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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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Phase difference

Considering two spring-and-block harmonic oscillators set up side by side with identical spring constants and block masses, if we observed that the blocks' positions were the same (moving in opposite ...
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The allowed energies of 3D harmonic oscillator [on hold]

I'm trying to calculate the allowed energies of each state for 3D harmonic oscillator. $$ E_n = (n_x+\textstyle\frac{1}{2})\hbar \omega_x+ (n_y+\textstyle\frac{1}{2})\hbar\omega_y+ (n_z+\textstyle\...
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Proving the raising and lowering of the raising and lowering operator

I am given a written proof of $\hat A^{\dagger}[u_n] = \sqrt{n+1} \ u_{n+1}$, and from it, and told to similarly prove $\hat A[u_n] = \sqrt{n} \ u_{n-1}$. However, in the written proof for $\hat A^{\...
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Solutions to an Underdamped Oscillator

In many of the books talking about damped Simple Harmonic Motion, the underdamped Oscillator is treated as follows: We have Newton's Second Law Equation as - $$m\ddot{x} + r\dot{x} + sx = 0 $$...
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Spring constant and dispersion relation

In order to calculate the dispersion relation (i.e $w(k)$) for the electrons and protons, I used the following relations: $ E = ℏω$, $p = ℏk$, and I substituted them in this formula for energy: $E = ...
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Simple pendulum and energy [on hold]

a simple pendulum with length l is known to oscillate with a period T seconds. If the length of the pendulum is increased to 2l and the maximum displacement remains the same, the total energy of the ...
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Magnetic force and field involving simple harmonic motion

Is there a simple harmonic motion when the electromagnetic compass was made by suspending a coil. The coil align itself perpendicular to the horizontal components of the earth's magnetic field and ...
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Simple harmonic motion in physics [closed]

When can you say that an electromagnetic compass is experiencing simple harmonic motion?
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Step-by-step guide to finding the phase constant in simple harmonic motion

Working on some simple harmonic motion problems involving an oscillating spring/mass system ... the usual. I never really understood exactly how to find the phase constant for the $$x(t)=Acos(wt+phase ...
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Period of coupled cycloidal pendula

If you were to couple two pendula following cycloidal paths would they still show similar properties to a single cycloidal pendulum? For example, could you in some sense still say that the period is ...
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Harmonic oscillators in fluids and driven oscllations

If given a normal spring/mass system and letting the mass oscillate in a fluid say water, would it be possible for the motion of the fluid, if the fluid is moving to create a driven oscillation and ...
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QFT Path integral for Harmonic Oscillator derivation [closed]

I am currently working through Srednicki's QFT, and am stuck on a step he uses in Eq. 7.3 to derive the path integral for the harmonic oscillator. He writes $$H = \frac{1}{2m}P^2 + \frac{1}{2}m \...
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Expressing the Hamiltonian of two coupled quantum harmonic oscillators as two independent oscillators [closed]

Shankar's Principles of Quantum Physics describes separating the Hamiltonian of two oscillators in the form $$H={p_1^2\over 2m}+{p_2^2\over 2m}+{m \omega^2 \over 2}(x_1^2+x_2^2-(x_1-x_2)^2) \, .$$ ...
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Equation of coupled springs : where does this potential come from?

In this document, we try to derive the equation of two coupled springs as in this picture. At the bottom of the page 2, they say : it would be more efficient to introduce the potential energy ...
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When to use sine or cosine when computing simple harmonic motion

For simple harmonic motion (SHM), I am aware you can start of using either sine or cosine, but I am a bit confused as to when you would start off with sine rather than cosine. I know that a sine graph ...
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What would be the minimum velocity of a particle performing S.H.M.?

We were asked a simple question on a test: What is the maximum and minimum velocity of a particle performing an SHM? Note here that we're talking about a generic standard SHM here. If the maximum ...
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Energy of harmonic oscillators

I've calculated the energy of a classical harmonic oscillator (HO) as: \begin{align*} \overline E = \overline{E_K} + \overline{E_P} = \frac{\overline{p^2}}{2m} + \frac{k\overline{x^2}}{2} = \frac{...
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Why does the anisotropic harmonic oscillator has no coupling between different directions?

