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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108 views

SHM of a rigid body

In the analysis of SHM of a point sized bob oscillating with small angular displacement we can analyse the SHM in both linear and angular terms and arrive at the same answer and this should be true ...
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Why two vibrations of different frequencies and amplitudes need to be commensurable when the resultant vibration is formed from their superposition?

I was reading chapter 2 of AP French's Vibrations and Waves. In the section "Superposed Vibrations of Different Frequency, Beats", this paragraph confused me :- "Unless there is some ...
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Relative motion of a particle of a moving wave with respect to the wave itself

When deriving the equation for the velocity of a transverse wave in a string of tension $T$ and linear mass density $d$, we assume that a small part of the wave is undergoing circular motion with ...
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Can the equivalence principle be safely used in non-relativistic mechanics?

Imagine an ideal pendulum in a train. While the train is in uniform motion, Newton's laws apply within the train, and we can easily write down the equations of motion for the pendulum. Now assume the ...
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Question regarding the expected value of the momentum of a Gauß distribution

Given the position amplitude $$\psi(q) = \frac1{\sqrt{L \sqrt{\pi}}} e^{-\frac1{2}\left(\frac{q-q_0}{L}\right)^2} $$ I get the expected value for the position $$\langle \hat{q} \rangle = q_0$$ which ...
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Time-dependent Schrödinger equation of the harmonic oscillator

hello I have the time dependent Schrödinger equation of a harmonic oscillator $$i \hbar \frac{\partial}{\partial t} \Psi(q,t) = - \frac{\hbar^2}{2M} \frac{\partial^2}{\partial q^2} \Psi(q,t) + \frac{M ...
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222 views

Quantum Harmonic Oscillator - $x^4$ expectation value [closed]

I need to evaluate $\langle n| \hat{x}^4 |n\rangle$ for the quantum harmonic oscillator. By replacing $\hat{x} = \sqrt{\frac{\hbar}{2m\omega}}(\hat{a}^{\dagger}+\hat{a})$ I've reduced the problem to ...
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Average energy of an SHM

Why do we usually calculate the average potential or kinetic energy of a simple harmonic motion with respect to time, why not with respect to position? Why even calculate average energy for an SHM? ...
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Units of angular frequency in a simple harmonic oscillator [closed]

The equation of a simple harmonic motion can be $x=A \cos(\omega t)$. $\omega$ therefore has units of $radians/sec$. I was solving some problems when I found a statement on my notes $x=\left(1+\...
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Are Spring Oscillations Goldstone Modes?

When considering a spring-like object, it seems almost intuitive that it would be "springy", i.e., harmonically oscillate. However, these oscillations occur at a macroscopic scale, unlike ...
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Hooke's law and atomic scale

Hooke's law is derived in this answer by Taylor expanding an energy potential with arbitrary functional form. This is dependent on the displacements involved being "small". By considering ...
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1answer
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Deriving system properties from energy spectrum [duplicate]

To what extent can we derive the properties of a system given the existence of a hermitian operator with a particular spectrum? For example, if we know that there exists a hermitian operator with ...
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Isn't the potential energy of a spring in SHM at equilibrium zero? [closed]

I was just solving some problems on SHM and there's a question which asked the PE of the body at equilibrium, the question: A body of mass 2 kg suspended through a vertical spring executes simple ...
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Harmonic oscillator with potential multiplied by a constant [closed]

I know the solution for the classic problem; a particle in an harmonic oscillator with the given potential; $ V = m\omega^2x^2/2$. Let's denote the eigenfunctions by $ \phi_n $ and the eigenvalues by $...
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What happens to $E_n$ of quantum harmonic oscillator if the potential changes?

As we all know there is the normal quantum harmonic oscillator with potential $V= \frac{m\omega^2x^2}{2}$ and we get $E_n =\hbar\omega(n+1/2)$ What is my $E_n$ when the potential $V= 2m\omega^2x^2$ or ...
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Adiabatic Invariant in Variable Mass Oscillator

Suppose you have an harmonic oscillator whose mass is adiabatically changing such that $T\frac{dm}{dt}\ll m$ where $T$ is the period of the motion. It could for example be an ice ball slowly melting ...
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Expectation value of the anticommutator of the bosonic creation and annihilation operator

