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)

0
votes
1answer
298 views

Difference between harmonic oscillator & coupled oscillators

Coupling, according to wiki, is the condition of two systems when they interact with each other. Now, I came across the terms harmonic oscillator and coupled oscillators. Now,what is the difference ...
7
votes
2answers
1k views

What will be the equation of motion of driven pendulum for amplitudes beyond the small angle approximation?

When finding the period of a pendulum beyond the small angle approximation, we have to use integration for small interval of $\theta$ and elliptical integration. I was trying to apply this situation ...
48
votes
5answers
6k views

Why is the harmonic oscillator so important?

I've been wondering what makes the harmonic oscillator such an important model. What I came up with: It is a (relatively) simple system, making it a perfect example for physics students to learn ...
1
vote
4answers
707 views

Spring stiffness calculation problems

I am trying to model a 1 DoF electromagnetic vibration sensor (geophone) analytically and with finite elements. A geophone consists of springs, a permanent magnet and coils. The coils are suspended ...
0
votes
0answers
210 views

Harmonic oscillator coherent state expected values

I'm looking to calculate the expected values of a coherent state (of a harmonic oscillator) evolving in time. I know that the $x$ and $p$ expectation values are as in classical motion, but I'm ...
1
vote
1answer
269 views

How to determine angular frequency of this system?

I am self studying on harmonic motion and springs. One of the problems is: Two identical objects of mass m are placed at either end of a spring of spring constant k and the whole system is placed on a ...
1
vote
1answer
4k views

Does damping force affect period of oscillation?

In my physics notes, it has been given that the damping force increases the period of oscillation. I am unable to understand this part. How is this possible? The only relation I know is that as the ...
8
votes
3answers
730 views

Will a damped harmonic oscillator, with no initial amplitude, oscillate if there was background “noise”?

Suppose I have a damped harmonic oscillator which is at rest, sitting comfortably with no initial amplitude, obeying the equation $$\ddot{x} + \frac{1}{Q}\dot{x} + x = 0$$ where x is the vertical ...
2
votes
1answer
66 views

What's the closed-form of the sum relating to the DOS of simple harmonic motion?

In order to calculate the density of states of single particle in the simple harmonic potential, we would calculate that $$ D(\epsilon)=\sum_{n}\delta(\epsilon-\epsilon_n) $$ where $\epsilon_n=(n+1/2)\...
1
vote
1answer
547 views

Why does $k/m=\omega^2$ for harmonic motion? [closed]

Can anyone please give me a proof for $k/m=w^2$ in simple harmonic motion? I have tried energy conservation and Newton's laws as follows : In the case of a mass-spring system, $$F=ma =-kx\\ F=ma = ...
0
votes
0answers
153 views

For a bob on a pendulum following simple harmonic motion, what causes the bob to accelerate towards the centre of equilibrium?

*The position of equilibrium being the position of the bob when the string is taut and vertically downwards. When I draw a simple diagram, I see that the tension of the string, which always acts ...
0
votes
1answer
67 views

Watertank waves

Say we have a rectangular tank of water and we push it lengthwise. Suppose the surface stays planar. What would be the trajectory of the centre of mass?
1
vote
2answers
305 views

Is it possible to write explicitly the exact solution for forced damped harmonic oscillator?

Preamble Consider a damped harmonic oscillator, with his well know differential equation \begin{equation*} m \ddot{x} + c \dot{x} + kx=0 \end{equation*} and let's find the solution that satisfies $x(...
1
vote
2answers
640 views

Number operator in quantum field theory?

The number operator, counting the number of quanta is defined as follows: $$ N = \int \frac{d^3 p}{(2\pi)^3} \hphantom{ii} a^{\dagger}_pa_p$$ with the momentum eigenstates being defined as $\lvert ...
3
votes
3answers
209 views

Quantum simple harmonic oscillator interpretation

I am just wondering what does the SHO system from quantum mechanics actually physically represent? Is it just a SHO of a quantum particle, seems a little too obvious for quantum theory? I'm from a ...
0
votes
2answers
155 views

Coupled oscillators and Normal Modes

Consider we have a system consisting of 2 arbitrary masses and 3 arbitrary springs connecting them horizontally and between fixed walls, and we want to obtain the motion of each mass after we input ...
4
votes
0answers
227 views

Cubic perturbation to coupled quantum harmonic oscillators

I recently came across this two-dimensional problem of a particle in a potential of the form $$V = \displaystyle{\frac{1}{2}m \omega^2} \big(y^2 + x^2y \big) - \alpha y,$$ where $x$ and $y$ are known ...
0
votes
1answer
83 views

How can I prove this inequality for a harmonic oscillator?

I need a hand with this problem. I have to prove that for a particle in any quantum state in an harmonic potential $$ \langle X\rangle \leq2\Delta E\Delta P/(m \omega^2 \hslash) $$ Here's my ...
0
votes
1answer
98 views

Period of swinging incomplete hula-hoop

I was working on a problem where I had to calculate the period of a swinging incomplete hula-hoop given its center of mass and radius. It only swings with very small amplitude so I considered the ...
0
votes
2answers
134 views

How to check if a system is in SHM in two spring systems?

I must determine if the system (a) is in SHM. I must arrive to the conclusion that there is a linear restoring force such that $a=-w^2x$. The solution's manual affirms that according to Newton's ...
0
votes
1answer
335 views

How do I incorporate friction and mass when analyzing spring motion?

Problem 12 of section B of this PDF file reads Two springs with spring constant k1 and k2 are attached to a body of mass (m) in two different configurations in 2 cases (A and B) as shown. ...
2
votes
2answers
436 views

What are the forces acting on a Pendulum Tuned Mass Damper?

Upon researching tuned mass dampers, I came across this free body diagram of a pendulum tuned mass damper. However, I don't understand where many of the forces come from. What exactly are the forces ...
1
vote
2answers
512 views

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

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 ...
3
votes
1answer
114 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 $[...
2
votes
1answer
109 views

Energy in harmonic oscillator [closed]

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 ...
0
votes
3answers
2k views

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 $(\text{...
1
vote
1answer
535 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 $...
10
votes
3answers
2k views

Definition of the $Q$ 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 ...
2
votes
0answers
145 views

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 of ...
0
votes
3answers
319 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 ...
0
votes
3answers
134 views

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 ...
2
votes
2answers
334 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 ...
0
votes
1answer
275 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 ...
0
votes
1answer
178 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 ...
0
votes
2answers
278 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 ...
0
votes
3answers
478 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 ...
2
votes
2answers
4k 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?
0
votes
1answer
68 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} = \...
0
votes
1answer
91 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 SHM....
1
vote
1answer
171 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 f}{\...
0
votes
2answers
137 views

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 $\ddot{x}+2\...
2
votes
3answers
219 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 ...
-1
votes
1answer
767 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 is ...
4
votes
3answers
295 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 ...
3
votes
0answers
94 views

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 ...
-3
votes
1answer
1k 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 ...
2
votes
0answers
109 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 ...
3
votes
3answers
293 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. The ...
0
votes
3answers
165 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 ...
2
votes
1answer
530 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} ...