The hamiltonian of the anisotropic HO e.g. in 2d is typically written as $$H=\frac{1}{2m}\left(p_x^2+p_y^2\right)+\frac{1}{2}m(\omega_x^2 x^2+\omega_y^2y^2)$$ What I wonder is why there is no ...
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3-dim Hyperboloid harmonic function [closed]

I have a question about this paper. In this paper, we will solve the Equation of motion of scaler field on open de Sitter space. So we use the hyperboloid coordinate. The eigenvalue equation of the ...
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Oscillation experimetns [closed]

So currently I have trouble evaluating my experiment because I do not have literature values for my experiment. So the experiment was, I found the relationship between the surface area of the damping ...
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Why do we nondimensionalize the Schrödinger equation when solving the quantum harmonic oscillator?

I read about how to solve the Schrödinger equation for the quantum harmonic oscillator in one dimension. It started with the Schrödinger equation, $$ \frac{p^2}{2m}\psi(x, t)+\frac{1}{2}m\omega^2x^2\...
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What is an anisotropic harmonic oscillator?

I can't find any explanation of it anywhere in the internet. How is it different from an isotropic harmonic oscillator?
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How is damping force related to frictional force?

I have a problem and I want to determine the friction coefficient using diffential equations i.e. solving the equation (this is from a Under-damped Oscillator) $$ m\ddot x+c\dot x+kx=0. $$ Suppose I ...
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Forced Oscillations and Complex Representation

An oscillating force $F \cos \omega t = \Re\{Fe^{i\omega t}\}$, where $F$ is real, is applied to a mass $m$ on the end of a spring with spring constant $k$. The displacement, $x$, of the particle can ...
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Newtonian mechanics: pendulum spring system [closed]

I am thinking about the following pendulum-spring system. But something is off with my equation of motion. Problem We have a uniform rod of length L with mass m pivoted at one end. We ...
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Centripetal force and energy

I have a question related to centripetal force. Say there is a pendulum, then \begin{align} mg[h_{max} - h_{current}] = \frac{1}{2}mv^2 = \frac{1}{2}\ell F_{cp} \, . \end{align} Solving for ...
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Find the drag coefficient from a ln(acceleration) v time graph [closed]

I don't have a huge amount of information to go on, but I basically have the decay of a pendulum v time, and then taking the decay of this graph, I have a plot of ln(Theta max) v time. From this I ...
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Potential energy of and forces within a massive spring

So, I've been doing problems related to calculating the time period of SHM for a spring mass system where the mass is M, and the spring has a finite mass m. The standard method of dealing with this ...
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Wave Equation Using Symmetry Argument

In Howard Georgi's The Physics of Waves, one chapter discusses space-invariant symmetry on the classic beaded string. From the symmetry matrix, we can then derive complex normal modes and the ...
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Two different definitions of ladder operator for Harmonic Oscillators

As it happened, I accidentally referred to two different editions of Introduction to QM by Griffiths. In the second chapter, while defining the ladder operator for harmonic oscillators, he used ...
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Simple Harmonic Motion of a Pendulum Derivation Questions [closed]

A simple pendulum with mass m on a string of length l is released from rest at an angle of $\theta_0$ to the vertical. (i) Assuming that the pendulum undergoes simple harmonic motion, find an ...
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2D isotropic quantum harmonic oscillator: polar coordinates

This may come a bit elemental, what I was working on a direct way to find the eigenfunctions and eigenvalues of the isotropic two-dimensional quantum harmonic oscillator but using polar coordinates: $...
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What is the Hamiltonian in the “energy basis” for a simple harmonic oscillator?

