# What is Oscilator: Definition and 32 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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6. ### I Electron wave funtion harmonic oscillator

As we see in this Phet simulator, this is only the real part of the wave function, the frequency decreases with the potential, so lose energy as moves away the center. we se this real-imaginary animation in Wikipedia, wave C,D,E,F. Because with less energy, the frequency of quantum wave...
7. ### Second order ODE into a system of first order ODEs

Homework Statement The harmonic oscillator's equation of motion is: x'' + 2βx' + ω02x = f with the forcing of the form f(t) = f0sin(ωt)The Attempt at a Solution So I got: X1 = x X1' = x' = X2 X2 = x' X2' = x'' ∴ X2' = -2βX2 - ω02X1 + sin(ωt) The function f(t) is making me doubt this answer...
8. ### How Does the Virial Theorem Help Determine Position Uncertainty in a C-H Bond?

Homework Statement Using paper, pencil and the Virial theorem, calculate the position uncertainty (an estimate of the vibration amplitude) of the H atom in its ground state C-H stretching mode. In more precise language, calculate the bond length uncertainty in a C-H bond due to the C-H...
9. ### Finding mechanical energy of simple harmonic oscilator

Homework Statement A simple harmonic oscillator consists of a block of mass 45 g attached to a spring of spring constant 240 N/m, oscillating on a frictionless surface. If the block is displaced 3.5 cm from its equilibrium position and released with an initial velocity of 2.5 m/s, what is its...
10. ### Energy of a Forced, Damped Oscilator

Homework Statement A forced damped oscilator of mass ##m## has a displacement varying with time of ##x=Asin(\omega t) ## The restive force is ## -bv##. For a driving frequency ##\omega## that is less than the natural frequency ## \omega_{0}##, sketch graphs of potential energy, kinetic energy...
11. ### Time dependent perturbation theory of the harmonic oscilator

Homework Statement A 1-d harmonic oscillator of charge ##q## is acted upon by a uniform electric field which may be considered to be a perturbation and which has time dependence of the form ##E(t) = \frac{K }{\sqrt{\pi} \tau} \exp (−(t/\tau)^2) ##. Assuming that when ##t = -\infty##, the...
12. ### Ground state of hamonic oscilator

When i take a coherent state ##|\alpha>## if ##\alpha -> 0## then the limit is the Fock state for n = 0. so ##|n = 0> = |\alpha = 0>## The problem is that they seem to have different http://www.iqst.ca/quantech/wiggalery.php: Where is the error? Thanks. Edit sorry, in the link the W function is...
13. ### Harmonic Oscillator Homework: Graph, Spring Constant & Weight

Homework Statement A test was made with a basket and 20 gram weights. they were put in the basket which hang on a spring, the basket was raised and released. the period was measured for a few number of weights in the basket. the results are as follows. the first of every pair is the number of...
14. ### Pendulum with Spring: Finding Position and Period

Homework Statement A system consists of a spring with force constant k = 1250 N/m, length L = 1.50m, and an object of mass m = 5.00kg attached to the end. The object is placed at the level of the point of attachment with the spring unstretched, at position yi= L, an then is released so that it...
15. ### Harmonic oscilator Tamvakis

Homework Statement ## H=\frac{p^2}{2m}+\frac{1}{2}m\omega^2x^2## Show that ##[H,[H,x^2]]=(2\hbar\omega)^2x^2-\frac{4\hbar^2}{m}H## Homework Equations ##[x,p]=i\hbar## The Attempt at a Solution I get ##[H,x^2]=-\frac{i\hbar}{m}(px+xp)## what is easiest way to solve this problem?
16. ### Hamiltonian of linear harmonic oscilator

Could hamiltonian of linear harmonic oscilator be written in the form? ##\hat{H}=\sum^{\infty}_{n=0}(n+\frac{1}{2})\hbar\omega |n\rangle \langle n| ##
17. ### Basic harmonic oscilator problem (but I'm having troble solving it)

Homework Statement I have small block of mass m=1kg on top of a bigger block mass M=10kg The friction coefficient between the blocks μ=0.40 No fricton between the big block and the ground. There is a spring with k=200N/m attached to the bigger block. The problem asks what is the maximum...
18. ### Help a Physics Student with Two Dim. Oscilator Problem

Hello! I am new here... I am a physics student, seeking for help. There is something I just can't seem to understand... Your help will be much appreciated... In my exercise, I have the following Hamiltonian: H=\frac{p_x^2}{2m}+\frac{p_y^2}{2m}+\frac{1}{2}m \omega^2 (x^2+y^2)+\lambda xy I was...
19. ### Why Does the Quantum Harmonic Oscillator's Equation Yield a Gaussian Curve?

