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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How would you go about scientifically comparing the physics of a game and real-life through gravity?

A bit of a hypothetical situation for y'all here. Let's say you want to test a video game's accurateness with regards to its physics; by seeing how closely the game's physics matches up with real-life ...
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Direct Series Solution Attempt of the Quantum Harmonic Oscillator

The non relativistic Schrodinger equation of the harmonic oscillator in dimensionless variables is $$\frac{d^2 \Psi}{d \xi^2} = (\xi^2 - k)\Psi$$ where $$k \equiv \frac{2E}{\hbar \omega}$$ According ...
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What is the single-mode quartic nonlinear oscillator evolution of a Fock state?

My question is how can I find the resultant state of for example Fock state $|n\rangle$ under single-mode quartic nonlinear oscillator evolution for time t? Attempt: I tried to find the relevant ...
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What factors would cause a change in oscillation period as a result of a change in mass of pendulum bob?

I'm running an experiment in a video-game simulator (ish) called 'G-MOD'. I setup a typical pendulum setup, and measured the change in the period as a function of the pendulum's mass. I obviously ...
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Slow parameter oscillations in coupled simple harmonic oscillator

In a paper by Turaev, he studies systems with a slow variation of parameters. The following is on page two, right column: He first discusses the one dimensional particle in a box, with ends at $x=-1$ ...
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Transition probability between two harmonic oscillator states (Hemite polynomial integration)

Objective: Show that $$ \int^{\infty}_{-\infty} x e^{-x^2} H_n(x) H_m(x) dx = \pi^{1/2} 2^{n-1} n! \delta_{m,n-1} + \pi^{1/2} 2^n (n+1)! \delta_{m,n+1} $$ My attempt at this is: \begin{eqnarray*} \...
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Quantum Mechanics: Harmonic Oscillator - Evaluating the expectation value to obtain the generating function for the moments [closed]

I don't know how to start evaluating the expression. I keep ending up with a messy integrand. Please help.
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Transfer operator spectrum of the discretized harmonic oscillator

I'm reading "An introduction to quantum fields on a lattice" by Jan Smit. In chapter 2, the transfer operator 𝑇̂ is defined and shown to be equal to $$ \hat{T} = e^{-\omega^2 \hat{q}^2/4} ...
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What is the physical significance of ladder operators?

If i have a quantum harmonic oscillator system, say, a Quantum Optics system or a crystal where i have some $\Psi$ in occupation number respresentation in energy eigenbasis. $$\Psi=|n_1 n_2 n_3...\gt $...
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Simplifying logarithmic decrement equation

I'm currently writing a paper on underdamped oscillatory systems where I'm using the logarithmic decrement equation: $\delta = \ln\frac{x(t_n)}{x(t_n+T)}$ Where $T$ is the period of the system. I ...
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Lagrangian for a series of $N$ one-dimensional coupled oscillators

The Problem I have been given the following Lagrangian of a series of $N$ one-dimensional coupled oscillators, with distance a. I have also been given the boundary conditions: $y_0=0=y_{N+1},$ but ...
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Question on the State Vector of the Quantum Harmonic Oscillator

My book states that the wavefunctions for the quantum harmonic oscillator are $$\psi_n(x)=(1/2)^{n/2}H_n \left(\sqrt{\frac {m\omega}\hbar}x \right) \exp \left( -\frac{m\omega}{2\hbar}x^2\right)$$ ...
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Angle of releasing a pendulum and the speed of the ball hit by it

I have conducted the experiment. My independent variable is the angle of releasing the pendulum and my dependent variable is the speed of the ball that is hit by it. I know the following quantities: ...
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Harmonic oscillator with ladder operators - proof for sum rule

I'm trying verify the proof of the sum rule for the one-dimensional harmonic oscillator: $$\sum_l^\infty (E_l-E_n)\ | \langle l \ |p| \ n \rangle |^2 = \frac {mh^2w^2}{2} $$ The exercise explicitly ...
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Quantum Field theory question just conceptually grasping from Sean Carroll's “Biggest Ideas in the Universe”!

Quantum Field Theory from Sean Carroll's Biggest Ideas in the Universe. I’m just checking to see if I’m on the right track of what he's explaining. He talks about a free field (non-interacting field), ...
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Hamilton-Jacobi Equation & Canonical Transformation

I am attempting to solve the Hamilton-Jacobi Equation in the case of a simple harmonic oscillator, to recover the associated generating function and the generated canonical transformation. Consider ...
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What is the difference between $\omega$ and $\Omega$ in the equation $\Omega = \theta_o\omega\cos(\omega t +\delta)$?

