# 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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### Driving a quantum system

I don't understand the mathematical form of an external driving force on a quantum system. For example, say we were driving a quantum harmonic oscillator. You would have the Hamiltonian for the ...
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### Linear harmonic oscillator in quantum mechanics, why we expand the potential into Taylor series?

We have started our first problem in QM by solving Schrödinger equation for the linear harmonic oscillator potential. I noticed that the first thing we did was to expand the potential near the stable ...
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### What are the maximum spring lengths of a double spring pendulum?

NOT a duplicate of Maximum length stretch of vertical spring with a mass?, I am asking about a system with two connected springs, as shown in this diagram For a single spring, you can simply equate ...
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### Power Spectral Density of vacuum fluctuations in harmonic oscillator

I was thinking about the question whether vacuum fluctuations in the non relativistic quantum harmonic oscillator actually have a power spectral density. The power spectral density is (to my knowledge)...
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### Effect of shortening length of pendulum on angular amplitude

A simple pendulum has a point like bob of mass $m$ and a light inextensible string. Suppose, by means of some arrangement, we shorten the string extremely slowly, so that at any instant, the ratio of ...
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### Orbital angular momentum as sum of harmonic oscillators

On section 7.3 of Ballentine's "Quantum Mechanics: A Modern Development" there is a really nice argument on why the eigenvalues of the total angular momentum operator must be integer, cf. e....
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### Why is max power below natural frequency

I can't understand why intuitively the max power is at the natural frequency. We were told in our notes that the power absorbed is exactly equal to the rate at which energy is dissipated. Yet, if we ...
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### Why does the Klein-Gordon harmonic oscillator have non-normalizable states?

The equation given here is the vector Klein-Gordon equation for the Harmonic oscillator potential. I have read about it in some papers. As far as I know, it gives the correct energy eigenvalues but ...
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### An approximation for path integral kernels [closed]

The kernel in the path integral formulation for a one dimensional system is given by \begin{equation} \label{eq:kernel1} K(x_b, t_b; x_a, t_a) = \lim_{N \to \infty} \sqrt{\frac{m}{2\pi i \...
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### For what configuration of oscillators on a ring the ring will not move?

Once upon a time, on a two-dimensional plane. Imagine we place pairs of oscillators on an incompressible ring. The ring has mass $M$ and radius $R$ (though I don't think this is of any importance), ...
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### Key on a string: double pendulum & consequences of not using a bob on a string?

While trying to do an at home experiment about a single pendulum, I used a key on a rope and let it swing. I came to the conclusion that this could not be a single pendulum, because of the extra '...
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### Difference between reactive (coherent) and dissipative coupling in open quantum systems?

I am unsure about the physical interpretation of the different types of coupling, I understand that reactive coupling is manifested by a term within the Hamiltonian and dissipative is via a term in ...
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### Is the amplitude of vibration of all particles equal in a sound wave?

[Assume ideal conditions and simple harmonic wave] My book showed resemblance in a lot of equations related to both transverse and longitudinal waves although they work very differently. the ...
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### Mass-spring damper system subjected to harmonic excitation [closed]

The following graphs represent the steady state responses shown in red of a mass-spring-damper system subject to harmonic forcing shown in blue in the graphs below. How can you determine which of the ...
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### Temporal stability of multimode coherent states

For the standard quantum harmonic oscillator, the coherent states $\{|\alpha\rangle, \alpha\in \mathbb{C}\}$ are temporally stable. That is, $$e^{-iH t}|\alpha\rangle = |e^{-i t} \alpha\rangle,$$ ...
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### Stopping time of a damped pendulum

I'm interested in how long it would take for a damped pendulum to stop moving. I've found that the general form of a function $x(t)$ which returns the angular displacement of the pendulum given some ...
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### Order of terms & harmonic approximation

I have a question regarding some basics about determining the order of a term in the context of the harmonic approximation. I am regarding the equation \begin{align} 2ml\dot{l}\dot{\phi} + ml^2\ddot{\...
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### What does 'seconds squared per meter' mean?

I was wondering about the meaning of seconds^2 / meters (more specifically, period^2 / length - of a pendulum, after the square root graph of period vs length is linearized). I know that acceleration ...
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### How to perform the multi-scale analysis beyond harmonic oscillations?

I occasionally see this interesting method called multi-scale analysis. From what I understood, it is used to perturbatively solve a perturbed harmonic oscillator, meaning that the equation of motion ...
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### Pendulum on an accelerating train with changing length

Based on my own research, I found a general solution that can model a pendulum found on an accelerating train. The following solutions are based on small angle approximations. F=-mgsin θ F≈-mgθ, ...
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### Does a tuning fork's resonance frequency increase with a repulsive force on it in AFM?

I have a tuning fork sensor with a probe tip that would be used in atomic force microscopy. Am I correct that the quartz tuning fork's resonance frequency increases with a repulsive force on it and ...
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### Simple Harmonic Motion of a particle attached to a spring all inside a circle

I want to understand the following problem: The way I approached this question was using Hooke' Law. I managed to solve for $\lambda$ and later made a differential equation through which I got a ...
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### Overlap between relative motion eigenbasis and single-particle eigenbasis in harmonic oscillator

The energy eigenstates of two particles in a 2d isotropic harmonic oscillator can be described in terms of a product basis of one particle states (I'm using the angular momentum basis with angular ...
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### Along which axis is the moment of inertia of a harmonically oscillating body calculated?

I have been learning about oscillating bodies and recently stumbled upon physical pendulums. Now the problem is i don't understand about which axis is the MOI calculated while finding the TIME PERIOD(...
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### How to calculate the damping ratio of a structure with a pendulum tuned mass damper?

I'm a highschool student investigating the damping of an oscillating structure by a pendulum mass damper. The structure has an accelerometer at the top to measure the acceleration. Although I know it ...
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### Sidebands in spin-oscillator coupled devices

I am researching spin-oscillator hybrid devices (particularly those coupled to NV centers), and I have come across this paper investigating the strain field of a cantilever and its effect on the NV ...
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### What is the velocity of the surface used to project the motion in this experiment for SHM? [closed]

What is the velocity with which the projection surface moves in this experiment?
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### Showing a system is in Simple Harmonic Motion [closed]

A force of $F=(8-2x)$ is applied to a $2kg$ object in the $X$-direction. It is released from $6m$ away from the $x=0$ point. Show that the object is in a Simple Harmonic Motion. And derive a formula ...
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### Double Pendulum and newtonian mechanics [duplicate]

I am searching for a way by which I can find the displacement vector of a double pendulum as functions of time.( without using lagrangian mechanics)
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### Why does SHM happen in vacuum?

Very simply put, I don't understand why SHM happens. If a spring or a pendulum can attain lowest potential energy by stopping at the mean position, why don't both of them stop at the mean position ...
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### Differential equations of a forced coupled spring-pendulum system

Currently working on a problem and I can really figure out how to write the differential equations for it. Here's the situation: So we have a mass $m$ tied to the wall with a spring of constant $k$. ...
I tried to find the DOS of a 2D harmonic oscillator using $2$ different methods but the results aren't the same. The energy spectrum is: $$E_n=\hbar\omega(n+1)\tag{1}$$ and the degeneracy of the $n$-...
Suppose we have a system of $N$ masses connected by $N+1$ springs, where the stiffness of the springs alternates between $k$ and $2k$. We assume N to be an even number. Determine the forms of the ...