What is Harmonic oscillator: Definition and 741 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:






{\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. A

    I Where do the vibrational modes of molecules come from?

    Hello everyone. First, sorry for my english. Second, I have got question where vibration mode of H2+ molecule (I think it is the most simple molecule for this topic explanation) comes from. If I should get basics before asking this tell me :). By my count the most important factor behind "being"...
  2. ergospherical

    I Translating the harmonic oscillator

    Let's say I know the position space wavefunctions of the 1d harmonic oscillator ##\psi_n(x)## corresponding to the state ##| n \rangle## are known. I want to write ##\psi_m(x + a)##, for fixed ##m = 1,2,...##, in terms of all of the ##\psi_n(x)##. I know \begin{align*} \psi_n(x+a) = \langle x |...
  3. P

    I Thoughts about coupled harmonic oscillator system

    Same instruction was given while finding value of 'g' by a bar pendulum. In the former case,does the spring obeys hooke's law while it forms a coupled harmonic oscillator system?Does the bar pendulum somehow breaks the simple harmonic motion(such that we can't apply the law for sumple harmonic...
  4. J

    Modification to the simple harmonic oscillator

    I was assuming there could be something via perturbation theory? I am unsure.
  5. G

    I Driven harmonic oscillator

    This is an equation I found for the delta phase lag of a driven oscillator. W is the driving angular frequency and Wo is the natural angular frequency of the driven system. Of course this is just a small part of the solution to the differential equation. Now ... 1) when W is much smaller than Wo...
  6. P

    X^4 perturbative energy eigenvalues for harmonic oscillator

    The book(Schaum) says the above is the solution but after two hours of tedious checking and rechecking I get 2n^2 in place or the 3n^2. Am I missing something or is this just a typo?
  7. S

    I How to interpret complex solutions to simple harmonic oscillator?

    Consider the equation of motion for a simple harmonic oscillator: ##m\ddot {x}(t)=-kx(t).## The solutions are ##x(t)=Ae^{i\omega t}+Be^{-i\omega t},## where ##\omega=\sqrt{\frac{k}{m}}##, and constants ##A## and ##B##. Physically, what does it mean for a solution to be complex? Is it only the...
  8. Misha87

    B Harmonic oscillator and simple pendulum time period

    Hi, I have been thinking about pendulums a bit and discovered that a HO(harmonic Oscillator) will take the same time to complete one period T no matter which amplitude A/length l it has, if stiffness k and mass m are the same. But moving on to a simple pendulum suddenly the time period for one...
  9. 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...
  10. warhammer

    Question on Intro QM pertaining to Harmonic Oscillator

    Hi. I have attached a neatly done solution to the above question. I request someone to please check my solution and help me rectify any possible mistakes that I may have made.
  11. Salmone

    I Particle on a cylinder with harmonic oscillator along z-axis

    I need to know if I have solved the following problem well: A spin-less particle of mass m is confined to move on the surface of a cylinder of infinite height with a harmonic potential on the z-axis and Hamiltonian ##H=\frac{p_z^2}{2m}+\frac{L_z^2}{2mR^2}+\frac{1}{2}m\omega^2z^2## and I need to...
  12. Mr_Allod

    Position expectation value of 2D harmonic oscillator in magnetic field

    Hello there, for the above problem the wavefunctions can be shown to be: $$\psi_{n,l}=\left[ \frac {b}{2\pi l_b^2} \frac{n!}{2^l(n+l)!}\right]^{\frac12} \exp{(-il\theta - \frac {r^2\sqrt{b}}{4l_b^2})} \left( \frac {r\sqrt{b}}{l_b}\right)^lL_n^l(\frac {r^2b}{4l_b^2})$$ Here ##b = \sqrt{1 +...
  13. Mr_Allod

    Quantum Harmonic Oscillator with Additional Potential

    Hello there, I am trying to solve the above and I'm thinking that the solutions will be Hermite polynomials multiplied by a decaying exponential, much like the standard harmonic oscillator problem. The new Hamiltonian would be like so: $$H = - \frac \hbar {2m} \frac {d^2}{dx^2}\psi + \frac...
  14. Huzaifa

    B Why is a simple pendulum not a perfect simple harmonic oscillator?

