The term "harmonic oscillator" is used to describe any system with a "linear" restoring force that tends to return the system to a equilibrium state. There is both a classical harmonic oscillator and a quantum harmonic oscillator. Both are used to as toy problems that describe many physical systems.

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Driven-damped oscillator: deduce the phase and/or resonant freq from amplitudes at varying freqs

Suppose that we have a fairly standard driven-damped harmonic oscillator (i.e. linear spring restoring force, linear damping force, sinusoidal driving force, etc). The catch is: we don't know the ...
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solution of pendulum equation [closed]

I have the pendulum expression $$\ddot{\theta}+\omega_{o}^{2}\sin(\theta)=0,$$ where I used a Taylor expansion for the sine term: ...
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106 views

Constant of motion

An exercise from Goldstein (9.31-3rd Ed) asks to show that for a one-dimensional harmonic oscillator $u(q,p,t)$ is a constant of motion where $$ u(q,p,t)=\ln(p+im\omega q)-i\omega t $$ and ...
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Measure energy state of quantum harmonic oscillator

When discussing the quantum mechanical harmonic oscillator we are talking about energy eigenstates. How would one actually measure in which state an harmonic oscillator is in? Could you weigh it and ...
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The Hamiltonian for clocks?

I am rather a theoretician and looking for a formalism to represent biological clocks by Hermitian operators. The simplest thought model I am looking for is a formal representation of a clock (for ...
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Energy Levels of 3D Isotropic Harmonic Oscillator (Nuclear Shell Model)

One simple way of detailing the very basic structure of the nuclear shell model involves placing the nucleons in a 3D isotropic oscillator. It's easy to show that the energy eigenvalues are $E = ...
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What is the physical meaning of $a_{\vec{p}} \! \mid \! 0 \rangle$

$a^\dagger_{\vec{p}} \! \mid \! 0 \rangle = \mid \! p \rangle$ is interpreted as a creation of a particle with momentum $p$ from the vacuum. $a_{\vec{p}} \! \mid \! p \rangle = \mid \! 0 \rangle$ is ...
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Why are overtones forbidden within the harmonic approximation?

In vibrational spectroscopy only transitions between neighboring vibrational states ($\Delta \nu = \pm 1$, $\nu$ being the vibrational quantum number) are allowed within the harmonic approximation. ...
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How To Use Ladder Operators?

I'm studying for a test in quantum mechanics and I'm having a hard time understanding how to use ladder operators. There are no examples in my text book, only definitions that I can't understand how ...
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Hermite polynomials for expected value of harmonic oscillator

This was a problem on my final exam that has been really bugging me. Consider the quantum Harmonic oscillator prepared in an energy eigenstate, $\psi_n$(x). Calculate the expectation value of the ...
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Undamped oscillations. Why is the solution a linear combination of $\sin()$ and $\cos()$?

$ma = mg - cx$, where $x(0) = x_0 = 0$ is the position in which there is no tension in the rope. $dx/dt = v_0$ for $t = 0$; $v_0$ is a known constant. The discriminant of the characteristic ...
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Complex energy eigenstates of the harmonic oscillator

Given the Hamiltonian for the the harmonic oscillator (HO) as $$ \hat H=\frac{\hat P^2}{2m}+\frac{m}{2}\omega^2\hat x^2\,, $$ the Schroedinger equation can be reduced to: $$ \left[ ...
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Why does the acceleration $g$ due to gravity not affect the period of a vertically mounted spring?

For a vertically mounted spring, I was looking at the formula $ T= 2\pi \sqrt{m/k}$ for a period. Why doesn't the gravitational acceleration $g$ factor in?
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Unstable equilibrium in a pendulum

Consider a pendulum with a bob and a massless, rigid, hinged rod attached to the bob. The bob is at rest at the bottom most position. Neglecting friction, is it possible to impart such a velocity ...
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294 views

Harmonic Oscillator (Quantum Mechanics)

Griffiths uses an algebraic "brute force" technique to solve the harmonic oscillator. I'm somewhat confused regarding a few parts. $$\frac{1}{2m}[p^2 + (m \omega x)^2] \psi = E \psi$$ $H = ...
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Is it only the ground state of the quantum harmonic oscillator that has the minimum uncertainty product?

