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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The Sine pattern of variation [closed]
I'm trying to fully understand sin and cos as a certain pattern of variation, a particular variation that start's to decelerate smoothly towards the peaks and starts to accelerate abruptly towards the ...
2
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2answers
66 views
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 ...
3
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0answers
60 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 ...
13
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3answers
189 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 ...
1
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3answers
110 views
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:
$$
\left[
...
1
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1answer
59 views
SHM of floating objects
If we consider an object undergoing who has an acceleration proportional to the displacement of the object, it is going simple harmonic motion.
In terms of Newton's second law, this is $$ -\dfrac k ...
2
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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:
...
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2answers
57 views
Time period for spring connected body
Two identical springs with spring constant $k$ are connected to identical masses of mass $M$, as shown in the figures above. The ratio of the period for the springs connected in parallel (Figure 1) ...
1
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1answer
46 views
Energy eigenvalues of a Q.H.Oscillator with $[\hat{H},\hat{a}] = -\hbar \omega \hat{a}$ and $[\hat{H},\hat{a}^\dagger] = \hbar \omega \hat{a}^\dagger$
I just finished deriving the commutators:
\begin{align}
[\hat{H}, \hat{a}] &= -\hbar \omega \hat{a}\\
[\hat{H}, \hat{a}^\dagger] &= \hbar \omega \hat{a}^\dagger\\
\end{align}
On the ...
1
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1answer
52 views
Eigenfunctions in a harmonic oscillator
This assignment is about the one dimensional harmonic oscillator (HO).
The hamiltonian is just as you know from the HO, same goes for the energies, but I get that the wavefunction of the particle, at ...
1
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1answer
76 views
Pendulum in an elevator
Suppose we have a pendulum tied to the ceiling of an elevator which is at rest. The pendulum is oscillating with a time period $T$, and it has an angular amplitude, say $\beta$. Now at some time ...
2
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1answer
53 views
Pendulum Wave Period
Recently I've seen various videos showing the pendulum wave effect. All of the videos which I have found have a pattern which repeats every $60\mathrm{s}$.
I am trying to work out the relationship ...
4
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2answers
94 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} &= ...
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2answers
51 views
How does one subtract two light beams?
From what I understand, it seems like you can only "add" beams together. You can use a beam combiner, basically using a beam splitter in reverse, to combine two beams. In homodyne detection, you use a ...
3
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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
138 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 ...
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1answer
34 views
What is $\gamma$ in the damping equation?
$x''+\gamma x'+w_0^2x=0$
That is the general equation for damped harmonic motion. What is the term or name that describes $\gamma$?
Is it called the damping constant? I know its the ration between ...
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0answers
34 views
Quantum harmonic oscilator - book that does it all right [duplicate]
I am dealing with quantum harmonic oscillator.
In every single book or video i have checked out i can read how the mathematical technique for solving this Schrödinger equation:
$$
W\psi = - ...
0
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1answer
35 views
Harmonic oscillator with light damping
My textbook gives the following for x as a function of time for a lightly damped harmonic oscillator: $$ x = Ae^{- \gamma t} \cos (\omega \, t)$$
for $\gamma = \dfrac b {2m}$.
It says this implies ...
4
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2answers
92 views
Why is the damping force on a spring oscillator linearly dependent on velocity?
If you consider the damping force is friction like in:
then the force should be $$F=\mu N$$ where $\mu$ is the coefficient of kinetic friction. Why then is the damping force assumed to be linearly ...
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0answers
27 views
Find Resonance Frequencies [closed]
How can I find the resonance frequencies for the harmonic dumped oscillator when it is written in this form?
$$y''\left(t\right)+2\zeta y'\left(t\right)+y\left(t\right)=\sin{(\omega t+\phi)}$$
where ...
2
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0answers
61 views
Relativistic genarization of Quantum Harmonic Oscillator
I am trying to find out relativistic description of a quantum harmonic oscillator.
For a classical relativistic oscillator mass is a function of co-ordinates(http://arxiv.org/abs/1209.2876).
...
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0answers
37 views
Compound pendulum clarification?
I read in a book the following about compound pendulum and small displacements:
There are two points only for which the time period is minimum.
there are maximum 4 points for which the time ...
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1answer
64 views
A good theoretical approximation for a magnetically damped pendulum
In a laboratory course we had to perform an experiment with a pendulum (just an iron weight on a wire) and play around for some time with its wire's length and so on.
This was quite boring and we ...
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1answer
72 views
Standing Waves: finding the number of antinodes
A string with a fixed frequency vibrator at one end forms a standing wave with 4 antinodes when under tension T1. When the tension is slowly increased, the standing wave disappears until tension T2 is ...
