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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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 ...
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What is the difference between solutions to 2nd order homogeneous ODE?

I’m studying Vibrations, and we have two forms to the 2nd order homogeneous ODE: $$mx ̈+kx ̇=0$$ $$x(t)=C_1 e^{iw_n t}+C_2 e^{-iw_n t}$$ and $$x(t)=A\cos(w_n t)+B\sin(w_n t)$$ Even though I can use ...
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What is meant by finite harmonic oscillator?

What does it mean to take finite harmonic oscillator, In research article "http://iopscience.iop.org/article/10.1088/1367-2630/17/11/113015 ", we were finding effective number of cobosons in ...
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Why does an object in simple harmonic motion have kinetic energy at its equilibrium point? [closed]

While an object is undergoing simple harmonic motion, its kinetic energy tends to vary with its position. This kinetic energy is highest when it's at the point where the forces on it are at ...
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If $\psi$ acting on $a_+$ and $a_-$ operator just moves up and down the ladder, why is $[a_-, a_+] = 1$ and not 0? [duplicate]

If $\psi$ is acted upon by both the operators one by one, it should return the same wave function. Thus order in which you increase or decrease the energy shouldn't matter. Then why is it so that the ...
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Volume in first octant

Here we are finding number of state less than energy E for particle trapped in 3D harmonic potential .For large value of E as compare to hw , we assumed energy levels as continuous.And we introduces ...
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Why are there two different averages for the kinetic energy in a harmonic oscillator?

Question: A particle of mass m executes simple harmonic motion with amplitude a and frequency v. The average kinetic energy during its motion from the position of equilibrium to the end is? ...
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Harmonic Oscillator and Shifts in Derivative Operators

What symmetries/symmetry breaking arises from shifts in the derivative operators? To explain what I mean let's study an example. The classical one particle one dimensional harmonic oscillator has the ...
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The creation and annihilation operators in quantum mechanics

What is the result of the commutation relation between the creation operator and a power of the annihilation operators in simple harmonic oscillator problem?
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Quantum Harmonic Oscillator propagator in Sakurai

In Sakurai the derivation of the propagator leads to the expression $$u_n(x)\exp{\left(\frac{-iE_nt}{\hbar}\right)} = \left(\frac{1}{2^{n/2}\sqrt{n!}}\right) \left(\frac{m\omega}{\pi\hbar}\right)^{1/...
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Is critical damping same as resonance condition?

Because frequency of external force is equal to that of the natural frequency of oscillator...can we call it resonance condition...if yes or no why? If no,then when do we get resonance?
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Linear combination of eigenstates in a potential [closed]

Linear combination of a set of vectors is only defined for a finite set of vectors even though the set might be an infinite set. In quantum mechanics we take infinite linear combination of all ...
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Motion along length of a spring [closed]

What's relation between velocity of each part of a massive spring undergoing Simple Harmonic Motion.
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What is a complex phase shift?

In a complex methods course I am taking, we were given an equation for a particular driven harmonic oscillator where the driving force is trigonometric. I have worked out the math and obtained an ...
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Is the harmonic oscillator approximation valid in occasion of very powerful fields?

I noted that in physics, to study electromagnetic wave phenomena when there is a sinusoidal behaviour, often is used the approximation of harmonic oscillation. I tried to understand the basics of why ...
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Degeneracy of anisotropic oscillator

I was working on the 3D isotropic harmonic oscillator and I found that the energies are given by: $$E=\hbar\omega(n_x+n_y+n_z+3/2)$$ Which has a degeneracy of $\tfrac12(n+1)(n+2)$. However, when ...
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173 views

Which equation to use for SHM?

Usually every simple harmonic question starts with the line: The block at equilibrium shifted to a position $X_0$ and released. I found this from the website: https://study.com/academy/lesson/...
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Large Damped Harmonic Oscillator misunderstanding

So I'm confused, here with what is highlighted. When the book says of "order $1/y_-$" you will reduce the displacement by a factor of $1/e$. Does of order mean when the time is equal to $1/y_-$, if ...
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How LC oscillator is used for generating signals?

