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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Classical probability of harmonic oscillator

I am trying to derive the classical probability density function to find the harmonic oscillator at position $x$. I am confused between the random variables involved here $x, t$ and not able to ...
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Books on waves with Fourier Transforms

There are many waves and oscillations books out there that also include Fourier analysis but very few give the subject a thorough treatment, they just pass it in a few pages. If anybody has any ...
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10 Harmonic Oscillators - Probability of finding one in state n=0 [on hold]

Given are 10 harmonic oscillators with a total energy of $E=2h\nu$. Note that the ground states are not included, since the calculations do not need them! e.g. $E_n= \hbar \omega (1/2 + n)\approx ...
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Spring-mass system with variable stifness $m \ddot{x}+k(x)x=0$, time period is known, stifness is unknown

This question is somewhat related to my previous question: What is the time period of an oscillator with varying spring constant? In that question, time period of mass-spring system with variable ...
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Transition from the second excited state to the ground state in 3d oscillator [on hold]

The problem: 3d harmonic oscillator is in a second excited state. Suddenly a perturbation is applied which depends only on the length of the position vector $|\vec{r}|$. Can the oscillator fall into ...
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35 views

What is the equation for a vibrating musical string? [closed]

Can one define the equations of motion for a vibrating musical string? If so, how? For simplicity, assume this string vibrates as an ideal sine wave and ignore the complexity of actual timbres like a ...
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144 views

Why is the wave equation so pervasive?

The homogenous wave equation can be expressed in covariant form as $$ \Box^2 \varphi = 0 $$ where $\Box^2$ is the D'Alembert operator and $\varphi$ is some physical field. The acoustic wave ...
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Conventionally, how many amplitudes does a (harmonic) oscillator pass through in one full cycle? [closed]

I don't know the typical scientific convention. My book says there are 4 amplitude. But no matter where I start the oscillator , the answer is at most 3.
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Harmonic oscillator ensemble

Suppose we have an ensemble of classical 1D harmonic oscillators, with displacement $x = A \cos(\omega t+\phi)$, where the phase angle $\phi$ is equally likely to be any angle between $0$ and $2 ...
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35 views

Harmonic oscillator :Two masses are attached to one unfixed spring from both sides (vertically) [closed]

while ($t<0$) the system is still ($\Sigma$ F=0). Mass $m_2$ is held while $t<0$. Mass $m_1$ is located $h_0$ meters above the ground and the spring is currently stretched $L$ meters. The ...
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What is the time period of an oscillator with varying spring constant?

It is well known that the time period of a harmonic oscillator when mass $m$ and spring constant $k$ are constant is $T=2\pi\sqrt{m/k}$. However, I would be interested to know what the time period ...
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Confusion about the resonance

I get confused with the concept of resonance. Many materials suggest that in order to achieve resonance, the system must undergo simple harmonic force ($F=F_0\sin(\omega t)$), and at the natural ...
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50 views

Eigenstates of a harmonic oscillator

Using ladder operators, I can find eigenstates $\psi_n$ with eigenenergies $$E_n=\hbar\omega\left(n+\frac{1}{2}\right). $$ In my textbook, ladder operators work like $$ a\psi_n = c_n \psi_{n-1}$$ $$ ...
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Period for small oscillations is like simple harmonic motion

In Arnold's book on mechanics there is the following problem: Consider the period of oscillations near a minimum $E_0$ of the potential energy function $U$. Then he says to compute the limit of ...
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101 views

Hooke's Law and Simple Harmonic Motion

Can anyone explain how the following formula is derived and what it means? $$x'' = -n^2x $$ I was reading through my high school physics textbook and I stumbled upon a section on Hooke's Law. It says ...
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1answer
35 views

Do I have some freedom when I define the quantum SHO ladder operators? [closed]

I tried to solve the quantum harmonic oscillator via the operator method. After doing it and looking up the solution I noticed that for some reason the ladder operators got an additional factor of (i) ...
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Is it true that spring has more force acting on it at its positive maximum amplitude than than at the negative one?

Am I missing something? It seems obvious to me that at $+A$ and $-A$, the spring has restorative forces equal in magnitude but opposite in direction. But since gravity is always pulling it down, ...
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Is harmonic oscillator continuous variable system?

