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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575 views
Why don't tuning forks have three prongs?
I was reading Why tuning forks have two prongs?. The top answer said the reason was to reduce oscillation through the hand holding the other prong.
So if having 2 prongs will reduce oscillation loss, ...
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votes
3answers
143 views
Hilbert space of harmonic oscillator: Countable vs uncountable?
Hm, this just occurred to me while answering another question:
If I write the Hamiltonian for a harmonic oscillator as
$$H = \frac{p^2}{2m} + \frac{1}{2} m \omega^2 x^2$$
then wouldn't one set of ...
10
votes
1answer
493 views
Exact energies of spherical harmonic oscillator in Dirac equation
The potential is given by:
$$
V(r) = {1\over 2} \omega^2 r^2
$$
and we are solving the radial Dirac equation (in atomic units):
$$
c{d P(r)\over d r} + c {\kappa\over r} P(r) + Q(r) (V(r)-2mc^2) = E ...
8
votes
2answers
950 views
Proof that the One-Dimensional Simple Harmonic Oscillator is Non-Degenerate?
The standard treatment of the one-dimensional quantum simple harmonic oscillator (SHO) using the raising and lowering operators arrives at the countable basis of eigenstates $\{\vert n \rangle\}_{n = ...
7
votes
3answers
293 views
Is the quantization of the harmonic oscillator unique?
To put it a little better:
Is there more than one quantum system, which ends up in the classical harmonic oscillator in the classial limit?
I'm specifically, but not only, interested in an ...
5
votes
2answers
405 views
What's a good reference for this classical picture Feynman's talking about?
I have a mathematics background but am trying to educate myself a little about physics. At the beginning of Feynman's QED book (not the popular one) is the following:
Suppose all of the atoms in ...
5
votes
2answers
393 views
Expectation value of time-dependent Hamiltonian
I'm trying to solve a problem in QM with a forced quantum oscillator. In this problem I have a quantum oscillator, which is in the ground state initially. At $t=0$, the force $F(t)=F_0 \sin(\Omega t)$ ...
5
votes
1answer
176 views
Finding coefficient of proportionality
Recently in my AP Physics class I did a lab in which I measured k for a spring by setting up an oscillating system with it, and timing the period, repeating for different masses. Since ...
5
votes
1answer
203 views
Basis transformation between eigenstates of harmonic oscillators with different frequency
Given two harmonic oscillators with frequencies $\Omega$ and $\Omega'$, the eigenstates themselves are exactly known. Let's call them $\Psi_n$ and $\Psi'_n$.
Is there a compact expression for the ...
5
votes
2answers
188 views
Why are the solution coefficients for a harmonic oscillator proportional to minors of the determinant?
I'm studying the oscillations of systems with more than one degree of freedom from Landau & Lifshitz's Mechanics Third Edition (for those who have the book, my question corresponds roughly to ...
5
votes
2answers
481 views
Tricky spring on a surface question
I have this relative simple-looking question that I haven't been able to solve for hours now, it's one of those questions that just drive you nuts if you don't know how to do it.
This is the scenario:
...
4
votes
9answers
3k views
Explanation: Simple Harmonic Motion
I am Math Grad student with a little bit of interest in physics. Recently i looked into the wikipedia page for Simple Harmonic Motion.
Guess, I am too bad at physics to understand it. Considering me ...
4
votes
3answers
379 views
Can someone please derive $T=2\pi\sqrt{l/g}$ or prove it without using calculus?
I don't know much calculus, but I want to know that how one derives the formula to find time period $T$ of a simple pendulum.
4
votes
2answers
88 views
Proof for commutator relation $[\hat{H},\hat{a}] = - \hbar \omega \hat{a}$
I know how to derive below equations found on wikipedia and have done it myselt too:
\begin{align}
\hat{H} &= \hbar \omega \left(\hat{a}^\dagger\hat{a} + \frac{1}{2}\right)\\
\hat{H} &= ...
4
votes
3answers
2k views
How to determine phase angle for a sinusoidal motion?
If I have an over-damped mechanical system that is excited with a sinusoidal motion. That sinusoidal motion starts with a determined frequency then increases frequency over time.
Of course, it is ...
4
votes
1answer
614 views
Evolution operator for time-dependent Hamiltonian
When i studyed QM I'm only working with non time-dependent Hamiltonians. In this case unitary evolution operator has the form $$\hat{U}=e^{-\frac{i}{\hbar}Ht}$$ that follows from this equation
$$
...
4
votes
1answer
354 views
Plotting a SHO in matlab
I have no prior experience of using matlab. My teacher want me to solve this question. I have been trying for a couple of hours now with no luck, please help!
The mass of 100 g hanging in a spring ...
4
votes
2answers
82 views
Why is the damping force on a spring oscillator linearly dependent on velocity?
If you consider the damping force is friction like in:
then the force should be $$F=\mu N$$ where $\mu$ is the coefficient of kinetic friction. Why then is the damping force assumed to be linearly ...
3
votes
2answers
163 views
Definition of “Quantizing”
Could anyone explain to me what "quantize" means in the following context?
Quantize the 1-D harmonic oscillator for which
$$H~=~{p^2\over 2m}+{1\over 2} m\omega^2 x^2.$$
I understand that the ...
3
votes
1answer
104 views
The issue on existence of inverse operations of $a$ and $a^{\dagger}$
I have asked a question at math.stackexchange that have a physical meaning.
My assumption: Suppose $a$ and $a^\dagger$ is Hermitian adjoint operators and $[a,a^\dagger]=1$. I want to prove that ...
3
votes
2answers
38 views
Probability of position in linear shm?
The problem that got me thinking goes like this:-
Find $dp/dx$ where $p$ is the probability of finding a body at a random instant of time undergoing linear shm according to $x=a\sin(\omega t)$. ...