I have been trying to understand some practical applications of LC oscialltors and I dont seem to find much information available on net. One consistent application that I see is "LC circuits are used ...
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Why is the restoring force not zero at the equilibrium position of simple harmonic motion?

The restoring force is applied in order to take the body it's equilibrium. Then in a SHM why in the mean position restoring force maximum rather than being zero as it has reached equilibrium. (After ...
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$\sqrt{-1}$ coefficient in a function

In a simple harmonic oscillator with $\ddot{x} = -\omega^2x$, it can be shown through differentiation that one solution can be given by $\dot{x} = i \omega Ae^{i \omega t}$. What does the factor of $i$...
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Tension in pendulum [closed]

I am asked to calculate the tension in the rope of a pendulum at (a) its initial position as well as at (b) its lowest position. $L = 3 m$ $α = 10^o$ $mass = 2kg$ (a) For the intial point I used ...
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Bose-Einstein condensation in 1D harmonic oscillator and its density of states

I have troubles understanding how (and whether) Bose-Einstein condensation works in 1-D harmonic oscillator. From my calculation it seems that in limit of infinite number of particles, almost all of ...
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Does “the initial phase of oscillation is 45°” mean a time dependence of the form $\sin(\omega t+\pi/4)$ or $\cos(\omega t+\pi/4)$? [closed]

A point particle of mass 0.1 kg is executing SHM with an amplitude 0.1 m. When the particle passes through the mean position, its kinetic energy is $8 \times 10^{-3}$ Joule. Obtain the equation of ...
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Rod connected to springs. Can this oscillate if the rod has no mass?

My thoughts led me to change the question's title. Here's what I've tried in solving the problem and led me to ask the question in the title. We have the following diagram where the beam has no mass ...
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Equation of motion from potential energy [closed]

Given that the potential energy of a particle in 2D space is $$V(x, y) = \frac{1}{2}k(x^2 + y^2),$$ find the equations of motion and show they are circular orbits. Substituting $r^2 = x^2 + y^2$, I ...
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How to derive this expression for the free scalar field in QFT? (Peskin & Schroeder)

In the introductory text to quantum field theory by Peskin & Schroeder, they state that in analogy to the simple harmonic oscillator in quantum mechanics, the free scalar field can be expressed as:...
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Why does $g$ show up in the frequency of this oscillation?

The problem diagram is given in the picture below: Having looked at this question Why does the acceleration $g$ due to gravity not affect the period of a vertically mounted spring? something troubles ...
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Physical meaning of commutation

I was reading the solution to quantum harmonic oscillator by J.J. Sakurai. He uses the annihilation and creation operators and there's a key step (I think) which is $$[a,a^{\dagger}]=1$$ I know we can ...
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398 views

Simple harmonic motion with pulley with mass

The problem is presented in the following diagram I'm refreshing things I've already learnt and I know I have some major gaps but I've searched and haven't managed to find a similar problem. The axis ...
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389 views

A harmoinic oscillator in the Heisenberg picture

Considering the Hamiltonian of a harmonic oscillator \begin{equation} H=\frac{p^2}{2m}+\frac{m\omega^2 x^2}{2}, \end{equation} the time evolution of the Heisenberg picture position and momentum ...
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Is there a proof for why the acceleration of an object undergoing simple harmonic motion related to angular velocity squared?

Many textbook says the defining equation of the acceleration of an object undergoing simple harmonic motion is $$a= -\omega^2 \times x.$$ Is there a reason as to why acceleration is related to $\...
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A particle is in a harmonic oscillator potential. Which are the possible spreads for a simultaneous measurement of the momentum? [closed]

A particle is in a harmonic oscillator potential, not in the ground state. The position of the particle is known with an rms spread of $1\mathring{\text{A}}$. Which of the following are possible ...
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What is the effect of mass on resonance amplitude?