    Khan Academy claims that a simple pendulum not a perfect simple harmonic oscillator. Why is it so?
  15. J

    Discretizing a 1D quantum harmonic oscillator, finding eigenvalues

    ##x## can be discretized as ##x \rightarrow x_k ## such that ##x_{k + 1} = x_k + dx## with a positive integer ##k##. Throughout we may assume that ##dx## is finite, albeit tiny. By applying the Taylor expansion of the wavefunction ##\psi_n(x_{k+1})## and ##\psi_n(x_{k-1})##, we can quickly...
  16. R

    Weakly interacting Bosons in a 3D harmonic oscillator

    1. Since N is large, ignore the kinetic energy term. ##[-\mu + V(r) + U|\Psi (r)|^2]\Psi (r) = 0## 2. Solve for the density ##|\Psi (r)|^2## ##|\Psi (r)|^2 = \frac{\mu - V(r)}{U}## 3. Integrate density times volume to get number of bosons ##\int|\Psi (r)|^2 d\tau = \int \frac{\mu -...
  17. koustav

    Find Ground State Energy of 3D Harmonic Oscillator

    Summary:: I am trying to find the exact ground state energy of the hamiltonian.kindly help me with this
  18. anaisabel

    Density of states of one three-dimensional classical harmonic oscillator

  19. K

    I Stimulated emission in harmonic oscillator

    Hello! Is stimulated emission possible for a harmonic oscillator (HO) i.e. you send a quanta of light at the right energy, and you end up with 2 quantas and the HO one energy level lower (as you would have in a 2 level system, like an atom)?
  20. S

    I How to solve 2nd order TDSE for a Gaussian-kicked harmonic oscillator?

    Consider the gaussian kick potential, ##\hat{V}(t) = \hat{x} \exp{(\frac{-t^2}{2 \tau^2})}## where ##\hat{x} = a+a^\dagger## in terms of creation and annihilation operators. Then we define the potential in the interaction picture, ##\hat{V}_I(t) = e^{i\hat{H}t}\hat{V}(t)e^{-i\hat{H}t}## I...
  21. TRB8985

    I Question on Harmonic Oscillator Series Derivation

    Good afternoon all, On page 51 of David Griffith's 'Introduction to Quantum Mechanics', 2nd ed., there's a discussion involving the alternate method to getting at the energy levels of the harmonic oscillator. I'm filling in all the steps between the equations on my own, and I have a question...
  22. docnet

    1-D Harmonic Oscillator

    Consider a one-dimensional harmonic oscillator. ##\psi_0(x)## and ##\psi_1(x)## are the normalized ground state and the first excited states. \begin{equation} \psi_0(x)=\Big(\frac{m\omega}{\pi\hbar}\Big)^{\frac{1}{4}}e^{\frac{-m\omega}{2\hbar}x^2} \end{equation} \begin{equation}...
  23. docnet

    Simple harmonic oscillator Hamiltonian

    We show by working backwards $$\hbar w \Big(a^{\dagger}a+\frac{1}{2}\Big)=\hbar w \Big(\frac{mw}{2\hbar}(\hat{x}+\frac{i}{mw}\hat{p})(\hat{x}-\frac{i}{mw}\hat{p})+\frac{1}{2}\Big)$$...
  24. Q

    Harmonic Oscillator With and Without Friction (mass on a spring)

  25. J

    I Zero-point energy of the harmonic oscillator

    First time posting in this part of the website, I apologize in advance if my formatting is off. This isn't quite a homework question so much as me trying to reason through the work in a way that quickly makes sense in my head. I am posting in hopes that someone can tell me if my reasoning is...
  26. Lo Scrondo

    I Different invariant tori in the case of a 2D harmonic oscillator

    Hi everyone! Both sources I'm currently reading (page 291 of Mathematical Methods of Classical Mechanics by Arnol'd - get it here - and page 202 of Classical Mechanics by Shapiro - here) say that, in the case of the planar harmonic oscillator, using polar or cartesian coordinate systems leads...
  27. patric44