We know that the uncertainty product of general states is bounded by the inequality described by Heisenberg's uncertainty relation. And the ground state of the quantum harmonic oscillator falls under ...
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Infinite period in Simple Harmonic Motion

I'm studying the Simple Harmonic Motion, and I am hesitant about, how to get mass values for infinite period? When mass is 0. When mass is infinite. With $\tau=2\pi/\sqrt{k/m}$.
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Organs & Oscillations: An Analysis on the Temperature Dynamics of Solids

Does temperature have an influence on the frequency of an oscillating organ pipe?
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Physical reason behind having greater amplitude when driving frequency$ < $ natural frequency than that when driving frequency $>$ natural frequency

This is quoted from A.P.French's Vibrations & Waves: If the driving force is of low frequency relative to the natural frequency, we would expect the particle to move essentially with the ...
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Meaning of “vacuum state”?

I just learned about $|0\rangle$ and siblings $|0_\gamma\rangle$ and $|0_\infty\rangle$ while studying coherent and squeezed states in a QM class, and I have a question about the meaning of ...
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100 views

How is a string in string theory different from a harmonic oscillator or a point?

I am reading String Theory and M-Theory: A Modern Introduction by Becker, Becker and Schwartz. I've tried to read this book before but not succeeded because I didn't know enough math or physics. This ...
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What are the forces acting on a Pendulum Tuned Mass Damper?

Upon researching tuned mass dampers, I came across this free body diagram of a pendulum tuned mass damper. However, I don't understand where many of the forces come from. What exactly are the forces ...
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Quantum harmonic oscillator solved by analytic method using Schrödinger equation and wave function

I'm having trouble understanding the recursion formula. Using $\xi \equiv \sqrt{m\omega/\hbar}x$ and $K = 2E/\hbar\omega$, the time-independent Schrödinger equation becomes $$\frac{d^2\psi}{d\xi ^2} ...
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103 views

Does the average momentum vanish for an eigenstate of the simple harmonic oscillator?

Suppose we have a simple harmonic oscillator, let's consider the ground state, $|0\rangle$ and the first excited state $|1\rangle$. $\langle 0|\hat p|0 \rangle$ is zero right? Since the particle can ...
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Harmonic oscillator

Let $|0\rangle,...$ be the states of the harmonic oscillator. Then a squeezed state was defined as $|\xi\rangle =S(\xi)|0\rangle $, where $S(\xi):=e^{\frac{1}{2}( \xi (a^{ \dagger ^2}-a^2))}$, where ...
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Why are particles in harmonic motion in normal modes?

Why do we assume that in normal modes, particles oscillate in form cos (wt) ? How do we know that the general motion of particles can be expressed as a superposition of normal modes? In both French ...
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Probability for harmonic oscillator outside the classical region

I'm having some trouble finding an expression for the probability to find the particle outside the classical area in the harmonic oscillator. I have a wavefunction defined as: $\psi \left( x,\,t ...
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Expected number of quanta in harmonic oscillator states

I'm working my way through A Squeezed State Primer, filling in details along the way. Let $a$ and $a^\dagger$ be the usual annihilation and creation operators with $[a,a^\dagger]=1$ and ...
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Degeneracy of states in mixed infinite square well, harmonic oscillator

I'm trying to determine the degeneracy of states given by $g(\epsilon)=g_{0} \epsilon$ for a system that is trapped in a quite specific potential. In two dimensions, the particle has a potential as ...
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What's the closed-form of the sum relating to the DOS of simple harmonic motion?

In order to calculate the density of states of single particle in the simple harmonic potential, we would calculate that $$ D(\epsilon)=\sum_{n}\delta(\epsilon-\epsilon_n) $$ where ...
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Can we represent Simple Harmonic function as triangular waves?

