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.

learn more… | top users | synonyms (1)

2
votes
2answers
144 views

Undamped oscillations. Why is the solution a linear combination of $\sin()$ and $\cos()$?

$ma = mg - cx$, where $x(0) = x_0 = 0$ is the position in which there is no tension in the rope. $dx/dt = v_0$ for $t = 0$; $v_0$ is a known constant. The discriminant of the characteristic ...
3
votes
0answers
1k views

Effective mass in Spring-with-mass/mass system

Suppose you have a particle of mass $m$ fixed to a spring of mass $m_0$ that, in turn, is fixed to some wall. I'm trying to calculate the effective mass $m'$ that appears in the law of motion of the ...
22
votes
3answers
1k views

Hilbert space of harmonic oscillator: Countable vs uncountable?

Hm, this just occurred to me while answering another question: If I write the Hamiltonian for a harmonic oscillator as $$H = \frac{p^2}{2m} + \frac{1}{2} m \omega^2 x^2$$ then wouldn't one set of ...
2
votes
3answers
383 views

Complex energy eigenstates of the harmonic oscillator

Given the Hamiltonian for the the harmonic oscillator (HO) as $$ \hat H=\frac{\hat P^2}{2m}+\frac{m}{2}\omega^2\hat x^2\,, $$ the Schroedinger equation can be reduced to: $$ \left[ ...
1
vote
1answer
2k views

SHM of floating objects

If we consider an object undergoing who has an acceleration proportional to the displacement of the object, it is going simple harmonic motion. In terms of Newton's second law, this is $$ -\dfrac k ...
2
votes
1answer
112 views

2nd order pertubation theory for harmonic oscillator

I'm having some trouble calculating the 2nd order energy shift in a problem. I am given the pertubation: $\hat{H}'=\alpha \hat{p}$, where $\alpha$ is a constant, and $\hat{p}$ is given by: ...
0
votes
2answers
882 views

Time period for spring connected body

Two identical springs with spring constant $k$ are connected to identical masses of mass $M$, as shown in the figures above. The ratio of the period for the springs connected in parallel (Figure 1) ...
1
vote
1answer
143 views

Energy eigenvalues of a Q.H.Oscillator with $[\hat{H},\hat{a}] = -\hbar \omega \hat{a}$ and $[\hat{H},\hat{a}^\dagger] = \hbar \omega \hat{a}^\dagger$

I just finished deriving the commutators: \begin{align} [\hat{H}, \hat{a}] &= -\hbar \omega \hat{a}\\ [\hat{H}, \hat{a}^\dagger] &= \hbar \omega \hat{a}^\dagger\\ \end{align} On the ...
1
vote
1answer
415 views

Eigenfunctions in a harmonic oscillator

This assignment is about the one dimensional harmonic oscillator (HO). The hamiltonian is just as you know from the HO, same goes for the energies, but I get that the wavefunction of the particle, at ...
3
votes
1answer
1k views

Pendulum in an elevator

Suppose we have a pendulum tied to the ceiling of an elevator which is at rest. The pendulum is oscillating with a time period $T$, and it has an angular amplitude, say $\beta$. Now at some time ...
4
votes
2answers
349 views

Pendulum Wave Period

Recently I've seen various videos showing the pendulum wave effect. All of the videos which I have found have a pattern which repeats every $60\mathrm{s}$. I am trying to work out the relationship ...
4
votes
2answers
473 views

Proof for commutator relation $[\hat{H},\hat{a}] = - \hbar \omega \hat{a}$

I know how to derive below equations found on wikipedia and have done it myselt too: \begin{align} \hat{H} &= \hbar \omega \left(\hat{a}^\dagger\hat{a} + \frac{1}{2}\right)\\ \hat{H} &= ...
1
vote
2answers
130 views

How does one subtract two light beams?

From what I understand, it seems like you can only "add" beams together. You can use a beam combiner, basically using a beam splitter in reverse, to combine two beams. In homodyne detection, you use a ...
3
votes
2answers
266 views

Probability of position in linear shm?

The problem that got me thinking goes like this:- Find $dp/dx$ where $p$ is the probability of finding a body at a random instant of time undergoing linear shm according to $x=a\sin(\omega t)$. ...
3
votes
1answer
578 views

Schrödinger equation for a harmonic oscillator

I have came across this equation for quantum harmonic oscillator $$ W \psi = - \frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + \frac{1}{2} m \omega^2 x^2 \psi $$ which is often remodelled by defining a new ...
2
votes
1answer
60 views

What is $\gamma$ in the damping equation?