The number operator is given by: $$\hat{n}= a^{\dagger} a.$$ For a presentation, I have to derive the expectation value of the anticommutator of the bosonic operators $a$ and $a^{\dagger}$ : $$\...
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Need help understanding an equation of motion for a pendulum [closed]

I solved the Lagrangian of a simple pendulum (with help from online examples as this concept is new to me) and ended with the following: $$\ddot{\theta}+\omega^2\theta=0$$ But in the example I was ...
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How to turn the equations of motion into horizontal and vertical positions of the mass as functions of time? [closed]

We were given this physics problem: Let $z(t) = y(t) + \alpha$ and recall the Simple Harmonic Motion Position Equation $x(t) = Acos(\omega t + \phi)$. From the equations of motion, one can show that ...
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Is there a way to get the generating function of Hermite polynomials?

I would like to know if there is any physical model in which the generating function of the Hermite polynomials arises, I know the problem of the quantum harmonic oscillator but I have not found the ...
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1answer
74 views

Finding the equilibrium point of a system that rotates and oscillates [closed]

I'm currently learning physics and I don't always know the right approach in order to solve a problem. I ran into a problem with some sort of oscillation combined with a circular motion. Problem: An ...
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Deduce how much does damping affect the angular frequency of a harmonic oscillator

I am asked to deduce by how much does damping change the angular frequency of a harmonic system. I immediately thought to use the equation $\omega$ = $\sqrt{\omega_0^2 - \frac{\gamma^2}{4}}$. However, ...
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Calculation of the oscillation frequency of a rotating system that performs small oscillations

I'm currently learning physics and sometimes I don't know how to approach a problem. I ran into this problem with oscillations combined with a circular motion, and I can't really find the answer. ...
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1answer
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How to compute $\langle n'| (a^++a)^k | n \rangle$ for arbitray $k$? [duplicate]

I'm trying to compute the 2nd order correction to the energy spectrum of a 1D quantum harmonic oscillator when a perturbation of the form $\gamma\,\hat{x}^k$ (with $\gamma\ll1$) is added to the ...
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Need help creating the Lagrangian for a coupled pendulum [closed]

I know that for 2 separate single pendulums, the kinetic and potential energies are: $$KE = \frac{1}{2}I(\dot\theta_1^2 + \dot\theta_2^2)$$ $$PE = 2mgl - mgl(\cos\theta_1 + \cos\theta_2)$$ But I don't ...
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Selection rules in the harmonic oscillator

If I have a molecule with two different atoms and this molecule gets excited by EM radiation, we can only see transitions with $\Delta n=\pm1$ in the molecule spectrum. I found this in my physics ...
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Symmetry and Finite Coupled Oscillators

For an infinite system of coupled oscillators of identical mass and spring constant k. The matrix equation of motion is $\ddot{X}=M^{-1}KX$. and the eigenvectors of the solutions are those of the ...
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Relativistic energy of harmonic oscillator

What is the relativistic energy of an harmonic oscillator: $$\frac{m_0 c^2}{\sqrt{(1-\frac{v^2}{c^2})}}+\frac{1}{2}kx^2$$ Or $$\frac{{m_0 c^2}+\frac{1}{2}kx^2 }{\sqrt{(1-\frac{v^2}{c^2})}}$$ I think ...
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Harmonic oscillator in QFT

Given a single bosonic mode with frequency $\omega_0$, such that $\hat{H}=\hbar\omega_0(\frac{1}{2}+\hat{a}^{\dagger}\hat{a})$ how should one show the equivalence between the coherent state path ...
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Force versus time and force versus angular displacement graph - simple pendulum

How to graph Force versus time and force versus angular displacement graph in simple harmonic motion of a pendulum system (Force= net force directed towards equilibrium) It seems to me that force ...
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Optimal control of a damped harmonic oscillator through a lossy, dispersive medium

Say a damped harmonic oscillator is at the far end of a dispersive, lossy medium governed by, say, Debye relaxation. I would like to determine what the optimal arbitrary input waveform is to obtain ...
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Why is the time period of a pendulum with a spring of force constant $k$ and a bob of significant mass $m$ the same on the Moon as on the Earth?