3
votes
1answer
135 views
Schrödinger equation for a harmonic oscillator
I have came across this equation for quantum harmonic oscillator
$$
W \psi = - \frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + \frac{1}{2} m \omega^2 x^2 \psi
$$
which is often remodelled by defining a new ...
3
votes
1answer
144 views
How do eigenstates of harmonic oscillators with different frequencies compare?
Suppose I have a harmonic oscillator with frequency $\Omega_1$ and another one with frequency $\Omega_2$. Is there a simple relationship between the eigenstates of the two? Especially, how would the ...
3
votes
1answer
187 views
Question on Sakurai's treatment of the Harmonic Oscillator:
In Section 2.3 of the second edition of Modern Quantum Mechanics (which discusses the harmonic oscillator), Sakurai derives the relation $$Na\left|n\right> = (n-1)a\left|n\right>,$$ and states ...
3
votes
1answer
4k views
How do I solve for the phase constant given the amplitude and the angular frequency?
A piston (with mass M) in a car engine is in vertical simple harmonic
motion with amplitude A. The engine is running at a period T. Suppose
a small piece of metal with mass m were to break ...
3
votes
1answer
77 views
From the local Hooke's law to the global one
My system consist of a cylinder with axis Z that can contract and dilate along this axis. It obeys microscopically Hooke's law of elasticity:
...
3
votes
2answers
173 views
Constant magnetic field applied to a quantum harmonic oscillator
I have a spinless particle of mass $m$ and charge $q$ which is an isotropic harmonic oscillator of frequency $\omega_0$, then I apply a constant magnetic field in the $z$ direction. We can show the ...
3
votes
2answers
307 views
Dynamics of a Vertical Mass-Spring Simple Harmonic Oscillator with Gravity
I am having some trouble obtaining the elastic potential energy and gravitational potential energy of a simple mass spring system.
In this experiment, masses attached to a spring were dropped from a ...
3
votes
1answer
279 views
Writing equation for amplitude of driven harmonic oscillator in Lorentzian form
This harmonic oscillator is driven and damped, with the form:
$$\ddot{x} + \lambda \dot{x} + \omega_0^2 x = A \cos(\omega_d t)$$
Now, I have used the ansatz (guess): $x(t) = B \cos(\omega_d t + ...
3
votes
2answers
358 views
What are some interesting coupled harmonic oscillators problems?
That I could create as a classical mechanics class project? Other than the classical examples that we see in textbooks.
3
votes
2answers
792 views
How to determine viscous dampening coefficient of spring?
I'm trying to determine the viscous dampening coefficient of a spring (c). Read about it on Wikipedia here.
The two equations which I have are:
f=-cv and ma+cv = -kx
I know the spring constant ...
3
votes
0answers
46 views
Effective mass in Spring-with-mass/mass system
Suppose you have a particle of mass $m$ fixed to a spring of mass $m_0$ that, in turn, is fixed to some wall. I'm trying to calculate the effective mass $m'$ that appears in the law of motion of the ...
3
votes
0answers
1k views
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 $E = ...
2
votes
3answers
975 views
Why is the angle of a pendulum as a function of time a sine wave?
OK so I'm trying to understand why the angle of a pendulum as a function of time is a sine wave.
I can't really find an explanation online and when I do find something partial there are certain ...
2
votes
2answers
191 views
Why are overtones forbidden within the harmonic approximation?
In vibrational spectroscopy only transitions between neighboring vibrational states ($\Delta \nu = \pm 1$, $\nu$ being the vibrational quantum number) are allowed within the harmonic approximation. ...
2
votes
4answers
116 views
Why is linear independence of harmonic oscillator solutions important?
The equation of motion for the harmonic oscillator (mass on spring model)
$$\frac{d^2x}{dt^2} + \omega_0^2 x = 0$$
with $\omega_0^2 = D/m$, $D$ and $m$ being the force constant of the spring and the ...
2
votes
1answer
1k views
3D Quantum harmonic oscillator
For an isotropic 3D QHO in a potential $V(x,y,z)={1\over 2}m\omega^2(x^2+y^2+z^2)$. I can see by independence of the potential in the $x,y,z$ coordinates that the solution to the Schrodinger equation ...
2
votes
1answer
243 views
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 ...
2
votes
2answers
745 views
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 ...
2
votes
1answer
83 views
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 ...
2
votes
1answer
123 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 ...
2
votes
2answers
357 views
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 ...
2
votes
2answers
111 views
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 ...
2
votes
2answers
3k views
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 ...
2
votes
1answer
487 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 ...
2
votes
4answers
8k views
How to calculate viscous damping coefficient?
The damping of a spring is calculated with:
$$[\zeta] = \frac{[c]}{\sqrt{[m][k]}}$$
Where c is the 'viscous damping coefficient' of the spring, according to Wikipedia. m is the mass, k is the spring ...
2
votes
1answer
65 views
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 ...
2
votes
2answers
442 views
Using eigenvalues to determine the stability/behaviour of the system
first time I've been on physics.se but have used the math and cs before...
Anyway, here's my question:
If we have a damped pendulum described by the equation $$y'' + ay' + b = 0 , a>0$$ Using the ...
2
votes
1answer
156 views
Symbol for dashpot/damper (in a harmonic oscillator)
In diagrams that contain the dashpot symbol, sometimes the mass is attached to the "interior" end of the dashpot, other times the mass is attached to the "base" end.
For example, consider the ...
2
votes
1answer
80 views
2nd order pertubation theory for harmonic oscillator
I'm having some trouble calculating the 2nd order energy shift in a problem.
I am given the pertubation:
$\hat{H}'=\alpha \hat{p}$,
where $\alpha$ is a constant, and $\hat{p}$ is given by:
...
