Questions tagged [harmonic-oscillator]

The term "harmonic oscillator" is used to describe any system with a "linear" restoring force that tends to return the system to an 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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Why is the damping force on a spring oscillator linearly dependent on velocity?

If you consider the damping force is friction like in: then the force should be $$F=\mu N$$ where $\mu$ is the coefficient of kinetic friction. Why then is the damping force assumed to be linearly ...
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2D harmonic oscillator having 4 constants of the motion and superintegrability

A 2D harmonic oscillator \begin{align} H=p_x^2+p_y^2+x^2+y^2 \end{align} has 4 constants of the motion: $E$ the total energy, $D$ the energy difference between coordinates, $L$ the angular momentum ...
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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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Velocity and acceleration in SHM

Can velocity and acceleration reach maximal values during the SHM simultaneously? Can you explain why?
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Forced oscillation resonance frequencies [closed]

Given this forced oscillator: $$ \ddot{x}+\gamma\dot{x}+\omega_0^2x=\frac{F(t)}{m} $$ Where $F(t)$ is: $$ F(t)=\sum_{n=1}^{\infty}\frac{4F_0}{n\pi}\sin\left(\frac{2n\pi t}{T}\right) \hspace{0.4cm} \...
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Individual particle states in Fock space

I am currently learning QFT, and after watching the wonderful lectures by Leonard Susskind (https://theoreticalminimum.com/courses/advanced-quantum-mechanics/2013/fall), I am still struggling to see ...
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Ladder Operators for a nonlinear oscillator

I was wondering if there was a way to construct the ladder operators for a nonlinear oscillator given by the Hamiltonian $$H=x^2+p^2+\lambda x^4$$ If we were to just calculate scattering amplitudes, ...
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Factorising the supersymmetric oscillator Hamiltonian: what (anti)commutes?

In this paper on supersymmetry, the Hamiltonian for the supersymmetric oscillator is given: $$H = \frac12 p^2 + \frac12 \omega^2 x^2 + \omega\bar\psi\psi.$$ Furthermore, its factorisation is given as $...
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Quantum Harmonic Oscillator propagator in Sakurai

In Sakurai the derivation of the propagator leads to the expression $$u_n(x)\exp{\left(\frac{-iE_nt}{\hbar}\right)} = \left(\frac{1}{2^{n/2}\sqrt{n!}}\right) \left(\frac{m\omega}{\pi\hbar}\right)^{1/...
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Is the energy really infinitely large in a measurement of the energy immediately after the measurement of the position?

For instance, assume the particle rests on the ground state $\psi_0 (x)$ of a one-dimensional simply harmonic oscillator around the origin of axes $ x=0 $, and once we measure the position of the ...
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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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Hamiltonian related to Riemann zeta function [closed]

using the eigenvstates of the Harmonic oscillator could we give a meaning to the Hamiltonian $$ H=\log(a.a^{+}+1) $$ here $ a$ and $ a^{+}$ are the creation/anihilation operators with commutation ...
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A pendulum in an elevator - looking upside down

If I have a pendulum connected to the floor of an elevator by a string, and the elevator is falling in an acceleration greater than g - can I just "rotate" the system and look at it as a regular ...
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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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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 dropped ...
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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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Energy Conservation of waves at a boundary

Consider a wave traveling on a string with velocity $\upsilon$ and mass density $\rho$ having unit length so that the mass of the string is $\rho$. Considering the string to be a simple harmonic ...
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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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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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Is there a spatial representation of the fermionic harmonic oscillator?

An answer to another question derives a Hamitonian of the fermionic harmonic oscillator in terms of a pair of position-like and momentum-like operators. These operators are, as expected, defined in ...
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Modelling a pendulum with physical restrictions on it's range of motion

I'm currently working on a project based on suspension bridges and their oscillations. I've got an equation of motion for the movement of a pendulum as shown in the first image, I then wanted to be ...
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Position and momentum eigenstates in terms of creation and annihilation operator? [closed]

Consider a simple harmonic oscillator; the position operator is $\hat{x}=(a^\dagger+a)/\sqrt{2}$ and the momentum operator is $\hat{p}=-i(a-a^\dagger)/\sqrt{2}$. One may verify that the eigenstates ...
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QFT-Style Lagrangian for a system of two symmetrized bosons

I'm wondering if anybody may have suggestions regarding the following problem. The Hamilton operator of the quantum harmonic oscillator (QHO) can be written as follows: $$ \hat{\mathcal{H}}_{QHO} = \...
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Damped quantum harmonic oscillator - evolution of coherent state

I am trying to solve the following Master equation (also similar to damped quantum harmonic oscillator): $$\frac{d\hat{\rho}}{dt} = \frac{\Gamma}{2}\left(2\hat{a}\hat{\rho}\hat{a}^{\dagger} - \hat{a}^...
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Are Black Holes set to take over the Harmonic Oscillator in the 21st century? [closed]

A few years ago I attended a talk given by Andy Strominger entitled Black Holes- The Harmonic Oscillators of the 21st Century. This talk, http://media.physics.harvard.edu/video/?id=...
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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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Relativistic genarization of Quantum Harmonic Oscillator

I am trying to find out relativistic description of a quantum harmonic oscillator. For a classical relativistic oscillator mass is a function of co-ordinates(http://arxiv.org/abs/1209.2876). $$m(x)=m-\...
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Spring pendulum - why is it possible to use this equation?

