The term "harmonic oscillator" is used to describe any system with a "linear" restoring force that tends to return the system to a 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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3D harmonic oscillator problem [on hold]

I'm stuck with this problem, that says Find the angular moment $l $ that can be measured for the energy state $5/2 \hbar \omega$, providing the probability of each value. Also build the wave ...
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47 views

Deriving the particular solution for a damped driven harmonic oscillator [on hold]

Consider a damped driven harmonic oscillator, for which $\beta = \omega_0/4$ and the driving force is given by $F = F_0\cos\omega t$ ($\omega_0$ and $F_0$ represent initial condition of those ...
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89 views

Proving $[a_k^\dagger, a_q^\dagger]=0$

I am trying to prove the commutation relations between the creation and annihilation operators in field theory. I was already able to show that $[a_k, a_q^\dagger]=i\delta(k-q)$. I want to show that ...
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17 views

Quantum harmonic oscillator doughnut shape

When phase-space trajectory is plotted for classical harmonic oscillator for p(t)=mx0ωcos(ωt +δ0), a circle is obtained. When done same for the quantum harmonic oscillator, why do we get a doughnut ...
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47 views

Energy in harmonic oscillator [on hold]

The expectation value of the potential energy is exactly half the total according to Griffiths. Is that case always true for quantum harmonic oscillator? Is that the case also for classical harmonic ...
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Pendulum in Accelerating Elevator

I have been looking for this for quite some time now. A simple pendulum behaves in SHM. Let's put that pendulum in an upward accelerating elevator. The component of the force that acts in SHM ...
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1answer
100 views

Classical action of the simple harmonic oscillator

I have been calculating the classical action of the harmonic oscillator, the problem I have is that I am only able to solve it if I set the integration limits of the action integral to be $t=T$ and ...
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1answer
39 views

Definition of the quality $(Q)$ factor?

According to Wikipedia, the Q factor is defined as: $$Q=2\pi\frac{\mathrm{energy \, \, stored}}{\mathrm{energy \, \,dissipated \, \, per \, \, cycle}}$$ Here are my questions: Does the energy ...
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23 views

Translational acceleration of cylinders [closed]

Two cylinders with total mass $M$ and radius $R$ are connected by a massless rod along their axis of rotation and rest on a horizontal surface. A frictionsless ring at the center of the rod is ...
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1answer
28 views

How to find amplitude?

What are the general methods of finding amplitude in various cases of simple harmonic motion containing spring block system? I always try with the basic sine formula of SHM to find amplitude.
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Question about massive spring and SHM [closed]

A mass $M$ is resting on the end of a spring with constant $K$. The mass of the spring is $m$, and the displacement of each element of the spring is proportional to the distance from the fixed end ...
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39 views

Pressure in Harmonic Oscillation

Classical Harmonic oscillator's energy depends on temperature as it equals $k_B$$T/2$. However, quantum harmonic oscillator energy is $(n+1/2)hf$. So, when T=0, quantum predicts motion. I have been ...
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Question about derivation of the Heisenberg Uncertainty Principle?

I am looking at the derivation presented here. The first thing I am unsure about is where the form of $\psi_0=Ae^{\frac{-m\omega x^2}{2\hbar}}$ came from. Also, is this form for all $\psi$, or just ...
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51 views

Is uniform circular motion an SHM?

I know the projection along a diameter is an SHM but is circular motion itself an SHM? If we consider the mean position to be the center of the circle then the centripetal acceleration is proportional ...
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42 views

Can the matrix element of the momentum operator be found in the momentum basis?

In Shankar's Quantum Mechanics example 7.3.4, the problem is to find $\langle n'\rvert P\lvert n\rangle$ for the harmonic oscillator. The answer contains imaginary parts; you can derive such an answer ...
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1answer
36 views

determining phase constants in SHM [closed]

A particle moves along the x axis. It is initially at the position $x$ of $0.300 m$, moving with velocity $v$ of $0.070 m/s$ and acceleration $a$ of $-0.330 m/s^2$. Suppose it moves with constant ...
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78 views

Simple harmonic motion frequency [closed]

I already did part a and b by using kinematics but I'm stuck for the next part A particle moves along the x axis. It is initially at the position 0.300 m, moving with velocity $0.070 m/s$ and ...
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3answers
45 views

Why do non-circular oscillations have angular frequency? [duplicate]

Why so the oscillations which are not circular also have angular frequency which is a quantity related to the circular motion? I have referred many articles related to simple harmonic motion where ...
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2answers
119 views

Why does the acceleration $g$ due to gravity not affect the period of a vertically mounted spring?

