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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Uniform Circular Motion vs Change in speed of $x$ & $y$ Components

If you are moving at a set rotational speed the x,y components are constantly accelerating and decelerating (aka simple harmonic motion), this is obvious in order to travel in a circular path. My ...
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Angular velocity vs angular frequency clarification

I can't seem to find a satisfactory answer on stack exchange for this question, so I will present an example which I would appreciate some clarification on. Let's say we have a pendulum with mass $m$ ...
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Why are sinusoids so common in nature? [duplicate]

When we are introduced to waves in school, we are often presented with a picture of a sinusoid (or a cosinusoid). Sinusoids can represent the way many physics phenomena behave, still.... Why are ...
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Simple harmonic motion on a vertical spring

Say we have a spring attached vertically to a wall. Now, let's assume that we attach a mass to the spring, but we do not let the spring extend just yet (we could hold the mass on our palm for example)....
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Proving that motion of an $n$ dimensional oscillator can be written as a linear combination of “sine waves”

Here is a related question which might provide some context: LINK. Let's consider an oscillator with equation of motion in $n$ dimensions: $$ \frac{d^2}{dt^2} \vec{x} = K \vec{x}. $$ Given that $\...
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Simple Pendulum in Cartesian Coordinates

Riffing on the question in Simple Pendulum Why Generalized Coordinate Always Angle? , I'm trying to write down Newton's law for a simple pendulum in Cartesian coordinates. (I'm doing this as an ...
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Kraus operators for two interacting harmonic oscillators: Problem with the calculation (Ex. 8.21 of Nielsen-Chuang)

I'm working with Exercise 8.21 of the Nielsen-Chuang book on quantum information. It illustrates the amplitude-damping quantum channel by the interaction between two harmonic oscillators (the first ...
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Experimental time-series for quantum particle-in-a-box or simple harmonic oscillator?

I would like to see experimental results for repeated measurement of a single-particle, quantum system that is approximately either particle-in-a-box or simple harmonic oscillator. If particle-in-a-...
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Amplitude of Shm if Constant External force is applied

In the attached picture, is the spring mass system in equilibrium with a constant force $F$? My question supposes that the system is slightly displaced from equilibrium (let's say to the left). Is ...
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Time evolution of harmonic oscillators with changing spring constant [on hold]

Let's consider a simple harmonic oscillator with spring constant $k$. It has hamiltonian $$ H=\frac{1}{2}(p^2+k^2x^2), $$ Where $k$ is time dependent. $k=k_0+\lambda t$. Now, imagine that at $t=0$, ...
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Intuition behind creation and annihilation operators? [duplicate]

Here I am talking about Harmonic Oscillators with Hamiltonian $$ H=\frac{1}{2m}(p^2+(m\omega x)^2), $$ with eigenstates $|1\rangle,|2\rangle,\ldots$ Many textbooks define the annihilation operator to ...
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Oscillating spring, speed close to the equilibrium: How is this answer not 1.5? [closed]

I have this question with the answer listed as $2.0\,\mathrm{m/s}$. "A $1.25\,\mathrm{kg}$ mass on a spring with a constant of $12.0\,\mathrm{N/m}$ is oscillating back and forth. Its maximum ...
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Bose–Einstein Condensation in one dimensional harmonic oscillator [duplicate]

I was given that there is Bose gas with spin 0 in one harmonic oscillator so the energy levels are: $\epsilon_{n}=\hbar\omega n$ Can this gas go through Bose Einstein Condensation? I think yes, ...
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Difference between Oscillatory motion and vibratory motion

What is the difference between oscillatory motion and vibratory motion. I have read in my book that "If the amplitude of oscillatory motion is extremely small,the motion is called vibratory motion". ...
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Deriving SHM equation

i have derived SHM equation till here $$\sin\left( \frac{\sqrt k}{\sqrt m}\cdot t \right)$$ Now all the solutions (youtube and textbook) I'm looking are just stating intuitively $\frac{k}{m} = \text{...
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Oscillatory Motion and SHM

My books states that SHM is that oscillatory motion which can be expressed in Simple sin and cos terms. What does simple actually means ? What's the criteria that make sin/cos expression simple ?
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Euler force for pendulum

Hello I have a question related to the Euler force. Why is this force never considered for a simple pendulum? As far as I understand, Euler force is given by (assume I would consider the 2d pendulum ...
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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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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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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 ...