In the literature I have seen that the notions "our system is continuous variable system", "Hilbert space of our system is infinite" were used as if they were equivalent. For example for harmonic ...
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Is the electron-hole pair a 1D quantum oscillator or 3D oscillator

I'm trying to use fluctuation dissipation theorem to describe spontaneous photon emission process by electron-hole recombination in semiconductor material. I notice that all the references using such ...
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Is there an analog to the Runge-Lenz vector for a harmonic potential?

The Runge-Lenz vector is an "extra" conserved quantity for Keplerian $\frac{1}{r}$ potentials, which is in addition to the usual energy and angular momentum conservation present in all central force ...
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Free energy of coupled classical harmonic oscillators

I'm looking to find the thermodynamic (NVT) free energy of a classical coupled harmonic oscillator system such as the one below: (image taken from ...
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Eigenstates of sum of creation and annihilation operators

Does the operator $a+a^\dagger$ have eigenstates? If yes, what are they?
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Derive frequency given potential using Newton's laws

A mass with mass $m$ has a potential energy function $U(x)$ and I'm wondering how you would find the frequency of small oscillations about equilibrium points using Newton's laws. I started by finding ...
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Simple Harmonic Motion in Special Relativity

I was trying to see what results I would get if I were to incorporate relativistic corrections into the case of a harmonic oscillator in one dimension. I thought that if the maximum velocity of the ...
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86 views

Hamiltonian related to Riemann zeta function [closed]

using the eigenvstates of the Harmonic oscillator could we give a meaning to the Hamiltonian $$ H=\log(a.a^{+}+1) $$ here $ a$ and $ a^{+}$ are the creation/anihilation operators with commutation ...
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Creating an arbitrary state of the quantum simple harmonic oscillator

Suppose $\mathcal{B}=\{|0\rangle, |1\rangle, |2\rangle, ... \}$ is the energy eigen-basis of a quantum simple harmonic oscillator. I want to create the state \begin{equation} |\Psi\rangle = ...
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Finding the wave function of a quantum harmonic oscillator [duplicate]

How can I find the wave function of a quantum harmonic oscillator? If I measure its energy several times, my measurements will change the state of a system. All I know are the possible states, given ...
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The classical correspondence in the harmonic oscillator

In the harmonic oscillator, the ground state is the one with minimum uncertainty (and also other squeezed states that satisfy the 'equal to' in the Heisenberg inequality). This should mean that these ...
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1answer
33 views

Undamped Resonance of a Classical Harmonic Oscillator

Consider an undamped harmonic oscillator. It may be driven at it's natural frequency, $\omega_0^2 = \frac{k}{m}$. According to Feynman, and other sources, were this to happen, the amplitude of the ...
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34 views

Creation and annhilation operator in the Heisenberg picture

I am trying to calculate the time evolution of the creation/anni. operator in the Heisenber picture. On this webpage http://quantummechanics.ucsd.edu/ph130a/130_notes/node191.html, they used the ...
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56 views

General solution of a mass spring system

This is the differential equation that describes small amplitude vertical oscillations of a mass $m$ that is hanging from a spring $$\frac{d^2x}{d t^{2}} + \frac{b}{m}\frac{dx}{dt} + \frac{k}{m} x = ...
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Oscillation frequency of a dipole

So I have found the potential to be: $U(x) = \frac{\mu_0 m_2 m}{4 \pi} (\frac{1}{(d+x)^3}+\frac{1}{(d-x)^3})$ Afterwards I have found the position which minimized the energy to be $x=0$. (By doing ...
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What is the physical reason that the undamped driven oscillator has mean power zero?

The instantaneous power absorbed by an undamped driven oscillator is given by:$$\mathbf{P} =-\omega\dfrac{F_o/m}{{\omega_0}^2 -{\omega}^2} F_0 \sin{\omega t}\cos{\omega t}$$. But my book says the mean ...
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Constructing a dispersion relation from the Hamiltonian

I'll begin by saying that I'm not entirely clear on if this is possible. I have a Hamiltonian of the form $$ \left( \begin{array}{cccc} \text{$\omega $1} & \text{J12} & 0 & \text{J14} \\ ...
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Periodically connected QHO's

I've recently been thinking about what happens when you connect quantum harmonic oscillators in a periodic way. I'm actually thinking about when you take a mass-spring system (which can easily be put ...
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Occurance and disappearance of degeneneracies in a periodic structure of (quantum) LC circuits

Introductory part I'm currently studying an analytical model of coupled LC circuits, in preparation for actually performing measurements on such structures. While the final goal will struggle with a ...
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How to find the period of a simple pendulum in an elevator going up with an acceleration?