$x''+\gamma x'+w_0^2x=0$ That is the general equation for damped harmonic motion. What is the term or name that describes $\gamma$? Is it called the damping constant? I know its the ration between ...
2
votes
0answers
47 views

Quantum harmonic oscilator - book that does it all right [duplicate]

I am dealing with quantum harmonic oscillator. In every single book or video i have checked out i can read how the mathematical technique for solving this Schrödinger equation: $$ W\psi = - ...
0
votes
1answer
128 views

Harmonic oscillator with light damping

My textbook gives the following for x as a function of time for a lightly damped harmonic oscillator: $$ x = Ae^{- \gamma t} \cos (\omega \, t)$$ for $\gamma = \dfrac b {2m}$. It says this implies ...
4
votes
2answers
1k views

Why is the damping force on a spring oscillator linearly dependent on velocity?

If you consider the damping force is friction like in: then the force should be $$F=\mu N$$ where $\mu$ is the coefficient of kinetic friction. Why then is the damping force assumed to be linearly ...
4
votes
0answers
340 views

Relativistic genarization of Quantum Harmonic Oscillator

I am trying to find out relativistic description of a quantum harmonic oscillator. For a classical relativistic oscillator mass is a function of co-ordinates(http://arxiv.org/abs/1209.2876). ...
2
votes
1answer
540 views

Compound pendulum clarification?

I read in a book the following about compound pendulum and small displacements: There are two points only for which the time period is minimum. there are maximum 4 points for which the time ...
2
votes
1answer
160 views

A good theoretical approximation for a magnetically damped pendulum

In a laboratory course we had to perform an experiment with a pendulum (just an iron weight on a wire) and play around for some time with its wire's length and so on. This was quite boring and we ...
-1
votes
1answer
1k views

Standing Waves: finding the number of antinodes [closed]

A string with a fixed frequency vibrator at one end forms a standing wave with 4 antinodes when under tension T1. When the tension is slowly increased, the standing wave disappears until tension T2 is ...
0
votes
2answers
2k views

Calculating phase difference of sound waves

An observer stands 3 m from speaker A and 5 m from speaker B. Both speakers, oscillating in phase, produce waves with a frequency of 250 Hz. The speed of sound in air is 340 m/s. What is the phase ...
0
votes
0answers
39 views

How can I find the frequency? [duplicate]

Grocery stores often have spring scales in their produce department to weigh fruits and vegetables. The pan of one particular scale has a mass of 0.5 kg, and when you place a 0.5 kg sack of potatoes ...
1
vote
0answers
343 views

Doubling the energy of an oscillating mass on a spring [closed]

From this question: Question 1. What do we need to change in order to double the total energy of a mass oscillating at the end of a spring? (a) increase the angular frequency by $\sqrt{2}$. ...
1
vote
2answers
226 views

Potential energy during vertical fall

Suppose I have a weightless spring connected perpendicularly to the ground, and it has on top of it some weightless surface. Now, I release some sticky object from height $h$ above the system of light ...
2
votes
1answer
434 views

Simple harmonic oscillator system and changes in its total energy

Suppose I have a body of mass $M$ connected to a spring (which is connected to a vertical wall) with a stiffness coefficient of $k$ on some frictionless surface. The body oscillates from point $C$ to ...
3
votes
1answer
238 views

The issue on existence of inverse operations of $a$ and $a^{\dagger}$

I have asked a question at math.stackexchange that have a physical meaning. My assumption: Suppose $a$ and $a^\dagger$ is Hermitian adjoint operators and $[a,a^\dagger]=1$. I want to prove that ...
1
vote
2answers
118 views

Does spatial coupling prohibit resonances due to an external source field?

The harmonic oscillator coupled to a sinodial external source $$\frac{\partial^2 x(t)}{\partial t^2}+\omega_0^2 x(t)=F_0\sin(\omega_\text{ext}\ t),$$ has the solution $$x(t)=x(0)\cos(\omega_0 t)+C ...
2
votes
1answer
463 views

Accessible microstates of harmonic oscillator in microcanonical enemble

While reading up on statistical physics, I am going through the calculation of the partition function of the harmonic oscillator in the microcanonical ensemble. The result for the partition function ...
2
votes
1answer
148 views

Why uncertainity is minimum for coherent states?

While reading for quantum damped harmonic oscillator, I came across coherent states, and I asked my prof about them and he said me it is the state at which $\Delta x\Delta y$ is minimum. I didn't ...
0
votes
1answer
158 views

Hyperbolic, parabolic, elliptical PDE related to under-, critical- and overdamped in harmonic osciallation

A damped harmonic oscillator has three cases for the damping: underdamped, critically damped and overdamped. With partial differential equations, I know the hyperbolic wave equation, the parabolic ...
2
votes
1answer
184 views

Coordinate representation of quantum ladder operator?