0
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2answers
92 views
Calculating phase difference of sound waves
An observer stands 3 m from speaker A and 5 m from speaker B. Both speakers, oscillating in phase, produce waves with a frequency of 250 Hz. The speed of sound in air is 340 m/s. What is the phase ...
0
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0answers
36 views
How can I find the frequency? [duplicate]
Grocery stores often have spring scales in their produce department to weigh fruits and vegetables.
The pan of one particular scale has a mass of 0.5 kg, and when you place a 0.5 kg sack of potatoes ...
0
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0answers
51 views
Frequency with the spring scale [closed]
Grocery stores often have spring scales in their produce department to weigh fruits and vegetables.
The pan of one particular scale has a mass of $0.5 kg$, and when you place a $0.5 kg$ sack of ...
1
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0answers
27 views
Doubling the energy of an oscillating mass on a spring [closed]
From this question:
Question 1.
What do we need to change in order to double the total energy of a mass oscillating at the end of a spring?
(a) increase the angular frequency by $\sqrt{2}$.
...
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2answers
69 views
Potential energy during vertical fall
Suppose I have a weightless spring connected perpendicularly to the ground, and it has on top of it some weightless surface. Now, I release some sticky object from height $h$ above the system of light ...
0
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1answer
76 views
Simple harmonic oscillator system and changes in its total energy
Suppose I have a body of mass $M$ connected to a spring (which is connected to a vertical wall) with a stiffness coefficient of $k$ on some frictionless surface. The body oscillates from point $C$ to ...
3
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1answer
105 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 ...
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2answers
48 views
Does spatial coupling prohibit resonances due to an external source field?
The harmonic oscillator coupled to a sinodial external source
$$\tfrac{\partial^2 x(t)}{\partial t^2}+\omega_0^2 x(t)=F_0\sin(\omega_\text{ext}\ t),$$
has the solution
$$x(t)=x(0)\cos(\omega_0 t)+C ...
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1answer
86 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 ...
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1answer
60 views
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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0answers
59 views
Quantum harmonic oscillator. Finding operators
Problem:
I'm trying to verify that $p_H(T)$ and $x_H(T)$ satisfy the following equations, (by solving the Heisenberg equation):
$x_H(t)=x_H(0)cos(\omega t)+(1/m\omega)p_H(0)sin(\omega t)$
...
0
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1answer
50 views
Hyperbolic, parabolic, elliptical PDE related to under-, critical- and overdamped in harmonic osciallation
A damped harmonic oscillator has three cases for the damping: underdamped, critically damped and overdamped. With partial differential equations, I know the hyperbolic wave equation, the parabolic ...
2
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1answer
124 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 ...
0
votes
1answer
75 views
FWHM in resonance amplitude square derivation
Consider a linear harmonic oscillator subject to a periodic force:
$$ \ddot x + 2 \beta \dot x + \omega _0 ^2x = f_0\cos \omega t$$
The solution tends to:
$$A \cos (\omega t - \delta)$$
where:
...
1
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1answer
39 views
Metronome synchronisation applied to swings
The movement of several metronomes can be synchronised when a movable floor is utilised which couples the movement of the different metronomes.
Is it possible to apply this sort of synchronisation to ...
4
votes
3answers
390 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.
0
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2answers
129 views
Why is simple harmonic motion called so?
Is the motion of a simple pendulum, a simple harmonic motion? It stops vibrating after sometime.
2
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1answer
67 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 ...
3
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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
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2answers
176 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 ...
2
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4answers
119 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 ...
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1answer
145 views
Shift operator (integral calculus involving Hermite polynomials) [closed]
I didn't know whether to pose this question on Physics.stackexchange or Math.stackexchange. But since this is the last step of a development involving the eigenfunctions of an Harmonic oscillator and ...
0
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0answers
98 views
Spring with mass [closed]
A block of mass $M$ is attached to a spring that has mass $m$ and the force constant $k$. The block is placed on a horizontal frictionless surface. Find the period of small-amplitude oscillations ...
0
votes
0answers
66 views
Mass spring system, increase mass [closed]
The question says that after a mass $m=M$ (attached to a horizontal spring) reaches its furthest point, so at its amplitude, the mass is doubled, $m=2M$.
What happens to the period, amplitude and ...
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3answers
483 views
Partition function for quantum harmonic oscillator
Hi guys I'm currently trying to solve a mock exam for an exam in a few days and am a bit confused by the solutions they gave us for this exercise:
Exercise:
A solid is composed of N atoms which ...
