# 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. ### I Doubt on Morse potential and harmonic oscillator

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25. ### Relativistic Harmonic Oscillator Lagrangian and Four Force

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26. ### QM: Writing time evolution as sum over energy eigenstates

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27. ### Solve a system of two linked harmonic oscillators

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28. ### Calculate the time out of sample points with set frequency

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29. ### I Why drop the vibrational ground state energy

This is from *Statistical Physics An Introductory Course* by *Daniel J.Amit* The text is calculating the energy of internal motions of a diatomic molecule. The internal energies of a diatomic molecule, i.e. the vibrational energy and the rotational energy is given by...
30. D

### I How is the CSCO in an harmonic oscillator?

Hi everyone, I have a great doubt in this problem: Let a mass m with spin 1/2, subject to the following central potencial V(r): V(r)=1/2mω2r2 Find the constants of motion and the CSCO to solve the Hamiltonian? This is my doubt, I can't find the CSCO in this potencial. Is a problem in general...
31. ### Linear perturbation to harmonic oscillator

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32. ### Anharmonic oscillator first-order correction to energy

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33. ### I Harmonic Oscillator equivalence

Hello, I'm studying the section 2.2 of "Introduction to Quantum Mechanics, 2nd edition" (Griffiths), and he shows this equation $$\frac{\partial^2\psi}{\partial x^2} = -k^2\psi ,$$ where psi is a function only of x (this equation was derivated from the time-independent Schrödinger equation) and...
34. ### Harmonic oscillator with 3 charged particles

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35. ### Mean number of oscilatory quanta?

A quantum mechanical oscillator with the Hamiltonian H1=p^2/2m +(m(w1)^2 x^2)/2 is initially prepared in its ground state (zero number of oscillatory quanta). Then the Hamiltonian changes abruptly (almost instantly): H1→H2=p^2/2m +(m(w2)^2 x^2)/2 What is the mean number of oscillatory quanta...
36. ### Perturbed Hamiltonian Matrix for Quantum Harmonic Oscillator

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37. ### I Mass on a string-harmonic oscillator

Hello, I encountered a mass on a string problem in which the mass, moved from the equilibrium, gets a harmonic motion. The catch, however, is that the mass of the string is not neglected. On the lecture, the prof. wanted to calculate, for some reason, the complete kinetic energy of the system...
38. ### I Harmonic Oscillator in 3D, different values on x, y and z

Hi, For a harmonic oscillator in 3D the energy level becomes En = hw(n+3/2) (Note: h = h_bar and n = nx+ny+nz) If I then want the 1st excited state it could be (1,0,0), (0,1,0) and (0,0,1) for x, y and z. But what happens if for example y has a different value from the beginning? Like this...
39. ### Estimate vibrational frequency of N2 molecule

Homework Statement Experimental data for the heat capacity of N2 as a function of temperature are provided. Estimate the frequency of vibration of the N2 molecule. Homework Equations Energy of harmonic oscillator = (n+1/2)ħω C=7/2kB Average molecular energy = C*T But this is an expression...
40. ### I Harmonic oscillator: Why not chaotic?

Hi. As far as I know, the movement of a harmonic oscillator normally is not considered to be chaotic. Why not? Since the angular frequency can never be known to absolute precision, an error in the phase builds up. I can see that this build-up is only linear in time (if we assume the angular...
41. ### Time period in harmonic oscillation.

Homework Statement Homework Equations Find Time Period. Find the error in my solution. The Attempt at a Solution Where i am wrong ?
42. ### I Why is there only odd eigenfunctions for a 1/2 harmonic oscillator

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43. ### Harmonic oscillator positive position expectation value?!

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44. ### Deriving hermite differential equation from schrødinger harm oscillator

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45. ### Ground-state energy of harmonic oscillator(operator method)

I studied this from Griffith Chapter 2, with the algebraic (raising and lowering operator) method, we reached the ground state by setting a_Ψ0 = 0 , then we got what the ground state is, and then plugged it in the Schrodinger equation to know the energy, and it turned out to be 0.5 ħω. My...
46. ### Hamiltonian of a 1D Linear Harmonic Oscillator

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47. ### How Fourier components of vector potential becomes operators

Hello. I'm studying quantization of electromagnetic field (to see photon!) and on the way to reach harmonic oscillator Hamiltonian as a final stage, sudden transition that the Fourier components of vector potential A become quantum operators is observed. (See...
48. ### Harmonic oscillator coherent state wavefunction

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49. ### Taylor series expansion of an exponential generates Hermite

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50. ### 1D Harmonic Oscillator in a Constant Electric Field

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