A question I came across in class today: How will the time period of a loaded spring change when it is taken to the Moon? What I've been told: The formula for the time period of a loaded spring $$ ...
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Uniqueness of phonon vacuum

Consider a quantum harmonic chain, described by the Hamiltonian: $$\hat{H}=\sum_{j=1}^{N} \frac{\hat{p_j}}{2M} + \frac{1}{2}K(\hat{u}_j-\hat{u}_{j+1})^2,$$ where $\hat{p}_j$ is the momentum and $\hat{...
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Approximation of Stable Orbits as Harmonic Oscillators

A textbook on classical mechanics I am currently reading considers the stable orbit (at $r_0$) of a body subject to the power law: $$\mathbf{F}(r)=-Kr^n\mathbf{\hat{r}},\quad n\in\mathbb{Z}$$ $$\...
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Postulate of a priori probability and harmonic oscillator

According to the fundamental postulate of a priori probability in Statistical Mechanics: An isolated system in equilibrium is equally likely to be in any of its accessible states. But for a ...
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Impact of mass on resonance

Two questions ...why is it that increasing the mass of a mass-spring system increases the resonance amplitude? And why is it that increasing mass causes its resonance curve to be 'narrower' - i.e. it ...
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Why is Simple Harmoic Motion in a Spring modelable with rotation without invoking calculus?

Is there a good reason why we can use uniform circular motion to get the equations for a mass on a spring, without invoking calculus? This relationship is often used to find the equations for the ...
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1answer
158 views

Block attached to spring oscillating on a surface with friction

Consider a block of mass $m$ moving with initial velocity $v_o$ attached to a spring with spring constant $k$, on a terrain which has a coefficient of kinetic friction $\eta$ and coefficient of static ...
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161 views

Deriving the Hamiltonian for a simple pendulum using mechanical momentum as a free parameter

So when we covered the derivation of a simple pendulum we , and from what ive found on the web, defined our free parameter as $q=L\theta$ and arrive at the Hamiltonian for a Harmonic oscillator. But ...
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1answer
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How the formula for mean energy of quantum harmonic oscillator is derived?

I mean this formula: $$\varepsilon ={\frac {h\nu }{2}}+{\frac {h\nu }{e^{h\nu /kT}-1}}$$ Is there full derivation somewhere?
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Simple Harmonic Motion example [closed]

https://en.wikipedia.org/wiki/Escalator https://en.wikipedia.org/wiki/Simple_harmonic_motion Can we classify escalator device as a Simple Harmonic motion example?
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Why is the center of 'percussion' called so?

I came across this word Center of Percussion while reading SHM from Resnick Halliday Krane Vol. 1 and couldn't figure out why it is called so. Please help me in doing so.
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Motion of a damped harmonic oscillator due to air resistance

I did a video (30 fps) of the motion of of a mass suspended vertically by just one spring in one side. This spring is held by a rope connected to a fixed rod, as shown in the picture. From the bottom ...
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Lagrangian of the Stretching Mode Vibration of the Acetylene Molecule

Actually, this is part of a homework question in my classical mechanics class. The question requires me to derive the eigenfrequencies of the acetylene molecule's bending and stretching modes under ...
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Harmonic motion of droplet of a non-viscous fluid

We are given that there is drop of density $d_1$ and radius $r_1$ with a surface tension $T$.Next we are told that the drop is made into an ellipsoidal shape. How do we find the time period of S.H.M ...
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Simple oscillator

When we are studying harmonic oscillations, we come across with this equation: $\ddot{x} + c^2x = 0 \tag{1}$ We know immediately that the period of the x oscillations is equal to $T = \frac{2\pi}{c}$...
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Liouville's theorem to show ergodicity?

a) Consider a harmonic oscillator with Hamiltonian $H=(1/2)(p^2+q^2)$ show that any phase space trajectory $x(t)$ with energy $E$, on the average, spend equal time in all regions of the constant ...
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2D harmonic oscillator trajectory

Consider the Hamiltonian for the classic planar harmonic oscillator: $$H = H_x + H_y$$ where $$H_x~=~\frac{1}{2}(p_x^2+x^2), \qquad H_y~=~\frac{1}{2}(p_y^2+y^2).$$ So it is possible to obtain a set of ...
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What variables can affect time period in a pendulum other than length?

I have been researching for almost a week looking for some variable that might affect the time period in a pendulum other than length and the only thing I found is the medium and that is due to ...
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2answers
251 views

Get the position of a spring as a function of time [closed]

If I have a spring that I displace by $x$ from its equilibrium and then let go, what position will the spring be at after $n$ time. (taking friction into account) I made a quick Illustration to ...

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