Harmonic oscillator Definition and 57 Discussions

In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional to the displacement x:







F




=

k



x




,


{\displaystyle {\vec {F}}=-k{\vec {x}},}
where k is a positive constant.
If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude).
If a frictional force (damping) proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator. Depending on the friction coefficient, the system can:

Oscillate with a frequency lower than in the undamped case, and an amplitude decreasing with time (underdamped oscillator).
Decay to the equilibrium position, without oscillations (overdamped oscillator).The boundary solution between an underdamped oscillator and an overdamped oscillator occurs at a particular value of the friction coefficient and is called critically damped.
If an external time-dependent force is present, the harmonic oscillator is described as a driven oscillator.
Mechanical examples include pendulums (with small angles of displacement), masses connected to springs, and acoustical systems. Other analogous systems include electrical harmonic oscillators such as RLC circuits. The harmonic oscillator model is very important in physics, because any mass subject to a force in stable equilibrium acts as a harmonic oscillator for small vibrations. Harmonic oscillators occur widely in nature and are exploited in many manmade devices, such as clocks and radio circuits. They are the source of virtually all sinusoidal vibrations and waves.

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  1. Salmone

    I Doubt on Morse potential and harmonic oscillator

    I have a little doubt about Morse potential used for vibration levels of diatomic molecules. With regard to the image below, if the diatomic molecule is in the vibrational ground state, when the oscillation reaches the maximum amplitude for that state the velocity of the molecule must be zero so...
  2. Mr_Allod

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  3. anaisabel

    Density of states of one three-dimensional classical harmonic oscillator

    ia
  4. S

    Calculating degeneracy of the energy levels of a 2D harmonic oscillator

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  5. Lo Scrondo

    I Time averages for a 2-dimensional harmonic oscillator

    I'm studying Ergodic Theory and I think I "got" the concept, but I need an example to verify it... Let's take the simplest possible 2D classical harmonic oscillator whose kinetic energy is $$T=\frac{\dot x^2}{2}+\frac{\dot y^2}{2}$$ and potential energy is $$U=\frac{ x^2}{2}+\frac{y^2}{2}$$...
  6. D

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    Hello everybody, new here. Sorry in advance if I didn't follow a specific guideline to ask this. Anyways, I've got as a homework assignment two cannonical transformations (q,p)-->(Q,P). I have to obtain the hamiltonian of a harmonic oscillator, and then the new coordinates and the hamiltonian...
  7. Vivek98phyboy

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  8. C

    A Please help me understand this HO energy in He4 gas

  9. H

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  10. M

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  11. Baibhab Bose

    Effects of KE & PE of a Harmonic Oscillator under Re-scaling of coordinates

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  12. PhillipLammsoose

    I Problem with the harmonic oscillator equation for small oscillations

    Hey, I solved a problem about a double pendulum and got 2 euler-lagrange equations: 1) x''+y''+g/r*x=0 2) x''+y'' +g/r*y=0 (where x is actually a tetha and y=phi) the '' stand for the 2nd derivation after t, so you can see the basic harmonic oscillator equation with a term x'' or y'' that...
  13. TheBigDig

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  14. D

    A position of stable equilibrium, and the period of small oscillations

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  15. G

    Finding the parameters for Harmonic Oscillator solutions

    Homework Statement Using the Schrödinger equation find the parameter \alpha of the Harmonic Oscillator solution \Psi(x)=A x e^{-\alpha x^2} Homework Equations -\frac{\hbar^2}{2m}\,\frac{\partial^2 \Psi(x)}{\partial x^2} + \frac{m \omega^2 x^2}{2}\Psi(x)=E\Psi(x) E=\hbar\omega(n+\frac{1}{2})...
  16. Baynie

    MATLAB Code: Stationary Schrodinger EQ, E Spec, Eigenvalues

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  17. Rabindranath

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  18. LarryC

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  19. Sushmita

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  20. K

    I Phase angle of a damped driven harmonic oscillation

    Hello, in every book and on every website (e.g. here http://farside.ph.utexas.edu/teaching/315/Waves/node13.html) i found for driven harmonic osciallation the same solution for phase angle:θ=atan(ωb/(k−mω^2)) where ω is driven freq., m is mass, k is spring constant. I agree with it =it follows...
  21. Safder Aree

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  22. S

    Entropy Contradiction for a Single Harmonic Oscillator

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  23. WeiShan Ng

    I Distribution of Position in classical & quantum case

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  24. D

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  25. G

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  26. M

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  27. Y

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  28. K

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  29. WeiShan Ng

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  30. D

    I How is the CSCO in an harmonic oscillator?

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  31. V

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  32. V

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  33. Adolfo Scheidt

    I Harmonic Oscillator equivalence

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  34. J

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  35. upender singh

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  36. L

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  37. L

    I Mass on a string-harmonic oscillator

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  38. O

    I Harmonic Oscillator in 3D, different values on x, y and z

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  39. S

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  40. G

    I Harmonic oscillator: Why not chaotic?

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  41. L

    Time period in harmonic oscillation.

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  42. thegirl

    I Why is there only odd eigenfunctions for a 1/2 harmonic oscillator

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  43. G

    Harmonic oscillator positive position expectation value?!

    So this is something that troubled me a bit- in Shankar's PQM, there's an exercise that asks you to find the position expectation value for the harmonic oscillator in a state \psi such that \psi=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle) Where |n\rangle is the n^{th} energy eigenstate of...
  44. G

    Deriving hermite differential equation from schrødinger harm oscillator

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  45. E

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  46. Patrick McBride

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  47. G

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  48. Q

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  49. C

    Taylor series expansion of an exponential generates Hermite

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  50. Entanglement717

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