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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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If friction is not proportional to velocity, why do we model it as such when considering damped oscillations? [duplicate]

Early in our study of mechanics, we learn that friction is usually proportional only to normal force, without dependence on velocity. However, during our studies of damped oscillations, we often model ...
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Why does $x_0V''(x_0)$, i.e. the constant force term of the $-V''(x_0)(x-x_0)$ term of a Taylor expansion, have no effect on the equation of motion? [closed]

My waves and vibrations class suggests that $F(x)\approx-V''(x_0)(x-x_0)$ is equivalent to $F(x)\approx-V''(x_0)x$ when considering only oscillations about the equilibrium position; i.e. that the term ...
Dominic Stewart-Guido's user avatar
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Why am I getting this derivation of time period of pendulum in an accelerated frame wrong? [closed]

We are working in the frame of the cart and we are trying to obtain the $\tau=k\theta$ form. So, let's write the $\tau=I_{axis}\alpha$ first for a small deviation $\theta$ from the vartical. (The ...
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Potentials increasing faster than harmonic oscillator

I'm reading a book which says: (HO stands for harmonic oscillator): The spectrum of the HO has equidistant energy eigenvalues. A potential that increases quicker than the HO has states which become ...
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Spherical quantum oscillator: Is energy smaller than the potential?

A particle with mass $m$ is inside the spherical quantum well $V(r)$: \begin{equation} V(r)= \begin{cases} -V_0, & \text{if}\ r<a \\ 0, & \text{otherwise} \end{cases} \...
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Understanding the dynamics of a perturbed quantum harmonic oscillator system

I'm trying to understand how quantum systems behave when they are perturbed, and I'm using the quantum harmonic oscillator as a model. I start by implementing a symmetric gaussian shaped bump in the ...
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Investigation Results of Damping of A Spring Showing Changing Phase Angle? Why?

In an experiment I've recorded the displacement of the spring over time, investigating underdamped simple harmonic motion. Using pre-existing formulae the data should conform to a curve of the form $$...
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Infrared regularizing the harmonic oscillator path integral

This is from Laine and Vuorinen’s Basics of Thermal Field Theory. I do not understand why the fact that the integral over $x(\tau)$ implies the following regularization scheme. That is, I don’t ...
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Oscillating body and Doppler effect

Say we have a body attached to a spring, oscillating with some frequency $\nu$. This is one of the simplest problems studied in elementary Physics, and yet I've noticed we always study it positioning ...
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Regarding to the asymptotic solution of quantum harmonic oscillator

In quantum mechanics, the radial equation of the SHO takes the form \begin{align} \frac{d^2 u}{dx^2}+\left(\epsilon-x^2-\frac{l(l+1)}{x^2}\right)u=0, \end{align} where $x=\sqrt{\frac{m\omega}{\hbar}}r$...
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Two Simple Harmonic Motion (S.H.M.) in Perpendicular Direction

Suppose a particle is moving under the superposition of two S.H.M in the perpendicular direction... The general equation for the trajectory for the resultant motion arising due to the two component S....
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Can a harmonic oscillator never be Raman active?

Assuming we have some harmonic oscillator \begin{equation} H = \omega_0 (a^\dagger a + \frac{1}{2}) = \frac{p^2}{2m} + k x^2 \end{equation} for which the excitations have even wavefunctions $\Psi_n(x)=...
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Closed expression for expected values of $\hat{p}\,\,^{2j}$ for the vacuum state

I am wondering if there is a closed expression for the expected value $\left<0\lvert \hat{p}\,\,^{2j}\lvert 0\right>$ with $j\in\mathbb{N}$, where $\left|0\right>$ is the vacuum state of the ...
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Quantum harmonic oscillator meaning

Imagine we want to solve the equations $$ i \hbar \frac{\partial}{\partial t} \left| \Psi \right> = \hat{H}\left| \Psi \right> $$ where $$\hat{H} = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial ...
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How can maximum kinetic energy not equal to total energy in SHM$?$ [closed]

A linear harmonic oscillator of force constant $2×10^6$$ \,\text{N}\,\text{m}^{-1}$ and amplitude $0.01 \,\text{m}$ has a total mechanical energy of $160 \,\text{J}$. Find ratio of maximum potential ...
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Generalizing Wave Equation to two strings connected at a point

Hi physics noob here with a question about strings. I saw that you can derive the wave equation assuming an increasing density of masses and increasing spring constants in a 1-dimensional system of ...
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Quantum Harmonic Oscillator With a Linear "Perturbation"

It is well known that the energy solutions for the unidimensional quantum harmonic oscillator $V(x) = \frac{1}{2}m\omega^2x^2$ are $E_n = (n + \frac{1}{2})\hbar\omega, n \in \mathbb{N}$. In particular,...
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What are the different types of resonances in forced oscillation systems?

