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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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Calculating exact energy levels of perturbed Hamiltonian

I wish to find the exact energy levels of the following perturbed hamiltonian. $$\hat{H}=\frac{p^2}{2m}+\frac{m\omega^2}{2}x^2+\alpha x+\beta p^2.$$ I believe that it can be solved by using the ...
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Distance of two indistinguishable particles

Consider: The wavefunction of a two-particle system (both Fermions and Bosons possible): $$ \psi_\pm(x_1,x_2) = \sqrt{\frac{1}{2}}[\psi_n(x_1)\psi_m(x_2) \mp \psi_m(x_1)\psi_n(x_2)] $$ And a ...
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Spring oscillation model

When a spring - in real world - is extended $Xo$ from its natural position, it oscillates and eventually decreasing it's amplitude with time, comes to a stop. Is this a damped system or no? If yes how ...
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If there are eigenstates of $L_z$ in a degenerate subspace, are there also eigenstates of $L^2$?

The question arises from an exercise but tackles deeper understanding of angular momentum operators. Suppose we have a 2D harmonic oscillator and an infinite square well in the third dimension: \...
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Centrifugal force and centripetal force in fixed coordinate frame

Hello I have a question related to centrifgual force. So in basic textbooks the movement of a particle in a rotating coordinate frame is derived. A starting point is to consider a fixed and a rotating ...
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A pendulum with an hollow bob full of flowing sand [duplicate]

THE PROBLEM Consider a pendulum with an hollow spheric bob of negligible weight filled with 1.1 kg of fine sand and hanging from an inextensible wire of negligible mass and $l= 30 m$ long. At the ...
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Creation and Annnihilator Operators: generality and meaning

I am studying my fisrst course in quantum mechanichs where we treated the example of the Harmonic Oscillator through the Weyl Heisenberg Spectrum Generating Algebra Method. In that context we ...
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Basics of simple harmonic motion

I was told in my class that simple harmonic motion (SHM) describes the motion of the projection onto a straight line of the motion of a particle undergoing uniform circular motion (i.e. with constant ...
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Harmonic oscillator path integral: regularizing the functional determinant

From Polchinski's Vol. 1 Appendix A, we can reduce the Euclidean path integral for the 1D harmonic oscillator to computing $(\det\frac{\Delta}{2\pi})^{-1/2}$ where $$\Delta = -\partial_u^2 + \omega^2.$...
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Charged particle and SHM [on hold]

A punctual charge +Q is fixed at the bottom of a vertical cylinder. Another particle of mass m and charge +q is placed in the cylinder above +Q, so its bound to move vertically without friction. Show ...
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I can not solve this quantum harmonic oscillator question in Cohen's book [closed]

If anyone can help me, I'm very grateful. For a) I tried to made this: U^dagger(t, 0) a e^{-iHt/(hbar)} | psi_n > = U^dagger(t, 0) a e^{-i (hbar)omega (n + 1/2) t/(hbar)} | psi_n > = e^{-i omega (...
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At what amplitude is maximum tension produced? [closed]

While I was studying angular SHM, a question struck me. Suppose there were two masses $m$ connected by a light rod at a distance $\frac{l}{2}$ from the centre of the light rod. If the centre of the ...
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Simple Harmonic Motion - magnitude of restoring force [closed]

The bob of a simple pendulum of mass 0.012kg swings with amplitude 51mm. It takes 46.5s to complete 25 oscillations. Calculate: A) length of pendulum B) magnitude of the restoring force that acts on ...
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Motivation behind action when deriving ''Strings as Harmonic oscillators" in Zwiebach's book on String theory

Page 248 gives us this action and he simply says that we will assume it correct. $$ S=\int d \tau d \sigma ~\mathcal{L}=\frac{1}{4 \pi \alpha^{\prime}} \int d \tau \int_{0}^{\pi} d \sigma\left(\dot{X}...
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Confusion on kinetic energy quadratic forms and eigenfrequencies

I am new to the idea of expressing kinetic energy in terms of the quadratic form. I noticed that online, people often express the kinetic energy as: $$T = \frac{1}{2} \dot q^T M \dot q \tag{1}$$ ...
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Simplified computation of matrices for normal modes?