I can't seem to figure out how to derive the coordinate representation of the $a_+$ ladder operator in quantum mechanics. I know that $a_-$ is $\sqrt{\frac{1}{2mwh}} (mwx + i\dot{p}) $ in which where ...
0
votes
2answers
496 views

FWHM in resonance amplitude square derivation

Consider a linear harmonic oscillator subject to a periodic force: $$ \ddot x + 2 \beta \dot x + \omega _0 ^2x = f_0\cos \omega t$$ The solution tends to: $$A \cos (\omega t - \delta)$$ where: ...
4
votes
1answer
195 views

Metronome synchronisation applied to swings

The movement of several metronomes can be synchronised when a movable floor is utilised which couples the movement of the different metronomes. Is it possible to apply this sort of synchronisation to ...
4
votes
3answers
4k views

Can someone please derive $T=2\pi\sqrt{l/g}$ or prove it without using calculus?

I don't know much calculus, but I want to know that how one derives the formula to find time period $T$ of a simple pendulum.
1
vote
2answers
814 views

Why is simple harmonic motion called so?

Is the motion of a simple pendulum, a simple harmonic motion? It stops vibrating after sometime.
2
votes
1answer
108 views

Sitting on the bob of a pendulum

Walter Lewin's best performance was the pendulum demonstration, and I copy the transcript now: Would the period come out to be the same or not? [students respond] Some of you think it's ...
3
votes
1answer
146 views

From the local Hooke's law to the global one

My system consist of a cylinder with axis Z that can contract and dilate along this axis. It obeys microscopically Hooke's law of elasticity: ...
4
votes
2answers
936 views

Constant magnetic field applied to a quantum harmonic oscillator

I have a spinless particle of mass $m$ and charge $q$ which is an isotropic harmonic oscillator of frequency $\omega_0$, then I apply a constant magnetic field in the $z$ direction. We can show the ...
2
votes
4answers
318 views

Why is linear independence of harmonic oscillator solutions important?

The equation of motion for the harmonic oscillator (mass on spring model) $$\frac{d^2x}{dt^2} + \omega_0^2 x = 0$$ with $\omega_0^2 = D/m$, $D$ and $m$ being the force constant of the spring and the ...
1
vote
1answer
327 views

Shift operator (integral calculus involving Hermite polynomials) [closed]

I didn't know whether to pose this question on Physics.stackexchange or Math.stackexchange. But since this is the last step of a development involving the eigenfunctions of an Harmonic oscillator and ...
1
vote
3answers
2k views

Partition function for quantum harmonic oscillator

Hi guys I'm currently trying to solve a mock exam for an exam in a few days and am a bit confused by the solutions they gave us for this exercise: Exercise: A solid is composed of N atoms which ...
16
votes
1answer
1k views

Why don't tuning forks have three prongs?

I was reading Why tuning forks have two prongs?. The top answer said the reason was to reduce oscillation through the hand holding the other prong. So if having 2 prongs will reduce oscillation loss, ...
0
votes
3answers
8k views

How to derive the period of spring pendulum?

So I wanted to find out how to (simply, if that's possible) derive the formula for a period of spring pendulum: $T=2\pi \sqrt{\frac{m}{k}}$. However, Google doesn't help me here as all I see is the ...
3
votes
2answers
557 views

Dynamics of a Vertical Mass-Spring Simple Harmonic Oscillator with Gravity

I am having some trouble obtaining the elastic potential energy and gravitational potential energy of a simple mass spring system. In this experiment, masses attached to a spring were dropped from a ...
0
votes
1answer
1k views

Estimate the damping coefficient of my car

I was wondering how I can estimate the damping coefficient of my car by doing the hand bouncing the car body and watching the motion of the car? I just need a rough estimate of the damping ...
4
votes
1answer
477 views

Writing equation for amplitude of driven harmonic oscillator in Lorentzian form

This harmonic oscillator is driven and damped, with the form: $$\ddot{x} + \lambda \dot{x} + \omega_0^2 x = A \cos(\omega_d t)$$ Now, I have used the ansatz (guess): $x(t) = B \cos(\omega_d t + ...
0
votes
2answers
2k views

Finding an equation for velocity and acceleration

I'm trying to derive an equation for the velocity and acceleration of an object undergoing simple harmonic motion. I have the equation for displacement: $x = A\sin (2 \pi ft)$ If I differentiate the ...