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:
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.
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"...
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 |...
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...
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...
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?
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...
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...
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...
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.
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...
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 +...
2d
Expectation
Expectation value
Field
HarmonicHarmonicoscillator
Landau
Magnetic
Magnetic field
Oscillator
Position
Quantum mechanics
Value
Wavefunction
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:
$$H = - \frac \hbar {2m} \frac {d^2}{dx^2}\psi + \frac...
##x## can be discretized as ##x \rightarrow x_k ## such that ##x_{k + 1} = x_k + dx## with a positive integer ##k##. Throughout we may assume that ##dx## is finite, albeit tiny.
By applying the Taylor expansion of the wavefunction ##\psi_n(x_{k+1})## and ##\psi_n(x_{k-1})##, we can quickly...
1. Since N is large, ignore the kinetic energy term.
##[-\mu + V(r) + U|\Psi (r)|^2]\Psi (r) = 0##
2. Solve for the density ##|\Psi (r)|^2##
##|\Psi (r)|^2 = \frac{\mu - V(r)}{U}##
3. Integrate density times volume to get number of bosons
##\int|\Psi (r)|^2 d\tau = \int \frac{\mu -...
Hello! Is stimulated emission possible for a harmonic oscillator (HO) i.e. you send a quanta of light at the right energy, and you end up with 2 quantas and the HO one energy level lower (as you would have in a 2 level system, like an atom)?
Consider the gaussian kick potential,
##\hat{V}(t) = \hat{x} \exp{(\frac{-t^2}{2 \tau^2})}##
where
##\hat{x} = a+a^\dagger## in terms of creation and annihilation operators.
Then we define the potential in the interaction picture,
##\hat{V}_I(t) = e^{i\hat{H}t}\hat{V}(t)e^{-i\hat{H}t}##
I...
Good afternoon all,
On page 51 of David Griffith's 'Introduction to Quantum Mechanics', 2nd ed., there's a discussion involving the alternate method to getting at the energy levels of the harmonic oscillator. I'm filling in all the steps between the equations on my own, and I have a question...
Consider a one-dimensional harmonic oscillator. ##\psi_0(x)## and ##\psi_1(x)## are the normalized ground state and the first excited states.
\begin{equation}
\psi_0(x)=\Big(\frac{m\omega}{\pi\hbar}\Big)^{\frac{1}{4}}e^{\frac{-m\omega}{2\hbar}x^2}
\end{equation}
\begin{equation}...
We show by working backwards
$$\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)$$...
First time posting in this part of the website, I apologize in advance if my formatting is off.
This isn't quite a homework question so much as me trying to reason through the work in a way that quickly makes sense in my head. I am posting in hopes that someone can tell me if my reasoning is...
Hi everyone!
Both sources I'm currently reading (page 291 of Mathematical Methods of Classical Mechanics by Arnol'd - get it here - and page 202 of Classical Mechanics by Shapiro - here) say that, in the case of the planar harmonic oscillator, using polar or cartesian coordinate systems leads...
hi guys
i am trying to solve the Asymptotic differential equation of the Quantum Harmonic oscillator using power series method and i am kinda stuck :
$$y'' = (x^{2}-ε)y$$
the asymptotic equation becomes :
$$y'' ≈ x^{2}y$$
using the power series method ##y(x) = \sum_{0}^{∞} a_{n}x^{n}## , this...
I'm trying verify the proof of the sum rule for the one-dimensional harmonic oscillator:
$$\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...
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...
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...
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_{-}...
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 $$U=\frac{ x^2}{2}+\frac{y^2}{2}$$...
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...
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...
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...
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...
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...
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.
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.
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.
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...
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}...
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...
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 $$k=x+y+z. But from that point on, I'm stuck. Any hints or reference...
An electric field E(t) (such that E(t) → 0 fast enough as t → −∞)
is incident on a charged (q) harmonic oscillator (ω) in the x direction,
which gives rise to an added ”potential energy” V (x, t) = −qxE(t).
This whole problem is one-dimensional.
(a) Using first-order time dependent perturbation...
##\newcommand{\ket}[1]{|#1\rangle}##
##\newcommand{\bra}[1]{\langle#1|}##
I have a momentum-shifting operator ##e^{i\Delta p x/\hbar}## acting on the ground state ##\ket{0}## of the QHO, and I want to compute the overlap of this state with the n##^{th}## excited QHO state ##\ket{n}##. Given...
As far as I know we can express the position and momentum operators in terms of ladder operators in the following way
$${\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}}.$$...