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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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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772 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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99 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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258 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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151 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
472 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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339 views

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
46 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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387 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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287 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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550 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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158 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
90 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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57 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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35 views

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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131 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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558 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
145 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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388 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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655 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 ...
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285 views

3DAnisotropic oscillator in Spherical Harmonic basis-States with $L_z=0$

I've been trying to prove a rather simple looking concept. I have a code that calculates states of a 3D anisotropic oscillator in spherical coordinates. The spherical harmonics basis used to expand ...
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362 views

Simplest explanation of pendulum having a constant time period at low angles

What is the simplest explanation for the pendulum having a constant time period at low angles?
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What is the fluctuations of the energy of a simple harmonic oscillator? [closed]

$$\begin{align} \varepsilon&=\frac{\vec{p}^{\,2}}{2m}+\frac{K}{m}\vec{q}^{\,2}\\ \rho(q,p)&=\biggl(\frac{\omega}{2\pi k_BT}\biggr)^3e^{-\frac{\varepsilon}{k_bT}} \end{align}$$ where ...
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108 views

Quantum Harmonic Oscillator - Normalizability of Annihilated Ground State

The common line of deductions in the operator analysis of the quantum harmonic oscillator goes something like this: It is derived that the action of the annihilation operator $a$ on an eigenfunction ...
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Coupled quantum harmonic oscillator

Given the following Hamiltonian for two identical linear oscillators with spring constant $k$ and interaction potential $\alpha x_1x_2$; I was asked to find the expectation value $\langle ...
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653 views

Bound states, scattering states and infinite potentials

I am doing my first semester of Quantum Mechanics and we're using Griffith's Introduction to Quantum Mechanics. As he is introducing the Dirac delta function potential he explains bound and scattering ...
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213 views

Creation and Annihilation Operators

Let $\widehat{a}^{+}_{i}$ and $\widehat{a}_{i}$ be the usual bosonic creation and annihilation operators. Consider $$\widehat{q}_{i} = \sqrt{\frac{\hbar}{2m_{i}w_{i}}}(\widehat{a}_{i}+ ...
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Sums of operators in practice

Consider a one dimensional harmonic oscillator. We have: $$\hat{n} = \hat{a}^{\dagger} \hat{a} = \frac{m \omega}{2 \hbar} \hat{x}^2 + \frac{1}{2 \hbar m \omega} \hat{p}^2 - \frac{1}{2}$$ And: ...
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355 views

Quantum field theory: field operators in terms of creation/annihilation operators

I am learning Quantum Field Theory and there is a step in my notes that I do not really understand. It starts with the classical definitions of position $q$ and momentum $p$: $$ q = ...
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360 views

Can we represent Simple Harmonic function as triangular waves?

Having studied the topic recently I found out that simple harmonic motion can represented well with sine and cosine functions.Take for example a pendulum swing which could look like : and the ...
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1answer
331 views

Given the spring constant & maximum kinetic energy; length of spring extension? [closed]

I need to understand the following question before i right my exam tomorrow. A body attached to a spring with spring constant 100 N/m executes simple harmonic motion. The maximum kinetiv energy of ...
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How can I derive the Hamiltonian of simple harmonic oscillator from this Lagrangian?

I'm working through Leonard Susskind's Theoretical Minimum: Classical Mechanics and I can't seem to understand how the Hamiltonian of a simple harmonic oscillator is derived from the following ...
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267 views

Why is $\omega = \sqrt{K/m}$ valid for a quantum oscillator?

I'm working in the 3rd edition of Modern Physics by Serway, Moses, and Moyer. In 6.6, it talks about a quantum oscillator. I don't fully understand how the definition of frequency works. Now, we ...
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Rational ratio of frequencies leads to isolating integral of motion

Padmanabhan's discussion of dynamics mentions that in general the two dimensional harmonic oscillator fills the surface of a two torus. He further notes that there will be an extra isolating integral ...
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Eigenstates of a shifted harmonic oscillator

Let's say I have a quantum harmonic oscillator $H = \omega a^\dagger a$, where $a^\dagger$ is the raising operator and $a$ is the lowering operator and $H |n\rangle = \omega n |n\rangle$. Now assume ...
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120 views

Derivation of $a_{j}$ coefficients in the quantum harmonic oscillator

In Griffiths' book page 53, when we derive the solution of the quantum harmonic oscillator by using the power series way, we have: $$a_{j+2} = \frac{2j+1-K}{(j+1)(j+2)}\, a_{j} .$$ And for large $j$, ...
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Deriving spring oscilation period [duplicate]

We have a spring attached to a wall and at the other part an object on a frictionless surface. I tried to calculate the spring oscilation period. I used the conservation of energy and kinematics ...
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1answer
2k views

Kinetic energy and potential energy variation over distance in SHM

When you compute the average potential energy of a horizontal spring mass system from the mean position to the positive amplitude A, the value comes out to be $\frac{1}{6}kA^2$. For the average ...
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Phase space derivation of quantum harmonic oscillator partition function

I would like to derive the partition function for the quantum Harmonic oscillator from scratch: $$\tag{1} Z = \int dp \, dx\, e^{-\beta H}.$$ The free particle appears in many textbooks. $H = ...
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Dilation operator in CFT viewed as 'hamiltonian'?

From the commutation relations for the conformal Lie algebra, we may infer that the dilation operator plays the same role as the Hamiltonian in CFTs. The appropriate commutation relations are ...
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In what sense is a quantum field an infinite set of harmonic oscillators?

In what sense is a quantum field an infinite set of harmonic oscillators, one at each space-time point? When is it useful to think of a quantum field this way? The book I'm reading now, QFT by ...
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189 views

Finding the minimum radius of the pivoted disc

Here is a question based on Simple Harmonic Motion that I tackled just now. However I think I am having an approach to tackle this but I am not sure about it. Ouestion: A uniform disc of radius ...
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Finding the total space that an oscillating body has gone through via complex analysis

I was solving my homework and I got to an exercise that stated: An harmonic oscillating body has an equation of $$y(t) = A \sin(t)$$ Find the total space that the body has travelled during $t \in ...
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Help explain how direction change relates to acceleration [duplicate]

I was doing some simple harmonic motion problems and I came across this picture describing the position, velocity and acceleration of a linear oscillator. At the moment in time when v is 0 the ...
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1answer
291 views

Questions related to resonance/standing-waves and sound

I understand resonance for a simple harmonic oscillator but not for more complex systems like standing waves. How can I be in resonance with the normal mode in an organ pipe? I understand that the ...
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207 views

How can one calculate the phase difference between two quantum harmonic oscillator (Hermite-Gauss) states?

The analytic solutions of a quantum harmonic oscillator are given by Hermite-Gauss states, which differ in the order $n$ of the Hermite polynomials. If two such states are plotted, there will be a ...
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186 views

What is the relationship between Q-Factor and viscosity for a Newtonian fluid based damper?

I am working with a rotary damper (paddle hanging off a shaft attached to the bottom of a rotary table and into a vat of fluid). I know the viscosity of the current Newtonian fluid, I know the flow is ...
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492 views

Why isn't a pendulum clock considered a perpetual machine?

Why do people say that a perpetual machine is one that runs for eternity? Wouldn't a machine that runs taking no energy from outside but sometimes needs to be restarted, such as pendulum clock, be ...
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Energy of damped harmonic oscillator begins to increase with very large Q in numerical integration

I have numerically integrated the (reduced) homogeneous equation of a damped harmonic oscillator in order to see how the error propagates. $$\frac{d^2 X}{d\phi^2} + ...