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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Pendulum in an elevator

Suppose we have a pendulum tied to the ceiling of an elevator which is at rest. The pendulum is oscillating with a time period $T$, and it has an angular amplitude, say $\beta$. Now at some time when ...
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Schrödinger equation for a harmonic oscillator

I have came across this equation for quantum harmonic oscillator $$ W \psi = - \frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + \frac{1}{2} m \omega^2 x^2 \psi $$ which is often remodelled by defining a new ...
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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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Is this a valid way for deriving the ODE for a lattice vibration of a one dimensional crystal?

Consider the following lattice: I want to derive a differential equation that describes the forces acting on the $n$-th atom in the lattice. Each atom is coupled to its neighbour ($n+1,n-1)$ by a ...
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Instantaneous energy eigenstates for forced harmonic oscillator

I'm interested in applying the adiabatic theorem to the forced harmonic oscillator with time dependent hamiltonian of the form: $$H(t) = \hbar \omega(a^{\dagger}a + \frac{1}{2}) - f(t)a - f^{*}(t)a^{\...
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Quantum mechanics: SHM expectation of $x^2$ time independent for one state but not superposition of 2 states?

my answers for the first bits $$\langle H\rangle =n\hbar\omega$$ $$\langle x\rangle =\sqrt\frac{\hbar n}{2m\omega}\cos(\omega t)$$ $$\langle p\rangle =-\sqrt\frac{\hbar m\omega n}{2}\sin(\omega t)$$ ...
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Harmonic oscillator with heat bath

I need to calculate the expectation value for a harmonic oscillator coupled to a heat bath using the trace method. I know that the density operator looks like: $$\rho = \frac{e^{-H / k_B T}}{\text{...
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Birkhoff Method for Harmonic Oscillator Perturbation

Problem: Given Hamiltonian $$H = \frac12 (p^{2}+q^{2})+q^{3}-3qp^{2}$$ make a perturbative canonical transformation $(q,p) \rightarrow (Q,P)$ such that the new Hamiltonian, apart from terms of degree ...
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Exact closed form solution to the quantum harmonic oscillator

I came across this question in Griffiths QM, which asked to show that this equation $$\Psi(x,t)=\left(\frac{m\omega}{\pi \hbar}\right)^{1/4} \exp\left[-\frac{m\omega}{2\hbar} \left(x^2+\frac{a^2}{2}(...
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The universality of the Stuart-Landau equation to describe nonlinear oscillators

I have read numerous papers which boldly suggest that the Stuart-Landau equation can be successfully used to model any weakly nonlinear oscillating system near a Hopf bifurcation. Even thought it has ...
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Classical Limit of the Quantum Harmonic Oscillator

The classical harmonic oscillator obeys an arcsine law in that the distribution of positions of the particle over a single time cycle is proportional to $\frac{1}{\sqrt{A^2-x^2}}$, $A$ being the ...
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Proving $[a_k^\dagger, a_q^\dagger]=0$

I am trying to prove the commutation relations between the creation and annihilation operators in field theory. I was already able to show that $[a_k, a_q^\dagger]=i\delta(k-q)$. I want to show that $[...
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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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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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Frequency of kinetic energy in shm

I am currently learning about simple harmonic motion. In a book I am reading it says frequency of kinetic energy is twice the frequency of velocity for a harmonic oscillator by showing velocity vs ...
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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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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: $${\cal{L}}=\frac12\rho\dot{u_z}^2-\frac12C_{zzzz}(\...
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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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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 $N=a^{\dagger}a$ ...
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What is meant by taking the partial derivative of the Hamiltonian in this situation?

