I showed in general that for a two-dimensional harmonic oscillator one can apply the recurrence theorem. So far so good.. now I would like to have a ...
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How can I find the motion equations of the 2-dim harmonic oscillator?

First of all: I am no physicist, so I am rather helpless. I need to find the moving equations of the 2-dim. harmonic oscillator. If it is possible it should be rather elementary, because, as I said, ...
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Nodes in wave functions outside of the classical turning point

When looking at the solutions of the classical harmonic oscillator for instance from wikipedia one can observe that there are no nodes in the wavefunction outside the classical turning points. But I ...
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Calculation of energy eigenvalues of $\hat{x}^4$

I would appreciate help in calculating the energy eigenvalues associated with $\hat{x}^4$, with $\hat{x}$ expressed using the ladder operators for harmonic oscillators. $\hat{x} = ...
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quick question about degeneracy

For two non-interacting particles, with eigenfunctions $\phi_{n1}(x1)$ and $\phi_{n2}(x2)$ in a one-dimensional potential well $V_{(x)}$ with n = 1,2,.... Consider two spinless non-identical ...
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What happens between two harmonics?

I know that standing waves in a simple harmonic system occur when the "echo" of a wave overlaps completely with the original wave at a certain point in time, doubling the amplitude. And then, because ...
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Does the average momentum vanish for an eigenstate of the simple harmonic oscillator?

Suppose we have a simple harmonic oscillator, let's consider the ground state, $|0\rangle$ and the first excited state $|1\rangle$. $\langle 0|\hat p|0 \rangle$ is zero right? Since the particle can ...
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104 views

Hamiltonian Operator for Harmonic Oscillator

I have been solving the harmonic oscillator problem in quantum mechanics using Algebraic Method and since then I am consulting the books of Tannoudji and Griffiths for that matter. While studying both ...
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74 views

Ladder operator on momentum basis

Since in Quantum mechanics momentum operator can be written in terms of ladder operators $$\widehat{p}=-i\sqrt\frac{{\hbar m \omega}}{2}(\widehat{a}-\widehat{a}^\dagger)$$ these operators operate on ...
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Quantum oscillator, position mean value problem

A quantum harmonic oscillator of mass $m$ and frequency $\omega$ is at time $t=0$ in the state: $$ \left|\psi(t)\right> = \sum_{n=N-\Delta N}^{N+\Delta N}\left|n\right>\frac{1}{\sqrt{2\Delta N ...
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Meaning of “Simple” in Simple Pendulum and Simple Harmonic Motion?

I have gone through the Phys.SE question Why is simple harmonic motion called so?. From the 1st answer of this Question it seems to me that another type of "Harmonic motion" is "Damped Harmonic ...
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solution of pendulum equation [closed]

I have the pendulum expression $$\ddot{\theta}+\omega_{o}^{2}\sin(\theta)=0,$$ where I used a Taylor expansion for the sine term: ...
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damped and undamped oscillation graph comparison

I have been trying to solve a problem which compares the two motion. If the undamped free harmonic oscillator motion is $ x=A\sin\omega_{o}t$. So the corresponding motion in case of damped motion will ...
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Simple harmonic motion [closed]

A uniform straight rod of length $L$ is hinged at one end. It is free to oscillate in vertical plane. Time period of oscillation with small angular amplitude when a point mass of mass equal to that of ...
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Forced harmonic oscillator with two springs

Consider a vertical system of two springs in series, with a mass(50 g) between them. From below the system is driven by a vibration generator. The setup is shown here, but the picture is taken while ...
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355 views

Nonzero ground state energy of the quantum harmonic oscillator [duplicate]

Since $\frac{1}{2}\hbar \omega$ is the zero point energy of the ground state of the harmonic oscillator, then there is no way to extract this energy. Therefore, in what way is this value different ...
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Raising and lowering 3D harmonic oscillator state

A good solution to the 3D harmonic oscillator is shown here. This gives the basis states $|n,\ell\rangle$ My question is if there are some operators comparable to the 1D SHO that will raise either n ...
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Finding Tension in an Elastic String?

I know that this is a homework type question and I'm not asking a particular physics question, but I'm really desperate for help. Here's the question: I tried to divide the string to 2 parts with ...
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Eigenstates of a density matrix of continuous variables

Consider a system of two entangled harmonic oscillators. The normalised ground state is denoted by $\psi_0(x_1,x_2)$. The reduced density matrix of the second oscillator is given by: $$\rho_2 = ...
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What are you studying when you study a Harmonic Oscillator in QM?