My textbook says that for a simple harmonic oscillator the Hamiltonian can be expressed in the "energy basis" in this way: $$\hat H=\hbar\omega\bigg(\hat a^{\dagger}\hat a + {1\over 2}\bigg).$$ I ...
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Harmonic Oscillator Trial Wavefunction

I was learning today about trial wave functions for a harmonic oscillator. We learnt that the solution to Schrödinger equation for a harmonic oscillator is a Gaussian curve, i.e. $$ f(x) = e^{-x^2} ....
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Significance of the complex component in the underdamped harmonic motion equation [closed]

The following differential equation represents the motion of a body of mass $m$ and displacement $x$ from the mean position, that is attached to a spring of force constant $a$ and viscous damping ...
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Energy basis to the X basis

On Shankar page 217 when going from the operator representation to the differential representation he starts with $$a|0\rangle = 0$$ And says that with a projection on the X basis we get $$|0\...
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Spacing of resonant modes in a (laser) cavity

I don't see why the frequency spacing between resonant cavity modes should decrease with higher frequency. Here's what I have: For a laser cavity of length $L$, from the condition of constructive ...
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Problem in harmonic motion

In the harmonic chapter of my book it is written that $\omega^{2}=\frac{k}{m}$ where $\omega$ is frequency and $k$ is the spring constant. But how did they get to this formula? What is its derivation? ...
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Particle coupled with $N$ harmonic oscillators

If I have a particle coupled with a bath of $N$ independent harmonic oscillators and that is subject to an harmonic potential (a sort of Ullersma's model), can I tell that my system is ergodic since ...
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Decomposing a prepared quantum harmonic oscillator state

Problem I'm attempting to decompose a system prepared in state $|\Psi_\mu\rangle$, defined by $$\Psi_\mu(x) = \left( \frac{m\omega}{\pi \hbar} \right)^{1/4} \exp \left( -\frac{m\omega (x-\mu)^2}{...
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Why does the length of a pendulum cause different natural frequencies of pendulums in Barton's pendulum?

In Barton's pendulum, the pendulum with string that is the same length, L, as the brass bob (source of driving frequency) has natural frequency equals to the bob's driving frequency. The pendulum ...
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Precision of measuring $N$-fold period one time vs single period $N$ times

One quick (I was told, trivial) question. Is it more precise to measure $N$-fold period one time, or to measure a single period, $N$ times? Ex. You could let the circular pendulum do 10 swings and ...
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1answer
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I need to find the drag coefficient of a pendulum bob [closed]

Is it possible to find the drag coefficient of a pendulum bob from the damping caused on it during swinging. I will be able to measure its displacement from the point of origin and plot it against ...
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2answers
92 views

Exact equation of exponential curves of underdamped harmonic motion

I was studying the underdamped harmonic motion and got curious about the fact that the decreasing exponentials $\pm Ae^{-\gamma t}$ are good approximations only for light damping $(\gamma<<\...
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1answer
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Transition amplitudes

We have a forced simple harmonic oscillator Lagrangian $$L = \frac{\dot{\phi}^2}{2} - \frac{m^2{\phi}^2}{2} + f(t)\phi \, .$$ The external force goes to $0$ as $t \to \pm \infty$. I'm trying to ...
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1answer
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How do we know which term to attach a phase factor to in a state equation?

I need to find the state of a particle in a one-dimesional harmonic oscillator where a measurement of the energy yields the values $\hbar\omega\over 2$ or $3\omega\hbar\over 2$, each with a ...
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Why does the position uncertainty of a harmonic oscillator not have the expectation value squared term?

My question is about the derivation for the uncertainty of a quantum harmonic oscillator in the non-zero ground state energy. In my textbook A modern Approach to Quantum Mechanics by John S. Townsend ...
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Why is the number of excited vibrational modes $g(\nu)d\nu$ proportional to $x^2e^{-x}$ in Debye's theory?

I come across a problem in Terrell Hill's "Introduction to statistical thermodynamics" saying that: In the Debye theory, the number of excited vibrational modes in the frequency range $\nu$ to $\nu+...
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1answer
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Angular SHM and center of mass

This has been confusing me for a while. Consider a solid, homogeneous rod of mass $m$ and length $l$, hanging from a fixed pivot. Its center of mass is located at $\frac{1}{2} l$, and its moment of ...
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Numerically Modeling Coupled Oscillators Point Masses

I seek to model the motion of two coupled oscillating point masses as shown below: Note that x1(t) models the leftmost point mass and x2(t) is the motion of the rightmost point mass. I would like to ...