I'm currently studying resonances in systems subjected to forced oscillations and have come across various terms and cases that I'd like to understand more clearly. Specifically, I am analyzing a ...
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Physical meaning of Zero-Point Energy

I know that a quantum system can never have 0 energy due the Uncertainty Principle, and its lowest energy is called the Zero point Energy. However, Energy is a relative quantity (atleast in classical ...
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Derivation of Differential Equation of a Simple Pendulum [closed]

This pretty much a simple question and i seem to be making a dumb error here, but nonetheless I can't get the correct answer for the general equation of a pendulum which is :$$\ddot\theta=-\frac{g}{L}...
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Question regarding the half Harmonic Oscillator

In the normal Quantum Harmonic Oscillator (QHO), we normally use the operator method (because it's to elegant), but I recently discovered the problem in Griffiths (prob 2.42) where they ask the same ...
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Velocity Formula in SHM

In Simple Harmonic Motion in one dimension, if we assume $$\text{Displacement}=x=A \text{sin} (\omega t+\phi)\implies \text{velocity}=v=A \omega \text{cos} (\omega t+\phi)$$ From here by substitution ...
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How can you have position basis and energy basis? [duplicate]

In Quantum Mechanics, my understanding is that we have a Hilbert space. If we to model a particle in space we consider the space defined by the basis $$|x\rangle$$ for each $x \in \mathbb{R}$ We then ...
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Interpretation of perpendendicularity of paths

Two particles are oscillating along two close parallel straight lines side by side, with the same frequency and amplitudes. They pass each other, moving in opposite directions when their ...
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How can one "encode" momentum into the wave-equation of a QM harmonic oscillator? [duplicate]

I am learning about Quantum Mechanics using Griffiths book and after reading the section about the quantum harmonic oscillator, I was left wondering how one can construct a solution to the Schrodinger ...
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Calculating the resonance frequency of a spring based on adding additional mass

I have a following problem. I have a spring of unknown spring constant and resonance frequency. I can measure only the force on the spring and the change in length of the spring. I can add mass and ...
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What is the name of the transformation from one harmonic oscillator basis to another centered elsewhere?

If I have a harmonic oscillator basis centered at $x=2$, how do I rewrite it in terms of the harmonic oscillator basis centered at $x=0$? To be more specific: If $|\Psi_n\rangle$ is the $n$th ...
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Mean squared displacement of a free Brownian particle moving in harmonic potential

For a free Brownian particle moving under harmonic potential ($\frac{1}{2}m\omega^2x^2$), the equation of motion can be written as, $$m\ddot{x}=-m\omega^2 x-m\gamma\dot{x}+R(t)\;,$$ where, $\gamma$ is ...
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Can motion be oscillatory but not periodic?

The equation of motion of a particle is $x = A \, \mathrm{cos}\left[(\alpha t)^2\right]$. What type of motion is it? The answer to this question in my textbook was: "Oscillatory but not periodic&...
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Why can we ignore the work done by gravity?

I am working through the problem above, starting with part (d). By the conservation of energy setting the spring in equilibrium as $y_0$ as the difference in length of the unstretched spring to the ...
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Closed expression of eigenfunctions of a two dimensional isotropic harmonic oscillator

Where can one find the closed expression of the eigenfunctions of the 2d isotropic harmonic oscillator? I saw something like this: $$ \psi_{n_r m }(r, \theta) \propto e^{im\theta} r^{|m|} e^{-r^2/2} F(...
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Shape of graph of energy in S.H.M

I'm confused to whether the graph of KE/PE of a simple harmonic motion system is sinusoidal or not those are my best sketches but if unclear, the blue one is in a shape of a sine wave. this question ...
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Probabilistic reformulation of classical Simple Harmonic Oscillator

As an interesting exercise, I was wondering whether we could reformulate classical mechanics in such a way that we could use the same mathematical paradigm we use in quantum mechanics. I'll expose it ...
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References describing how the initial angular displacement of a pendulum affects its damping ratio?