For a vertically mounted spring, I was looking at the formula $ T= 2\pi \sqrt{m/k}$ for a period. Why doesn't the gravitational acceleration $g$ factor in?
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Amplitude resonance

Why does amplitude resonance occur at a frequency lower than the natural frequency of a body? specifically, why is $w=\sqrt{w_0^2-2a^2}$ where $a=\frac{damping\space force}{2\cdot mass}$
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39 views

Integral over scalar product of eigenfunction of momentum operator and harmonic oscillator one

Recently I've met following expression: $$ \tag 2 \sum_{n}f(n)\int dp~ |\langle p | n\rangle|^{2} = 2\pi \sum_{n}f(n). $$ Here $|n>$ is eigenfunction of harmonic oscillator with energy $$E_{n} = ...
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1answer
32 views

Simple harmonic motion equation [duplicate]

I don't really understand this equation and was wondering if someone could help. The book says when the restoring force is directly proportional to the displacement the oscillation is called ...
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1answer
57 views

Analytical solution of Liouville's equation for classic harmonic oscillator - which book?

So the past five hours I've spend fruitlessly searching the web for any materials containing the analytical solution of the simple PDE: $$\frac{\partial f}{\partial t} - m\omega^2x\frac{\partial ...
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Damped Oscillations: Incoherence between a general solution and a specific one

In my 'Classical Dynamics of Particles and Systems, THORNTON/MARION, 5th Edition' book of classical mechanics it is given the following general solution for a damped oscillation solving ...
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2answers
73 views

What is the physical meaning of $a_{\vec{p}} \! \mid \! 0 \rangle$

$a^\dagger_{\vec{p}} \! \mid \! 0 \rangle = \mid \! p \rangle$ is interpreted as a creation of a particle with momentum $p$ from the vacuum. $a_{\vec{p}} \! \mid \! p \rangle = \mid \! 0 \rangle$ is ...
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38 views

The period of a mass-spring system including a pulley [closed]

A spring with spring constant $k$ is attached to a mass $m$ as illustrated in the following four set ups: Calculate the period of the motion for each of them. In cases (1) and (2) the period ...
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110 views

Good source for numerical simulations of Wigner function?

I'm interested in simulating the time evolution of a Wigner function for a harmonic oscillator (and possibly some other potentials) and I can't seem to find a good resource for that. My background in ...
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175 views

Why is $\pi$ used when calculating the value of $g$ in pendulum motion?

I am trying to intuitively understand why $\pi$ is used when calculating the value of $g$ using the harmonic motion of a pendulum: $$g ~=~\frac{4\pi^2L}{T^2}.$$ Does it have something to do with the ...
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Driven-damped oscillator: deduce the phase and/or resonant freq from amplitudes at varying freqs

Suppose that we have a fairly standard driven-damped harmonic oscillator (i.e. linear spring restoring force, linear damping force, sinusoidal driving force, etc). The catch is: we don't know the ...
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1answer
51 views

What could be the applications of Damped Oscillation? [closed]

I've been researching on Damped Oscillation for a few days for a research paper, however I couldn't find any of its applications on the web, though there are few examples of it, but they couldn't be ...
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53 views

A driven quantum harmonic oscillator (DQHO) [closed]

I'm trying to find the dispersion relation for the DQHO with Lagrangian $$ L(q,\dot q,t)=\frac{1}{2}\dot{q}^2-\frac{1}{2}\omega q^2+F(t)q $$ with $F(t)$ being non-zero for $0<t<T$. The ...
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165 views

What is the significance of clamping the center of the spring?