I'm doing a computation involving the quantum mechanical harmonic oscillator, and I have an expression of the form $\frac{\partial}{\partial \omega} \hat{H}$ where $$\hat{H} = \frac{1}{2m} \left( - \...
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Damping Coefficient of SHM

In lab for my physics of digital systems class, we were told to find the damping coefficient of a spring experiencing simple harmonic oscillation. We were given the formula $$x = A e^{\left( -\frac{...
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$N$ coupled quantum harmonic oscillators

I want to find the wave functions of $N$ coupled quantum harmonic oscillators having the following hamiltonian: \begin{eqnarray} H &=& \sum_{i=1}^N \left(\frac{p^2_i}{2m_i} + \frac{1}{2}m_i\...
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Computing $\langle0|T[Q(t_2)Q(t_1)]|0\rangle$

Given Hamiltonian $H=\frac{P^2}{2}+\frac{\omega^2}{2}Q^2$, compute $\langle0|T[Q(t_2)Q(t_1)]|0\rangle$, where $T$ is the time-ordering of the product, $|0\rangle$ is the ground state. Now set $\hbar=m=...
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Simplest explanation of pendulum having a constant time period at low angles

What is the simplest explanation for the pendulum having a constant time period at low angles?
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Quantum Harmonic Oscillator - Normalizability of Annihilated Ground State

The common line of deductions in the operator analysis of the quantum harmonic oscillator goes something like this: It is derived that the action of the annihilation operator $a$ on an eigenfunction ...
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Quantum field theory: field operators in terms of creation/annihilation operators

I am learning Quantum Field Theory and there is a step in my notes that I do not really understand. It starts with the classical definitions of position $q$ and momentum $p$: $$ q = \frac{1}{\sqrt{2\...
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373 views

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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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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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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Evaluating path integral

I am having some trouble remembering how to evaluate path integrals involving multiple particles. Suppose that I have two interacting particles with Lagrangian $$L= \frac{1}{2}m \dot y^2-\frac{1}{2}m ...
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State time evolution of a quantum harmonic oscillator with a Dirac-Delta function as an initial state [closed]

I have a question just like this Phys.SE problem here with a difference that our system is a harmonic oscillator (rather than a free particle). A particle with mass $m$ is connected to a string with ...
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Driven-damped oscillator: deduce the phase and/or resonant freq from amplitudes at varying freqs

Suppose that we have a fairly standard driven-damped harmonic oscillator (i.e. linear spring restoring force, linear damping force, sinusoidal driving force, etc). The catch is: we don't know the ...
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Coupled quantum harmonic oscillator

Given the following Hamiltonian for two identical linear oscillators with spring constant $k$ and interaction potential $\alpha x_1x_2$; I was asked to find the expectation value $\langle x_1x_2\...
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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: $$\ddot{\theta}+\omega_{o}^{2}\left(\theta-\frac{1}{6}\theta^3\...
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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 $\omega=(k/...
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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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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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What is the physical meaning of $a_{\vec{p}} \! \mid \! 0 \rangle$

$a^\dagger_{\vec{p}} \! \mid \! 0 \rangle = \mid \! p \rangle$ is interpreted as a creation of a particle with momentum $p$ from the vacuum. $a_{\vec{p}} \! \mid \! p \rangle = \mid \! 0 \rangle$ is ...
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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[ \frac{d^2}{dz^2}-\...
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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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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 equation ...
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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 $\alpha=\frac{d^2\theta}{dt^2}=-(\...
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Harmonic oscillator: if $E=\frac{1}{2} q \dot{\theta}^2+\frac{1}{2} s \theta^2$ then $\omega=\sqrt{\frac{s}{q}}$?

Consider an harmonic oscillator. Suppose that I manage to write the mechanical energy as a function of a quantity, like the angle $\theta$ in this way $$E=\frac{1}{2} q \dot{\theta}^2+\frac{1}{2} s \...
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Why does the acceleration $g$ due to gravity not affect the period of a vertically mounted spring?

For a vertically mounted spring, I was looking at the formula $ T= 2\pi \sqrt{m/k}$ for a period. Why doesn't the gravitational acceleration $g$ factor in?
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Unstable equilibrium in a pendulum

Consider a pendulum with a bob and a massless, rigid, hinged rod attached to the bob. The bob is at rest at the bottom most position. Neglecting friction, is it possible to impart such a velocity (...