In normal modes, we often refer the total potential energy of the system to be: $$V = q^T B q$$ where $V$ is the total potential energy, $q$ is the coordinates of the system and $B$ is just some ...
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Mechanism behind a simple pendulum [closed]

Show that the tension in a simple pendulum is of the form $$T(t)= T_0 + T_2 \cos(\omega t)$$ and find $T_0$ and $T_2$ in terms of the mass $m$ of the bob, the amplitude of the oscillation $θ_0$ ...
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Harmonic Oscillator (2DOF) - Do these results seem correct?

I solved a 2DOF system for a buried harmonic oscillator with a forcing function, but I'm not sure what I should be seeing in terms of resonant frequency shift & velocity. The resonant frequency of ...
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Coupled Harmonic Oscillator (Forced Vibration)

I derived two equations for a 2DOF harmonic oscillator system, declared state variable equations, and placed them into matrix form: $Ax' + Bx = C$. I have a Matlab script to determine the constants ($...
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In practice, how does one work with the phase states?

The phase states are defined usually by the finite sum $$ |\theta \rangle = (s+1)^{-1/2}\sum_{n=0}^s \exp(i n\theta) |n\rangle, $$ where $\theta = 2\pi k/(s+1)$ and $|n\rangle$ is the $n$-th ...
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Can someone please verify/correct my explanation? [closed]

2D harmonic oscillator Consider a 2D harmonic oscillator which is described by the Hamiltonian $$\hat H = \frac{\hat p_1^2 + \hat p_2^2}{2m} + \frac{k}{2}(\hat x_1^2+\hat x_2^2) \tag 1$$ where $\...
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Connection between algebraic and analytic method of quantum harmonic oscillator

I am studying Quantum harmonic oscillator, There are 2 methods to solve Harmonic oscillator one is algebraic method and another is analytic method , Wave functions derived from 2 methods are ...
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What does it mean: $[\langle(\Delta x)^2\rangle\langle(\Delta p)^2\rangle](t)$?

I got following expression regarding linear harmonic oscillator in quantum mechanics, and I don't understand what it means. $[\langle(\Delta x)^2\rangle\langle(\Delta p)^2\rangle](t)$ $\Delta x$ ...
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There exists at least one odd bounded eigenstate for finite depth parabolic potential?

My attempt : Let $$V(x) = \begin{cases} V_0 \left(\frac{x^2}{a^2}-1 \right), & \text{for } |x|≤a \\ 0, & \text{for } |x|>a \end{cases}$$ Suppose $\psi$ is an odd eigenfunction with ...
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Why does increasing the length of suspension strings increases a bifialr pendulum period of oscillation?

Given where L stands for the length of suspension springs. What is the physics behind the correlation between the period of a bifilar pendulum and the length of its suspension strings?
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How does the following variables affects the period of a bifilar pendulum?

Hey everyone, I am a highschool student from New Zealand and I have 4 questions concerning the period of a bifilar pendulum system. It is very much appreciated if you can answer in detailed ...
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Solution as the real part of a complex exponential from simple harmonic motion

From the book entitled Classical Mechanics written by John R Taylor, chapter no 5, Simple Harmonic Motion. I'm just citing the lines. $$x(t)=\text{Re }Ce^{i\omega t}=\text{Re }A e^{i(\omega t-\...
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Oscillation of 2 identical pendulums connected by a rubber band

While solving A level past papers, I came across the following question. For reference, this is the Edexcel GCE A2 Physics Paper 2 from June 2018. What does not make sense in my mind is the fact that ...
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Sign issue deriving SHM of Electric dipole in external uniform electric field

If we have an electric dipole as shown: Net torque on system = $Fdsin\theta$= Rate of change of angular momentum = $I \ddot \theta $ For small displacements along line of E field, $sin\theta \...
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How the length, flexural rigidity and position of attached mass affects the period of oscillaion of cantilever?

Hey everyone, I'm a highschool student from New Zealand and can someone please explained to me with physics principles in words: Why increasing the length of cantilever increases the period of ...
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Supersymmetry Perturbation Theory

Source:Mirror Symmetry p.198 I have the Hamiltonian $$H = \lambda\bigg( \frac{1}{2} \tilde{p} + \frac{1}{2}h''(x_i)^2(\tilde{x}-\tilde{x_i})^2 + \frac{1}{2}h''(x_i)[\overline{\psi}, \psi] \bigg) + \...
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Heisenberg picture of displacement

In a certain model the displacement operator for the normal modes in a lattice is given by $$u_{s}=\sum_{\textbf{k}}\left(\frac{\hbar}{2mn\omega_{k}}\right)^{1/2}(a_{k}e^{iksb}+a_{k}^{+}e^{-iksb}).$$ ...
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Derivation of 3D simple harmonic oscillator energies in spherical coordinates

I'm trying to show the permitted energies of the 3D simple harmonic oscillator (which is spherically symmetrical) are: $E_n = \hbar \omega(N + \dfrac{3}{2})$ In particular, $V(x) = \dfrac{1}{2} m \...
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Oscillating problems [closed]

I am practicing for my "Mechanics of continuous media" exam. There is two exercises I couldn't really do yet: A homogeneous meter rod at the 70 cm line is hooked up, and making small amplitude ...
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How to visualize the angular frequency in SHM?