I'm writing a research paper exploring how damping ratio of a simple pendulum relates on its initial angular displacement. In order to validate the findings of my paper, I am required to include a ...
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Directly integrating the Lagrangian for a simple harmonic oscillator

I've just started studying Lagrangian mechanics and am wrestling with the concept of "action". In the case of a simple harmonic oscillator where $x(t)$ is the position of the mass, I ...
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Movement of a mass on a spring in damped SHM

Suppose a ball is connected to a spring attached to a wall, and they are in space, i.e. assume no gravity. The ball is put into a fluid with Stoke's drag and oscillates backwards and forwards relative ...
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How does time period change with damping in oscillations?

Does time period increase or decrease with an increase in damping? I've had contradicting answers. My teacher has told me that time period increases with damping. It does make sense in a way because ...
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Ladder operators and creation & annihilation operators - different between $a$, $b$ and $c$ [closed]

Usually, the ladder operator denoted by $a$ and $a^\dagger$. In some case, people talk about the creation operator and denote it by $c$ and $c^\dagger$. Recently I see another notation, $b$ and $b^\...
Yohay Halfon's user avatar
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The interpretation of the Bose occupation factor

I was reading into the Oxford solid state basics, by Steven H.Simon and I stumbled upon a confusing interpretation of the Bose Occupation factor: $$n_B (x) = \frac{1}{e^x-1}$$ with: $$x = \beta \hbar \...
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Doubt obtaining the expected value of $x^2$ of a bidimensional harmonic oscillator

Just for the sake of context I'll add a little bit of introductory of the theory we were doing: Say we are in the context of a bidimensional isotropic harmonic oscillator with an energy found of $2\...
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Plot of Number of Oscillations of a Pendulum

I have been studying the oscillations of a ball attached to a string which is released from some initial angle $\theta$. The number of complete oscillations over a certain time interval $\Delta t$ is ...
Tom's user avatar
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Can we equate SHM with motion in a circle?

So in SHM, $v = r\omega\cos\omega t$. But we also know that $v = r\omega$ from circular motion. Then, we can write $$r\omega = r\omega\cos \omega t$$ $$1 = \cos\omega t \tag{1} \label{1}$$ $$0 = \...
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Ground state of an harmonic chain and Klein-Gordon vacuum

Consider the lagrangian of a system of classical coupled harmonic oscillators of mass $M$, connected with springs with elastic constant $\chi$ and connected to the background with springs of elastic ...
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What is the equation if that projection starts SHM on the $x$-axis from extreme position?

Consider A particle performing Uniform Circular Motion. We know that its projection on diameter performs SHM. Then, if that projection starts SHM on the y axis from mean position, then $y=A\text{sin}(...
NERD's user avatar
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Does it make sense to talk about individual energies of interacting quantum particles?

Does it make any sense to talk about energy of any one particle in an interacting system? For example if we talk about a system of two coupled quantum harmonic oscillators of same mass and frequency, ...
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Does the motion of a simple pendulum shifts to SHM, due to damping?

I have doubt in damping oscillations of a pendulum. Let's consider a simple pendulum in a drag medium like air. Suppose if the angular displacement (theta) is larger (say 40°), the oscillations start ...
Rajesh R's user avatar
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How to get the factor of $n^{-27/4}$ in number of open string states from the calculation in GSW's book?

In section 2.3.5 of Green, Schwarz, Witten's book on string theory (volume-1) pp. 116-118, the objective is to calculate an Asymptotic Formula for Level Densities $d_n$ for open bosonic string theory. ...
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How to solve the coupled equation of motion? [closed]

there we have the EOM: \begin{align*} \alpha q_{2} + \lambda - \ddot{q}_1=0 \\ \alpha q_{1} + \lambda - \ddot{q}_2=0 \end{align*} and $q_{i}$ is the canonical coordinates. Can I use the Fourier ...
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Eigenvalues and Normal Modes in SHM

I'm reading Symmetries part from the textbook provided by MIT OCW Physics3 8.03SC course, but have a question about the condition to find normal modes of SHM. In the book they mentioned $S$ - symmetry ...
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What defines weightlessness feeling during a fall?

So I know that we feel weightless during a free fall as there's no normal force acting on us or my other way of thinking is that considering a lift and both lift and I fall with $g$ then no normal ...
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