    The Harmonic Oscillator Asymptotic solution?

    hi guys i am trying to solve the Asymptotic differential equation of the Quantum Harmonic oscillator using power series method and i am kinda stuck : $$y'' = (x^{2}-ε)y$$ the asymptotic equation becomes : $$y'' ≈ x^{2}y$$ using the power series method ##y(x) = \sum_{0}^{∞} a_{n}x^{n}## , this...
  28. chocopanda

    Harmonic oscillator with ladder operators - proof using the Sum Rule

    I'm trying verify the proof of the sum rule for the one-dimensional harmonic oscillator: $$\sum_l^\infty (E_l-E_n)\ | \langle l \ |p| \ n \rangle |^2 = \frac {mh^2w^2}{2} $$ The exercise explicitly says to use laddle operators and to express $p$ with $$b=\sqrt{\frac {mw}{2 \hbar}}-\frac...
  29. Mayan Fung

    Perturbation from a quantum harmonic oscillator potential

    For the off-diagonal term, it is obvious that (p^2+q^2) returns 0 in the integration (##<m|p^2+q^2|n> = E<m|n> = 0##). However, (pq+qp) seems to give a complicated expression because of the complicated wavefunctions of a quantum harmonic oscillator. I wonder whether there is a good method to...
  30. VVS2000

    Time period of a harmonic oscillator

  31. K

    A Equipartition theorem and Coupled harmonic oscillator system

    Dear all, While simulating a coupled harmonic oscillator system, I encountered some puzzling results which I haven't been able to resolve. I was wondering if there is bug in my simulation or if I am interpreting results incorrectly. 1) In first case, take a simple harmonic oscillator system...
  32. S

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

    Too dim for this kind of combinatorics. Could anyone refer me to/ explain a general way of approaching these without having to think :D. Thanks.
  33. I

    Griffiths Problem 3.35. Harmonic Oscillator, Bra-ket notation

    Firstly, apologies for the latex as the preview option is not working for me. I will fix mistakes after posting. So for ##<x>## = (##\sqrt{\frac{\hbar}{2m\omega}}##) ##(< \alpha | a_{+} + a_{-}| \alpha >)## = (##\sqrt{\frac{\hbar}{2m\omega}}##) ##< a_{-} \alpha | \alpha> + <\alpha | a_{-}...
  34. 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 $$U=\frac{ x^2}{2}+\frac{y^2}{2}$$...
  35. D

    Phase space of a harmonic oscillator and a pendulum

    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...
  36. S

    Normalization constant A of a harmonic oscillator

    I've worked through it doing what I thought I should have done. I normalized the original wavefunction(x,0) and made it = one before using orthonormality to get to A^2(1-1) because i^2=-1 but my final answer comes out at 1/0 which is undefined and I don't see how that could be correct since A is...
  37. J

    I Atoms in a harmonic oscillator and number states

    I am confused about the relation between the number state ##|n\rangle## with the annhilation and creation operators ##a^\dagger## and ##a## respectively, and the number of atoms in the harmonic oscillator. I'll try to express my current understanding, I thought the number states represent the...
  38. JD_PM

    Working out harmonic oscillator operators at ##L \rightarrow \infty##

    Let's go step by step a) We know that the harmonic oscillator operators are $$a^{\dagger} = \frac{1}{\sqrt{2 \hbar m \omega}} ( -ip + m \omega q)$$ $$a= \frac{1}{\sqrt{2 \hbar m \omega}} (ip + m \omega q)$$ But these do not depend on ##L##, so I guess these are not the expressions we want...
  39. A

    A Piezoelectricity and the Lorentz Harmonic Oscillator?