How to find the period of a simple pendulum in an elevator going up with an acceleration of $a$. Don't just say, $T=2 \pi$ $\sqrt{ \frac l {g+a}}$ I want to know how the above equation is formed.
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Is a constant on the RHS of the equation of simple harmonic motion allowed? [closed]

I read at a STEP booklet that we have to know how to bring a simple harmonic motion's equation to the form: $$\frac{\mathrm{d}^2x}{\mathrm{d}t^2} + \omega^2x= c$$ where $c$ is a constant. We also ...
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The spring of air: Why does a piston undergo SHM when it is displaced lengthening the air-column inside a cylinder filled with air?

Let there be a cylindrical tube, closed at one-end, with a well-fitting but freely moving piston of mass $m$. [. . .] The piston has certain equilibrium position. If the piston is moved a distance ...
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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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Period of a spring in SHM (simple harmonic motion)

An object with unknown mass M is hanged on a vertical spring with unknown spring constant K, the spring is in rest and is 14 cm from its normal point (if it didn't had the mass hanged it had less 14 ...
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85 views

Density of states of 3D harmonic oscillator

for the first red box, shouldn't be $\epsilon^2 =\epsilon_{n_x}^2 +\epsilon_{n_y}^2 + \epsilon_{n_z}^2 + 2\epsilon_{n_x}\epsilon_{n_y} + 2 \epsilon_{n_x}\epsilon_{n_z} + ...
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Is it possible to write the fermionic quantum harmonic oscillator using $P$ and $X$?

The Hamiltonian of the quantum harmonic oscillator is $$\mathcal{H}=\frac{P^2}{2m}+\frac{1}{2}m\omega^2X^2$$ and we can define creation and annihilation operators ...
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87 views

Pendulum on the moon, (Highschool)

A simple pendulum used as a clock, set with the correct time at earth, was sent to moon, it was noticed that it is late 36mins for each hour on earth. Calculate the ratio between acceleration of ...
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Lax representation of the harmonic oscillator

Peter Lax showed that the differential operators $$L=-6\partial_x^2-u,\quad B=-4\partial_x^3-u\partial_x-(1/2)u_x$$ fulfilling the Lax equation $$\dot{L}+[L,B]=0$$ is equivalent to the KdV equation ...
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Cause of SHM of liquid-column in V-tube

Suppose there is a v-shaped tube filled with water. The left limb is at $\theta_1$ & the right limb is at $\theta_2$ with the horizontal base. Initially, the level of water in both the columns are ...
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Trick for reformulating in terms of centre of mass and relative variables

I am working through a problem that has caused me difficulties in the past. I have the Hamiltonian $$\mathcal{H}=\frac{p_1^2}{2m_1}+\frac{p_2^2}{2m_2}+\frac{k}{2}(q_1-q_2)^2$$ I want to express the ...
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Finding the phase space density of $N$ harmonic oscillators

Consider a system of $N$ identical harmonic oscillators in 1d. The Hamiltonian will be given by $$\mathcal{H}_N=\sum_{i=1}^N \frac{p_i^2}{2m}+\frac{m\omega^2}{2}q_i^2$$ Now supposedly the Hamiltonian ...
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Where is there posted dampening coefficients for viscous dampers?

Depending on the size of the viscous dampener the coefficient will change, however i am trying to model a "physical" system with calculating everything out, so that way i can test failure before it ...
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1answer
51 views

What are the units of the creation and annihilation operators?

The creation and annihilation operators - also known as ladder operators are; $ \hat{a}^\dagger$ and $\hat{a}$ respectively. Using the equation $\hat{H} = \hbarω\left(\hat{a}^\dagger \hat{a} + ...