It is known that, when we describe the spring pendulum, we are bound to use the formula $T = 2\pi \sqrt{m/k}$, however, we can go further and set $\omega = \frac{2\pi}{T}$ I ponder why is this ...
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Help understanding what the Hamiltonian signifies for the action compared with the Euler-Lagrange equations for the Lagrangian?

Consider the Lagrangian for a simple harmonic oscillator \begin{equation} L (x,\dot{x}) = \frac{1}{2}m\dot{x}^2 - \frac{1}{2}kx^2 \end{equation} Obviously we have \begin{align} \frac{\partial L}{\...
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Does damping force affect period of oscillation?

In my physics notes, it has been given that the damping force increases the period of oscillation. I am unable to understand this part. How is this possible? The only relation I know is that as the ...
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Hamiltonian of quantum harmonic oscillator with $\psi(x)=\delta(x)$: comparison to classical mechanics

I was just reading the question Why can't $\psi(x)=\delta(x)$ in the case of a harmonic oscillator? The accepted answer says that $\psi(x)=\delta(x)$ is a mathematically valid state, though it's not ...
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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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Number operator in quantum field theory?

The number operator, counting the number of quanta is defined as follows: $$ N = \int \frac{d^3 p}{(2\pi)^3} \hphantom{ii} a^{\dagger}_pa_p$$ with the momentum eigenstates being defined as $\lvert ...
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Why doesn't mass of bob affect time period?

The gravitation formula says $$F = \frac{G m_1 m_2}{r^2} \, ,$$ so if the mass of a bob increases then the torque on it should also increase because the force increased. So, it should go faster and ...
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Is it possible to get the SHO approximation of a pendulum without using energy conservation?

I tried to get the approximation for small angle of a simple pendulum using only $\sum \mathbf F = m\mathbf a$ and cartesian coordinates (that means only $x$'s and $y$'s, without $\theta$). After some ...
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What does the uncertainty principle tell us about the harmonic oscillator?

For the harmonic oscillator we have $\sigma_x \sigma_p = \hbar(n+1/2) $ and by the uncertainty principle $\sigma_x \sigma_p \geq \frac{\hbar}{2}$. In one of the exercises I was doing I was asked to ...
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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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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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How to derive the angular momentum operator for 3D harmonic oscillator?

In Angular momentum for 3D harmonic oscillator in two different bases Robin Ekman comes with the expression to $L_i$. I can't see how $$\epsilon_{ijk}(a_j^\dagger a_k^\dagger - a_j a_k) = 0$$ when ...
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Does the Fundamental Frequency in a Vibrating String NOT Necessarily Have the Strongest Amplitude?

I am doing some experiments on musical strings (guitar, piano, etc.). After performing a Fourier Transform on the sound recorded from those string vibrations, I find that the fundamental frequency is ...
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Why is the angle of a pendulum as a function of time a sine wave?

OK so I'm trying to understand why the angle of a pendulum as a function of time is a sine wave. I can't really find an explanation online and when I do find something partial there are certain ...
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Linear combination of eigenstates in a potential [closed]

Linear combination of a set of vectors is only defined for a finite set of vectors even though the set might be an infinite set. In quantum mechanics we take infinite linear combination of all ...
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Does the equation $x = a\sin^2\omega t$ represent a simple harmonic motion?

I've asked this question due a doubt in the following question. The displacement of a particle along the $x$-axis is given by $x = a\sin^2\omega t$. The motion of the particle corresponds to (1) ...
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What is the time period of an oscillator with varying spring constant?

It is well known that the time period of a harmonic oscillator when mass $m$ and spring constant $k$ are constant is $T=2\pi\sqrt{m/k}$. However, I would be interested to know what the time period ...
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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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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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How to properly find the spectrum of the Quantum Harmonic Oscillator? [duplicate]

We want to find the spectrum of the Harmonic Oscillator Hamiltonian: $$H=\frac{\hat{p}^2}{2m}+\frac{1}{2}m\omega ^2 \hat{x}^2$$ From what I have seen in many books the procedure is as follows: We can ...
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Do Hermite polynomials imply a weight for quantum harmonic oscillator wavefunctions?

I know that solutions of quantum harmonic oscillator can be expressed in the form of Hermite polynomials. But I recently came to know that Hermite polynomials are actually orthogonal polynomials ...
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How to calculate the classical on-shell action for a harmonic oscillator? [closed]

So, short and sweet, I've been reading the path integrals book by Feynman and Hibbs, and one of the elementary problems they ask is to calculate the classical on-shell$^1$ action of a harmonic ...
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Quantum simple harmonic oscillator interpretation

I am just wondering what does the SHO system from quantum mechanics actually physically represent? Is it just a SHO of a quantum particle, seems a little too obvious for quantum theory? I'm from a ...

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