Can anyone define how can i visualize the angular frequency(ω) in a SHM y(t)=R sin(ωt+ϕ) (where ω=2π/T).Bcoz we can visualize frequency(f=1/T) as number of times the process is repeated in 1 sec so ...
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Simple harmonic motion as projection of circular motion

Can we consider $\omega$ (angular frequency) in equation of simple harmonic motion (SHM) as the angular velocity of the object in circular motion, when we see simple harmonic motion as projection of ...
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What's the maximum compression of the spring? [closed]

I tried to use the conservation of energy to solve this problem, here's what I tried to do: $\require{enclose}$ $$\begin{align} \enclose{downdiagonalstrike} {\frac{1}{2}} m v^{2} &= \enclose{...
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Effect of magnetic field on time period of a spring pendulum

Let us consider that we have a spring pendulum, and a magnet nearby. Assuming that the spring is being attracted toward the magnet, does the period decrease? The formula of time period is = 2𝜋√m/k ...
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Resource recommendation for SHM & Wave [duplicate]

I am looking for good resources or books on Simple Harmonic Motion and Wave that doesn't require any prior knowledge of those topics. Books that consist even the derivation of even the very basic ...
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What is the equation of time period for a bent wire pendulum?

Just like there is a formula for time periods of a simple pendulum and a spring pendulum. Is there a formula for the time period of bent wire pendulum? There does not seem to be much research done on ...
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Analysis of time period of a complex spring system

My physics professor tells me the only way to solve for the time period of the give oscillating system is by shifting the axis of rotation to the point of contact of the CYLINDER and the surface (...
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Phase space harmonic oscillator area and probability

I want to find the probability of finding an oscillator between $x$ and $x+dx$. I calculated the volume $\frac{8\pi EdE}{\omega^2}$ enclosed in the phase space for the oscillator with energy between $...
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Expectation Value of displacement to the $n$th power quantum harmonic oscillator? [closed]

Is there a closed-form expression for $\langle x^n\rangle$ for the ground state quantum harmonic oscillator, where $n =$ even integer $>0$? I am attempting to pursue this with rising and lowering ...
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Why does the number operator gives the number of excitations?

Could somebody explain why the number operator (for a simple harmonic oscillator) gives the number of excitations? I understand its definition and its relation to the Hamiltonian, but I just can't ...
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Solution to two non-coupled quantum harmonic oscillators

Given the following Hamiltonian: $$\hat H = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x_1^2} -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x_2^2} + \frac{1}{2}m\omega_1^2x_1^2 + \frac{1}{2}m\...
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Simple harmonic motion momentum leads restoring force by 90 degrees [closed]

What is meant by "in a simple harmonic motion, momentum leads the restoring force of elasticity by 90 degrees" ??
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Angular velocity of a pendulum in Cartesian coordinates

Hello I have a problem how to write down equations for the pendulum correctly. Say I would use Cartesian coordinates $x, y$ representing the position of the mass. Then the velocities would be usually ...
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Properties of Expectation Values Under Variable Substitution [closed]

I am working on a homework problem from Griffiths QM (Problem 2.11, 3rd Edition). Specifically, I'm working on finding $\left<x^2\right>$ for the ground state and the first excited state of the ...
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What will be the kinetic energy expression in SHM with two bodies attached with spring?

I know Kinetic energy of two body system is $$KE = \frac{1}{2}\mu V^2_{rel}$$ where $\mu$ is the reduced mass $\mu = m_1 m_2 /(m_1+m_2)$ and Vrel is V1-V2 or the reverse .here I am not taking KE of ...
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Equation for Simple Harmonic Oscillator with moving base

It is known that the base of a simple harmonic oscillator moves according to a known function $u(t)$. Is the dynamics of this system given by $$m\ddot{x} = -\nabla V = -\nabla\frac{k}{2}|x - u(t)|^2 =...