Dimensionless equation for quantum harmonic oscilator in the lowest energy state is: d2u/dx2=(x2-1)u u means wave function and solution is: u = exp(-x2/2) As we can see, solution is the Gauss curve. But, what is special in the above equation that it give the Gauss curve? Maybe...
20. ### Harmonic Oscilator (driven-and-damped) problem

Homework Statement An Oscillator with free oscillation period \tau is critically damped and subjected to a periodic force with the 'saw-tooth' form: F(t)=c(t-n\tau) , (n-1/2)\tau<t<(n+1/2)\tau for integer n with c constant. Find the ratios of the amplitudes of oscillations at the...
21. ### Driven Harmonic Oscilator

1. Driven Harmonic Oscillator with an arbitrary driving force: f(t)=x"+2bx'+w^2 x Let x(t) be expressed by x(t)= g(t)*exp(a1*t), where a1 is a solution to the characteristic equation a^2 + 2ba+w=0 for the above second order differential equation. Find the ordinary differential equation that is...
22. ### Solving Damped Oscillator: Time to Reduce to 0.50 Energy

hi, i am supposed to solve this excerise and i don't even know where to start. A mass M is suspended from a spring and oscillates with a period of 0.880 s. Each complete oscillation results in an amplitude reduction of a factor of 0.96 due to a small velocity dependent frictional effect...
23. ### What Type of Oscillator Is Represented in This Schematic?

Homework Statement hello; i have this past paper question relevant to a oscillator they have given the following schematic and question is "identify the type of oscillator giving reasons" Homework Equations The Attempt at a Solution i tried lecture notes,some books but couldn't...
24. ### Harmonic Oscillator Analytical Mechanics: Creation/Annihilation Ops

Homework Statement The Hamiltonian for the one-dimensional harmonic oscillator is given by: H = \frac{p^2}{2m}+ \frac{mw^2q^2}{2} Homework Equations (a) Express H in terms of the following coordinates: a = \sqrt{\frac{mw}{2}} (q+i\frac{p}{mw}) a^* = \sqrt{\frac{mw}{2}} (q-i\frac{p}{mw})...
25. ### Is the Velocity Zero in Equilibrium Points of a Simple Harmonic Oscillator?

Hallo, Does the velocity i simple harmonic oscillator is zero in equilibrium points? if it's true how does it make sense with the fact that i suppose to get a maximum kinetic Energy in those points (stable ones) i would really appreciate if someone could clear this issue for me...
26. ### Uncertainty in simple harmonic oscilator

Hallo, Why can we assume (in the case of simple harmonic occilator) that at the maximum momentum ,pmax, we can evluate pmax=\Deltap? Thanks, Omri
27. ### [Q]Question about harmonic oscilator

Hi, Finally! I reached harmonic oscilator! Congratulation! Most of all QM textbook introduced this formula : Time independent energy eigenstate equation is ( - \frac{\hbar^2}{2m} \frac{\partial}{\partial x) + \frac{Kx^2}{2} )\varphi = E\varphi (1) \varphi_{xx} = -k^2 \varphi...
28. ### Question about quantum harmonic oscilator

Hi, I am preparing for a quantum mechanics exam, and I have this problem that I can`t solve: I have to find the complete energy eigenvalue spectrum of a hamiltonean of the form: H = H0 + c and also another of the form H = H0 + \lambdax^{2} Where in both cases, H0 is the...
29. ### DC to AC oscilator and transformer

Hello, can someone please help me out here, i want to step up the voltage of a 1.5V battery to 300V, i know that to do so you need an oscilatro circuit consisting of a transformer a small capacitor and a transistor. so my question is, does the type of transistor/capacitor i use matter?
30. ### Kinetic energy of harmonic oscilator

[SOLVED] Kinetic energy of harmonic oscilator Homework Statement Find the expectation value of the kinetic energy of the nth state of a Harmonic oscillator Homework Equations <T> = \frac{<p^2>}{2m} p_{x} = \frac{1}{i} \sqrt{\frac{m\hbar\omega}{2}} (\hat{a} -\hat{a}^\dagger)...
31. ### Linear Harmonic Oscilator - QM

Just a quickie: A particle is in the first excited Eigenstate of energy E corresponding to the one dimensional potential V(x) = \frac{Kx^2}{2}. Draw the wavefunction of this state, marking where the particles KE is negative. Now my question. The first excited state will be n=1 correct...
32. C

### Learning the harmonic oscilator

I'm currently taking graduate classes toward my phD in physics... when I was undergraduate I learn the harmonic oscillator just solving the schrodinger equation with such potential can be derive that: E=(n+1/2)hw, the wave functions (with hermite polynomial *e^-x2). that take to pages of...