Having studied the topic recently I found out that simple harmonic motion can represented well with sine and cosine functions.Take for example a pendulum swing which could look like : and the ...
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Concrete example of a two-dimensional harmonic oscillator

I am a student of mathematics and some time ago I showed in general that for a two-dimensional harmonic oscillator one can apply the recurrence theorem. So far so good.. now I would like to have a ...
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282 views

Harmonic oscillator - wavefunctions

I understand now how I can derive the lowest energy state $W_0 = \tfrac{1}{2}\hbar \omega$ of the quantum harmonic oscillator (HO) using the ladder operators. What is the easiest way to now derive ...
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Accessible microstates of harmonic oscillator in microcanonical enemble

While reading up on statistical physics, I am going through the calculation of the partition function of the harmonic oscillator in the microcanonical ensemble. The result for the partition function ...
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Why uncertainity is minimum for coherent states?

While reading for quantum damped harmonic oscillator, I came across coherent states, and I asked my prof about them and he said me it is the state at which $\Delta x\Delta y$ is minimum. I didn't ...
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196 views

Coordinate representation of quantum ladder operator?

I can't seem to figure out how to derive the coordinate representation of the $a_+$ ladder operator in quantum mechanics. I know that $a_-$ is $\sqrt{\frac{1}{2mwh}} (mwx + i\dot{p}) $ in which where ...
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Sitting on the bob of a pendulum

Walter Lewin's best performance was the pendulum demonstration, and I copy the transcript now: Would the period come out to be the same or not? [students respond] Some of you think it's ...
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Non-Degeneracy of Eigenvalues of Number Operator for Simple Harmonic Oscillator [duplicate]

Possible Duplicate: Proof that the One-Dimensional Simple Harmonic Oscillator is Non-Degenerate? I'm trying to convince myself that the eigenvalues $n$ of the number operator ...
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Force to use in harmonic oscillation through the inside of a planet

I am to find an equation for the time it takes when one falls through a planet to the other side and returns to the starting point. I have seven different sets of values - mass of object falling, mass ...
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When are Maximum Velocity and Acceleration acheived in Simple Harmonic Motion?

Im trying to get my head around SMH out of curiosity because it seems simple yet I'm not getting the concept behind some ideas. For a SMH equation : $$ x=a \sin(\omega t+\phi) $$ Under what ...
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817 views

Using Fourier Transforms to Solve Systems with springs of high frequency

I'm trying to numerically solve the differential equations of motion in a system with multiple springs of very high frequency. Because the solution is often a combination of rapidly-oscillating sine ...
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Is there a curve for which a particle restricted to move within it under the gravitational force will always exhibit a pure harmonic motion?

A simple pendulum, for example, is not isochronous for large amplitudes (that is, the frequency will depend on the amplitude). So a particle confined in a circumference will not always exhibit a ...
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The universality of the Stuart-Landau equation to describe nonlinear oscillators

I have read numerous papers which boldly suggest that the Stuart-Landau equation can be successfully used to model any weakly nonlinear oscillating system near a Hopf bifurcation. Even thought it has ...
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Why does the Bohr-Sommerfeld quantization for give the exact energy-levels for a harmonic oscillator?

Why does the Bohr-Sommerfeld rule for quantization give the exact energy-levels for a simple harmonic oscillator?
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Energy in harmonic oscillator [closed]

The expectation value of the potential energy is exactly half the total according to Griffiths. Is that case always true for quantum harmonic oscillator? Is that the case also for classical harmonic ...
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229 views

Simplest explanation of pendulum having a constant time period at low angles

What is the simplest explanation for the pendulum having a constant time period at low angles?
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Quantum Harmonic Oscillator - Normalizability of Annihilated Ground State

The common line of deductions in the operator analysis of the quantum harmonic oscillator goes something like this: It is derived that the action of the annihilation operator $a$ on an eigenfunction ...
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Eigenstates of a density matrix of continuous variables

Consider a system of two entangled harmonic oscillators. The normalised ground state is denoted by $\psi_0(x_1,x_2)$. The reduced density matrix of the second oscillator is given by: $$\rho_2 = ...
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Different hamiltonians for quantum harmonic oscillator?

The Hamiltonian for a classical simple harmonic oscillator is $$ H = \frac{p^2}{2m} + \frac{1}{2}m\omega^2x^2$$ With the usual choice of the ladder operators $$a = ...
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Harmonic Oscillator driven by a Dirac delta-like force

Consider that there is no damping for simplicity. As we know, a driving force of the form $\sin(\omega t)$ will make the oscillator at steady state vibrates at the external frequency $\omega$. What ...