7. A block is hung on a spring, and the frequency $f$ of the oscillation of the system is measured. The block, a second identical block, and the spring are carried in the Space Shuttle to space. ...
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85 views

Damped simple harmonic oscillator problem

I'm supposed to calculate and draw the phase space trajectory for this: for the two different cases when and . I've never done this sort of question before, how are they done? I've tried ...
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1answer
154 views

Quantum harmonic oscillator solved by analytic method using Schrödinger equation and wave function

I'm having trouble understanding the recursion formula. Using $\xi \equiv \sqrt{m\omega/\hbar}x$ and $K = 2E/\hbar\omega$, the time-independent Schrödinger equation becomes $$\frac{d^2\psi}{d\xi ^2} ...
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Decoupling the Hamiltonian by a Discrete Fourier transform

For $N$ coupled oscillators(periodic BC) whose Hamiltonian is given as $H=\sum\limits_{i=1}^N (\frac{p_i}{2m} + \lambda(x_{i+1} - x_i)^2)$ decoupling can be achieved by change of variables by using ...
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1answer
37 views

What is the dormant common ground between harmonic mechanical oscilators and electromagnetical ones?

When I learnt electromagnetic oscillators I couldn't help but notice that it has many common stuff with mechanical ones. I know that it had to have sinusoidal equations. I (firstly, without ...
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103 views

Unstable equilibrium in a pendulum

Consider a pendulum with a bob and a massless, rigid, hinged rod attached to the bob. The bob is at rest at the bottom most position. Neglecting friction, is it possible to impart such a velocity ...
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1answer
82 views

Simple harmonic oscillator, calculate the trajectory in real space

Potential of a simple harmonic oscillator: $$U=\frac{1}{2}k x^2$$ I'm asked to calculate the trajectory of a particle moving in this potential, with initial conditions $x(t=0) = 0$ and $v(t=0)=v_0$. ...
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91 views

Green function for simple harmonic oscillator

I'm interested in examples on how to use Green function (GF)for simple harmonic oscillator (SHO)? I am from undergrad physics, so I need a fundamental math and quantum mechanical application of GF ...
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107 views

Forced Quantum Harmonic Oscillator

I'm just starting my journey to QFT and Particles physics and I have a question about the problem of QHO witch we hit with a force $F(t)$ for $ t< t' $, for which the force is zero for $t>t'$. ...
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1answer
66 views

Harmonic oscillator differential equation solution [closed]

My math book explains how to solve second order equations like : $$\ddot{x} + \omega^2x = 0$$ but I end up with the general solution : $$A\cos(\omega t) + iB\sin(\omega t).$$ Now my physics book ...
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28 views

Under-damped oscillator - how to determine speed?

At t=0 a mass M is stretched from equilibrium by x with spring constant k and damping coefficient $\gamma$. How do I determine how fast a weight is moving when it passes through equilibrium on an ...
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Confusion regarding the trial solution taken in the mathematical treatment of forced oscillations, at steady state

In the text-book that I am currently using, it is given that in case of forced oscillations, the periodic external driving force is a complex-driving force, and is generally of the form $F_0e^{jwt}$. ...
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20 views

Under-damped oscillator determine number of oscillations

With an under-damped harmonic oscillator how do I determine the amount of oscillations that happen before the maximum displacement is less than some distance x?
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1answer
93 views

Evolution of harmonic oscillator in path integral formulation

The unnormalized ground state of the harmonic oscillator (choosing units such that $m = \hbar = \omega = 1)$ is $$\tag{1}\psi(q,t) = \exp(-q^2/2-it/2).$$ The transition function is ...
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A question about physical pendulums

Is there an intuitive way to explain why increasing the distance from the center of gravity increases the accuracy when trying to model the period for a physical pendulum with a small amplitude? The ...
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2answers
119 views

Is angular frequency dependent on time in damped harmonic motion?

I have a doubt regarding the angular frequency of a harmonic oscillator when there is damping involved. The frequency of the oscillation changes with time in the case of damping, but I haven't seen ...
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1answer
63 views

Why do we use sine/cosines in Simple Harmonic Motion? [duplicate]

For example, to calculate the displacement of the particle in an harmonic oscillator we do: $$x(t) = x_{\max} \cos(ωt+φ)$$ What do we find out taking the cosine of (ωt+φ)? Example Graph:
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115 views

How to derive the time period equation for a spring mass system taking into account the mass of the spring without involving energy analysis?

I want to know the way to derive the time period equation of a spring mass system accounting for the mass of the spring but not using the energy analysis method but by proceeding in the same way as we ...
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75 views

What do they mean by a pendulum losing seconds?

In many pendulum related question, a pendulum is taken do a different place where it loses seconds. For example: A second's pendulum is taken to a mountain and it loses 20 seconds per day. What ...