The standard equation for any object moving in a linear simple harmonic motion (SHM) is $$x=A\sin(\omega t +\delta)$$ where $A$ is the amplitude, $\omega$ is the angular velocity. Likewise, converting ...
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Is the natural frequency of the damping system equal to the natural frequency of the forced oscillator? For example a bridge and people walking

When resonance occurs, there is a maximum transfer of energy from the driver to the forced oscillator. This means that the damping system should have the same frequency as the forced oscillation ...
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Solutions of the Harmonic Oscillator are *not* always a Combination of Separable Solutions?

Are there solutions of the Schrödinger equation that are not a linear combination of separable solutions and how do we find them? In Griffiths, Quantum, Prob. 2.49, there is a solution of the (time-...
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On the 1D Quantum Mechanics Harmonic Oscillator

I was solving the P. 2.41 of Griffiths' Introduction to Quantum Mechanics. Nothing really new until I read a proposed solution (from Griffiths' himself) for the problem in which it states that I can ...
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Equivalent ways to evaluate the expectation value of the creation/annihilation operator in QM

stackexchange, Upon reading in Modern Quantum Mechanics by Sakurai & Napolitano, I fell upon a calculation of the expectation value of the annihilation/creation operators squared, with respect to ...
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Why we take vertical displacement as $A\sin(κx-ωt)$ not $A\sin(ωt-κx)$?

A general form of wave equation is $$Y = A \sin(ωt-kx+φ)$$ Now if $t = 0$ is taken at the instant when the left end is crossing the mean position from upward to downward direction, $φ = π$ and the ...
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Eigenvalues and Eigenstates of Number operator

I have been working through a problem. It has asked me to determine the eigenstates and corresponding eigenvalues of the number operator in a quantum harmonic oscillator; $$\hat{n}=\hat{a}_+\hat{a}_-$$...
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Where did this potential energy come from?

I was reading a paper about Wilberforce Pendulum: https://faraday.physics.utoronto.ca/IYearLab/WilberforceRefBerg2of8.pdf This is the potential energy of (spring+weight with moment of inertia) $V={{1}\...
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Equation of motion of harmonic oscillator with dissipation function of speed

We have a mass sustained by a spring $K$ and a damper $C$, with a base excitation. Let's call $s(t)$ the base excitation and $x(t)$ the mass motion. The differential equation of this system will be: $...
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Does maximum velocity change when vertical mass-spring system is used in different location on Earth in SHM?

Let me elaborate for you my concerning I am thinking of a example of a vertical mass spring system. Suppose i place my system at equator, let suppose a wall clock which uses a vertical spring mass ...
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Is gravitational Force is damping force in vertical mass-spring system of constant k in Simple Harmonic motion

When a block of mass m is suspended by vertical spring system,it for sure perform SHM in the absence of damping force,only force which act on the system is internal which is -kx, where k is spring ...
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What is the differnce of angular velocity and angular frequency in angular SHM

A body free to rotate about a given axis can make angular oscillation. This angular oscillation are called Angular simple harmonic motion, In derivation, Ohm = theta ✖ w ✖ cos(wt+ phi) Where omega is ...
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Is there a physical process that - ignoring measurement uncertainties - has a fixed duration?

I'm looking for a process that has an absolutely fixed duration, where all the variance obtained by measurement is just due to measurement uncertainty, and the obtained mean is the actual duration of ...
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Understanding difference between classical and quantum harmonic oscillator probability distributions

Here we have an example QHO wavefunction squared in blue overlayed an equivalent CHO probability distribution. I'm trying to understand intuitively why the QHO result has zeros, i.e. points where we ...
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Damped harmonic Oscillator Lagrangian equivalence

The objective is to prove that the Lagrangian: $$L'=\frac{2\dot x+\lambda x}{2\Omega x}\tan^{-1}(\frac{2\dot x+\lambda x}{2\Omega x})-\frac{1}{2}\ln(\dot x^2+\lambda \dot{x } x + \omega^2x^2), \qquad \...
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How can we be sure that the equation of SHM that works for one dimension of an object moving in circular motion works for all SHM?