When a system is undergoing forced oscillations, why does reducing the mass of the system cause the frequency response curve to shift downwards? I encountered this problem in a practice paper, but I ...
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Using a single harmonic oscillator to implement a quantum gate. Confusion over concept

I'm trying to simulate a quantum gate operation in mathematica using a harmonic oscillator and I have some confusion with how the physical system relates to the theory. This may be a bit long winded ...
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Can anyone explain the harmonic oscillator (in context to quantum mechanics) 2.3 (Griffiths) using Taylor series?

At the end he concludes $V(x) = V''(x_0)(x-x_0)^2$. How does he get to know that the rest are $0$? How does he conclude $V''(x_0) = k$. Please try to explain in easy ways and tough vocabulary. I don't ...
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Modelling a pendulum with physical restrictions on it's range of motion

I'm currently working on a project based on suspension bridges and their oscillations. I've got an equation of motion for the movement of a pendulum as shown in the first image, I then wanted to be ...
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Gravity’s effect on a vertical spring-block simple harmonic oscillator

I just found a question in my textbook which asked how the period of the vertical oscillation will change if the spring and block system is moved to the moon, and the gravity due to acceleration is ...
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meaning of “sufficiently small” in approximations to behaviour of a Harmonic Oscillator

So in my classical mechanics book it states: "For any sufficiently small displacement, any system of this kind behaves like a harmonic oscillator." When discussing SHO. So I am curious what is ...
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Why do materials obey Hooke's law? [duplicate]

Why do materials extend proportionally to the force exerted on them (Hooke's law)? I thought that when materials are compressed or extended under force, their atoms become closer or further apart; ...
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Justification for using wave equation for describing a phenomena

I have recently started learning about waves. We didn't really formally describe what a wave is, but instead started by looking at a concrete example namely harmonic sinusoidal waves in 1d. We then ...
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Would a spring ever stop?

It is not difficult to show from Newton's second law $$m\ddot x = -kx - b\dot x $$ that an underdamped spring has the equation of motion quantified by $$x(t) = c_1e^{-\beta t}\sin\left(\omega t\right) ...
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Mass hanging by a spring [closed]

I have a mass hanging from the ceiling by a spring, with costant $k$, and wih gravity $g$. Using analytical mechanics I got to the differential equation: $$ \ddot{x} + \frac{k}{m}x + g = 0 $$ The ...
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Is the time period of a fixed string always a resonant time period?

Consider a string held fixed at $x=0$ and $x=L$. This string has a harmonic series associated with it with each harmonic time period, $T_n$, given by: $$T_n=\frac{2}{n}\int\frac{dx}{v(x)},$$ where $v(...
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Thermal wave function of the harmonic oscillator - proving that it's a gaussian?

I'm a bit stumped trying to prove this. I've computed the probability density for a thermal density matrix for the quantum harmonic oscillator, namely $$ \rho(x) = \frac{\sum_n^\infty e^{-\frac{\hbar\...
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Determine period and angle of harmonic oscillator with $x = 2 \pi \sin(120 \pi t + 3.2 t)$ [closed]

A particle moving under simple harmonic motion has displacement $$x = 2 \pi \sin(120 \pi t + 3.2 t) \, .$$ How can I determine the period and the phase angle?
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Why are oscillations so ubiquitous in nature? [duplicate]

I'm aware that you can always approximate a potential by a quadratic term. But is this the most 'fundamental' reason for the pervasiveness for oscillations?
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What is the relationship between excitation and resonance?

From Resonance (particle physics) - Wikipedia: In particle physics, a resonance is the peak located around a certain energy found in differential cross sections of scattering experiments. These ...
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Do the ladder operators $a$ and $a^\dagger$ form a complete algebra basis?

It is easy to construct any operator (in continuous variables) using the set of operators $$\{|\ell\rangle\langle m |\},$$ where $l$ and $m$ are integers and the operators are represented in the Fock ...
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Can the precise form of a quantum field Hamiltonian be determined by harmonic oscillator analogy?

In one of my other questions What is the precise formal correspondance between an oscillator and a quantum field? , I was helpfully given the exact form of the well discussed analogy between the ...