    Hi! As I outlined in my https://www.physicsforums.com/threads/hello-reality-anyone-familiar-with-the-davisson-germer-experiment.985063/post-6305937, I'm curious to ask if there is anyone with knowledge on the theory of the piezoelectric effect on this forum? I think it's fascinating how a...
  40. M

    A critically damped simple harmonic oscillator - Find Friction

    c = Critically Damped factor c = 2√(km) c = 2 × √(150 × .58) = 18.65 Friction force = -cv Velocity v = disp/time = .05/3.5 Friction force = - 18.65 * .05/3.5 = -.27 N I am not sure if above is correct. Please check and let me know how to do it.
  41. G

    Harmonic Oscillator Ladder Operators - What is (ahat_+)^+?

    I know that ahat_+ = 1/sqrt((2*m*h_bar*w)) * (mw(xhat)+i(phat)) and ahat_- = 1/sqrt((2*m*h_bar*w)) * (mw(xhat)-i(phat)). But I'm not sure what (ahat_+)^+ could be.
  42. D

    A Understanding Harmonic oscillator conventions

    I don't quite understand how he got the line below. By using discrete time approximation, we can get the second order time expression. But i don't see how by combining terms he is able to get such expression.
  43. Vivek98phyboy

    I Why is this SHM the way it is?

    I know four different forms in which an SHM can be represented after solving the differential and taking the superposition acos(wt+Ø) asin(wt+Ø) acos(wt-Ø) asin(wt-Ø) where a- amplitude In the above image they took B as negative in order to arrive at acos(wt+e). If i already knew i wanted...
  44. T

    Simple Harmonic Oscillator Squeezing

    I'm working through https://ocw.mit.edu/courses/physics/8-05-quantum-physics-ii-fall-2013/lecture-notes/MIT8_05F13_Chap_06.pdf, and I'm stumped how they got from Equation 5.26 (##\vert 0_{\gamma} \rangle \equiv \frac{1}{\sqrt{cosh\gamma}} exp(-\frac{1}{2}tanh\gamma \hat{a^\dagger}\hat{a^\dagger}...
  45. I

    Time Derivatives of Expectation Value of X^2 in a Harmonic Oscillator

    I can show that ##\frac{d}{dt} \langle \psi (t) \vert X^2 \vert \psi (t) \rangle = \frac{1}{m} \langle \psi (t) \vert PX+XP \vert \psi (t) \rangle##. Taking another derivative with respect to time of this, I get ##\frac{d^2}{dt^2} \langle \psi (t) \vert X^2 \vert \psi (t) \rangle = \frac{i}{m...
  46. C

    A Please help me understand this HO energy in He4 gas

  47. Diracobama2181

    A Volume Element for Isotropic Harmonic oscillator

    I am currently having trouble deriving the volume element for the first octant of an isotropic 3D harmonic oscillator. I know the answer I should get is $$dV=\frac{1}{2}k^{2}dk$$. What I currently have is $$dxdydz=dV$$ and $$k=x+y+z. But from that point on, I'm stuck. Any hints or reference...
  48. Diracobama2181

    A Time Dependent Perturbation of Harmonic Oscillator

    An electric field E(t) (such that E(t) → 0 fast enough as t → −∞) is incident on a charged (q) harmonic oscillator (ω) in the x direction, which gives rise to an added ”potential energy” V (x, t) = −qxE(t). This whole problem is one-dimensional. (a) Using first-order time dependent perturbation...
  49. H

    A Overlap of nth QHO excited state and momentum-shifted QHO ground state

    ##\newcommand{\ket}[1]{|#1\rangle}## ##\newcommand{\bra}[1]{\langle#1|}## I have a momentum-shifting operator ##e^{i\Delta p x/\hbar}## acting on the ground state ##\ket{0}## of the QHO, and I want to compute the overlap of this state with the n##^{th}## excited QHO state ##\ket{n}##. Given...
  50. M

    A rather weird form of a coherent state

    As far as I know we can express the position and momentum operators in terms of ladder operators in the following way $${\begin{aligned}{ {x}}&={\sqrt {{\frac {\hbar }{2}}{\frac {1}{m\omega }}}}(a^{\dagger }+a)\\{{p}}&=i{\sqrt {{\frac {\hbar }{2}}m\omega }}(a^{\dagger }-a)~.\end{aligned}}.$$...