I have learned that a component of a uniform circular motion is an example of SHM. And I have no question about it, I totally understand that. I also understand how we can derive formulas like $\vec{...
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The Negative Energy in the Harmonic Oscillator Potential! [closed]

I'm self studying Quantum mechanics from Griffiths. Now I'm at the Harmonic oscillator potential. All my questions raised after defining the ladder operators $a_-$ and $a_+$. If $\psi$ satisfies the ...
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Why probability density for simple harmonic oscillator is higher at ends than that in middle?

I was watching this crash course by Geek Lesson on Quantum Mechanics specifically for Quantum Harmonic Oscillator and [at 1:54:54] when video shows the plot of probability density for different states ...
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Direction of displacement

Displacement is defined as the vector obtained by joining the final position to the initial position (head towards the final position). Well,i know this is silly but what are these final and initial ...
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Can the heating of the spring in oscillator be modeled by a velocity dependent force?

In a damped oscillator, the damping term is represented by a velocity dependent force $b \ \dot{x}$. This makes sense if the damping is due to viscosity of the medium. Is this modeling correct for the ...
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The position and potential energy of the masses in two mass rigid body pendulum lie on a cirlce

Suppose I have two mass rigid pendulum, both of whose masses are equidistant from the pivot point at P. All three points lie on a circle of diamater D and subtend an angle $\alpha$ at the pivot. let ...
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How does time translational invariance and linearity imply exponential solutions?

I'm currently studying "Waves and Oscillation". While going through the book The Physics of waves, from page 11-12. The author has mentioned that the differential equation being linear ...
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The quantisation of the harmonic oscillator applied to the free Klein-Gordon field

In David Tong's lecture notes on quantum field theory, at the bottom of page 23, we are applying the quantisation of the harmonic oscillator to the field to obtain expressions for the field operators ...
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Quantum Harmonic Oscillator and an instantaneous force that imparts a momentum

The question is as follows, Consider a simple harmonic oscillator in its ground state. An instantaneous force imparts momentum $p_0$ to the system. What is the probability that the system will stay in ...
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Simple harmonic motion problem

I have no problem with the solution provided but, I have a problem with understanding it's meaning. Shouldn't the two solutions for b) add up to be the period? If not, why?
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Speed of a standing wave

Standing waves are the waves in which disturbances do not simply propagate forward or backward, but rather the material particles are moving up and down continuously, with the particles between two ...
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Question about a relativisticaly accelerated harmonic oscillator

How can the speed of oscillation of a harmonic oscillator be affected if somehow it got accelerated to a relativistic speed perpendicular to its oscillation? Can this be compared with the effect on ...
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Solutions to damped harmonic oscillator?

For the damped harmonic oscillator equation $$\frac{d^2x}{dt^2}+\frac{c}{m}\frac{dx}{dt}+\frac{k}{m}x=0$$ we get that the general solution is $$x(t)=Ae^{-\gamma t}e^{i\omega_d t}+Be^{-\gamma t}e^{-i\...
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Resonance and standing waves on a bar [closed]

I'm having trouble solving this problem: By applying a harmonic force, acting on the end of a free bar of length $L$, a standing wave is formed due to multiple reflections: Where are the nodes of ...
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A spring with non-negligible mass

I see everywhere in the analysis of a spring-mass system of Simple Harmonic Motion, that each infinitesimal element on the spring of length $L$ is $\frac{vx}{L}$ where $v$ is the velocity of the block ...
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What does the disconuity in the equation for modulated phase mean when superposing two SHM?

So I was working out the result of the composition of two SHM that are in the same direction and have different frequencies and amplitudes. Turns out I found the following equation for the modulated ...
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Is displacement related to velocity or mass (in this question)? [closed]

Two blocks of masses $m_{1}$ and $m_{2}$ are kept on a smooth horizontal surface. A spring of mass $m$ and natural length $L$ connects the two blocks as shown in the figure. At t=0, $m_1$ and $m_{2}$ ...
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Energy transfer between oscillators [closed]

Suppose I have two mechanical oscillators $a(t), b(t)$, coupled through the interaction $V_\text{int} = \mu^2 a(t) b(t)$. Is there a simple way to express the rate of energy transfer from $a$ to $b$ ...
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Does the uncertainty $\Delta \hat a$ of the annihilation operator of the harmonic oscillator remain constant over time?

I'm supposed to prove that the uncertainty of the annihilation operator of the harmonic oscillator, given by $$\Delta \hat a=\sqrt{\langle\hat{a}^{2}\rangle-\langle\hat{a}\rangle^{2}} \tag1$$ doesn't ...

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