This probably is a naive question - so please forgive a self-studier. In the text I am studying, one builds a HO by placing a particle in a potential that increases quadratically from the origin. The ...
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Trace as integral

Consider a system of two entangled harmonic oscillators. The normalised ground state is denoted by $\psi_0(x_1,x_2)$. I've been taught that a density matrix is constructed as $\rho = ...
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Uncoupling a coupled oscillator Hamiltonian by change of variables

I'm working on the problem of two entangled harmonic oscillators with Hamiltonian: $$H = \frac{1}{2} [p_1^2 + p_2^2 + k_0(x_1^2 + x_2^2) + k_1(x_1 - x_2)^2].$$ Introducing the variables $x_± = ...
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Degeneracy, spherical harmonics

In a 3D oscillator, the energy levels are known to be $(n_x + n_y + n_z + \frac{3}{2})\hbar \omega = (n + \frac{3}{2})\hbar \omega$. Say for $n = 1$, any of the $n$'s can be $1$ and the rest are $0$. ...
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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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Resonance and a tuning fork

I carried out this experiment in class: I struck a tuning fork with a hammer. The sound lasted for some time. However, when I connected the tuning fork onto a wooden sounding box, the sound lasted ...
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Change of variable in harmonic oscillator time independent Schrodinger equation

I was revising the harmonic oscillator for my intro to quantum course and realised I'd sort of accepted a change of variable result without actually being able to get to it. It says: The stationary ...
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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 = ...
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Relation between shm and circular motion

I recently read in a book that combination of two simple harmonic motions of equal amplitude in perpendicular directions differing in phase by pi/2 is circular motion. I don't seem to understand this ...
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How to model “Doppler Distortion” of speakers?

Simple Model w/o Doppler I have a speaker driven by an electrical signal. The pressure at the sampling point is some linear operator acting on the input signal: $L[ s(t)]$. Where $L$ combines the ...
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Question on Quantum Harmonic Oscillator

My textbook claims that the uncertainty in position of the particle in a quantum harmonic oscillator is $\frac{A}{\sqrt{2}}$ and the uncertainty in the particle momentum is $\frac{p}{\sqrt{2}}$ ...
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Mean energy harmonic oscillator

I know that for a particle under the potential $$V(x,y,z)=\frac{k}{2}(x^2+y^2+z^2)$$ the equipartition theorem says that it contributes to the mean energy to $\frac{3k_BT}{2} $ (one half for each ...
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Normalisation of Linear Harmonic Oscillator - Ladder Operator Method

I was watching the following video on the harmonic oscillator using ladder operators : http://youtu.be/gRdCV9p8sAU?t=30m9s Clicking on the video above will take you to the exact point where my ...
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Quantum Mechanics: Momentum operator questions [closed]

I'm asked to determine $\hat{P}|\Psi_0\rangle$, $\langle{\hat{P}}\rangle$, and $\langle\hat{P}^2\rangle$ for $$\Psi_0(u) = \psi_0 + 2\psi_1$$ I understand how to make the matrix for $P$ in regards ...
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Is this oscillator driven?

A mass $m$ is attached to a vertical massless spring or a spring constant $k$. Originally, the spring was relaxed because the mass was held by a clip. Suddenly the clip was released. THe mass ...
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Faster than critical damping for harmonic oscillator?

The image below shows damping for spring oscillator with Hooke law F=-kx and damped with F=-cv where: k is spring constant x is oscillator position c is damping coefficient v is velocity of oscillator ...
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A block falling from a height on a block suspended by spring [closed]

The block suspended by the spring is hanging freely and its mass is M. The small block of mass m is dropped on the bigger block from height h. After the small block is dropped 》》》 I want help in ...
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Why the distance between peaks of the probability distribution function decreases when n increases?

In the solution of Schrödinger Equation for harmonic oscillator why the distance between peaks of the probability distribution function decreases when n increases? Is there a good reason for it or is ...
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335 views

Harmonic Oscillator potential, proof that Gaussians remain Gaussians?

I read in several papers that for a Harmonic Oscillator Hamiltonian in the time dependent Schrödinger equation a Gaussian wave packet remains Gaussian. Unfortunately I could not find any proof for ...
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340 views

Phase Plot for Harmonic Oscillator

This is probably gonna be a dumb question but I don't know exactly where I am making the mistake. I have been taught in highschool that simple harmonic oscillator phase plot is the $sin(\omega t)$: ...
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Heisenberg picture usage - Merzbacher 14.106

I am trying to understand a line in the quantum mechanics book by Merzbacher, specifically the second line of equation 14.106. The problem is a forced quantum harmonic oscillator. The Hamiltonian ...
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Interesting Harmonic Oscillator Solution

On page 89 of Griffith's QM book, an exact solution to the time-dependent SE equation for the harmonic oscillator is mentioned: $$ ...
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Why not drop $\hbar\omega/2$ from the quantum harmonic oscillator energy?

Since energy can always be shifted by a constant value without changing anything, why do books on quantum mechanics bother carrying the term $\hbar\omega/2$ around? To be precise, why do we write $H ...