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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Can a uniform circular motion be considered as simple harmonic motion? [duplicate]

The acceleration in a circular motion is directed towards the centre and is directly proportional to the radius of circle if it has uniform angular velocity. Is circular motion with uniform angular ...
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Oscillating block amplitude change when 2nd mass added [closed]

There is a oscillating block with amplitude $A$ and mass $M$. We add a mass $m$ with zero velocity and vertically.when the block is in this two conditions: ...
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Birkhoff Method for Harmonic Oscillator Perturbation

Problem: Given Hamiltonian $$H = \frac12 (p^{2}+q^{2})+q^{3}-3qp^{2}$$ make a perturbative canonical transformation $(q,p) \rightarrow (Q,P)$ such that the new Hamiltonian, apart from terms of degree ...
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25 views

Simple Harmonic Motion Derivatives, and the equation

If the velocity time graph of a SHM is the derivative of the Distance time graph, and the kinetic energy of the mass in the SHM is maximum when the displacement is 0, how can the maximum velocity be ...
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The “general uncertainty” of the harmonic oscillator defies the correspondence principle?

If you use the definition of $(\Delta x)^2 = \langle n | x^2 | n \rangle - \langle n | x | n \rangle^2$ and the same for $(\Delta p)^2$ to calculate $\Delta x \Delta p$ for the state $|n\rangle$ of a ...
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61 views

Harmonic oscillator and cyclic coordinates

I am reading goldstein there is some comment I don't understand. Consider the following hamiltonian $$H = \frac{p^2}{2m} + \frac{kq^2}{2}$$, which can be rewritten as follows $$H = \frac{1}{2m}(p^2 ...
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Period of small oscillations of liquid in a bottle

I have seen that water in a bottle of water when perturbed invariably starts oscillation back and forth, with the shape of the water surface remaining intact. I was wondering if there was any way to ...
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Deriving an equation for the mass of a pendulum (Follow up)?

Following this question: Deriving mass from simple pendulum which is summarized below Some mass $m$ is release from rest at a horizontal position. $m$ reaches the bottom of its path (so directly ...
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Why does my teacher say “the simple harmonic oscillator passes through the amplitude four times in one cycle”?

I asked him to explain it. But he simply told that where the sinusoidal graph the x axis counts as two times. I don't get why. I thought the answer should be three.
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Fluid filled harmonic oscillator

A vessel (preferably circular) filled with water is accelerating unidirectionally such that the level of water is higher on one end than the other. What I want to know is that if the vessel is ...
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236 views

Combination of Simple Harmonic Motions

Will the combination of 2 Simple Harmonic motions will be an SHM in itself? For example for simple functions such as $$\ f(t)=\sin\omega t-\cos\omega t$$ I can use trigonometry to show that it can ...
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31 views

AFM cantilevers driven below resonance?

Is there a physical reason why AFM cantilevers are driven below their resonance frequencies? In all of the AFMs I have used, once you measure the resonance frequency of the cantilever, it is set up ...
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Block-Spring System Kicked Into Motion

Given two blocks of identical mass on a frictionless surface and connected by a spring with spring constant k, I'm asked to find the motion of the blocks -- after one is kicked into motion with ...
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Is there an easy way to treat the anisotropic harmonic oscilator?

In Quantum Mechanics we can deal with the one-dimensional harmonic oscilator by using the trick of the ladder operators. In that case, the original Hamiltonian is $$H = ...
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Do you know the principle which says that connecting two sources of similar kind produces a waste and destruction? [closed]

There is a great article, called commutation cells, which states that you cannot transfer kinetic energy from one container to another immediately, bypassing the potential energy storage. Otherwise, ...
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90 views

Why $k/m$ in simple Harmonic motion equal $\omega^2$? [closed]

I've come across this thing in simple harmonic motion but never did I manage to find a reason why $k/m$ should equal $\omega^2$ and the theory behind it. People say it is done for convenience equating ...
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95 views

Why are harmonic oscillators quantized? [closed]

What physical reason is there for a mass on a spring to have discrete energy levels? And why are those energy levels equally spaced, i.e. why is $E \ \alpha \ f$? Personal background and ...
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27 views

I do not understand why $a_-a_+\psi_n=(n+1)\psi_n$ and not $\sqrt n(n+1))\psi_n$ [duplicate]

I do not understand why $a_-a_+\psi_n=(n+1)\psi_n$ and not $\sqrt n(n+1))\psi_n$ or how the Energy formula can help me understand this (I was told that it would). In the introduction to quantum ...
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63 views

I do not understand why $a_-a_+\psi_n=(n+1)\psi_n$

I do not understand why $a_-a_+\psi_n=(n+1)\psi_n$ and not $\sqrt n(n+1))\psi_n$ or how the Energy formula can help me understand this (I was told that it would). In the introduction to quantum ...
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65 views

Quantum harmonic oscillator, clarification about the period

We can write the general state |A> at time $t=0$ as $|A, 0>=\sum a_n |n>$ where |n> are the eigenvectors of the oscillator. In my textbook there is written that if the $a_n=0$ for every n=even ...
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32 views

Resonance of a system featuring a collection of individual resonators?

Suppose you had a number of harmonic oscillators, each with different resonant frequencies in a system. Does this imply that their is an overall system resonance that is dependent on the individual ...
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Total energy of a simple pendulum proportional to the square of the amplitude? [duplicate]

It is known that in simple harmonic motion, the total energy of the system is proportional the square of the amplitude, but how can I prove that for a simple pendulum where amplitude is the arc length ...
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54 views

Mismatch between underdamped and critically damped solutions

If you have a harmonic oscillator with damping $D$ (e.g. small angle pendulum) $$\ddot{\theta}+D\dot{\theta} + \theta=0$$ then the solution I get in the underdamped case ($D^2-4<0$) is: ...
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Perturbation theory of $\lambda q^4$ perturbed harmonic oscillator

For a perturbed Hamiltonian $$ H = H^{(0)} + H' $$ the perturbation theory approach $$ \Psi = \Psi^{(0)} + \lambda \Psi^{(1)} + ... \\ E = E^{(0)} + \lambda E^{(1)} + ... $$ leads to the equations ...
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170 views

Simple harmonic motion versus oscillations

I want to see whether certain oscillations in my daily life, such as the oscillation of violin strings when plucked, are simple harmonic motion or not. Can we identify whether an oscillation is simple ...
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Physical interpretation of Fourier $[x(t)]$ where $x(t)$ is the position of mass $m$ as a function of time?

If a macroscopic body of mass $m$ moves according to a certain law of motion like, for example, $$x(t)=A\cos(2\pi ft)$$ then what physical interpretation can be attributed to the Fourier transform of ...
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Coupling of LC circuits

I am reading an old article in a physics journal in which the author explains that the frequency between two identical, but capacitively coupled oscillators becomes ...
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45 views

Why does the potential in simple harmonic motion contain only even powers?

The lecturer was showing that any system (almost) will behave as SHM if we move it by a small $\alpha$ from its equilibrium point. For doing so, he wrote the potential of the motion as ...
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100 views

Spring-mass system with complex spring constant

Suppose a system containing a mass $m$ on frictionless surface, attached by a spring to a wall. The spring's constant is complex, given by $K = K_1 + K_2i$, with $K_1 \gg K_2$. Write the ...
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151 views

Exact closed form solution to the quantum harmonic oscillator

I came across this question in Griffiths QM, which asked to show that this equation $$\Psi(x,t)=\left(\frac{m\omega}{\pi \hbar}\right)^{1/4} \exp\left[-\frac{m\omega}{2\hbar} ...
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87 views

Why does $\omega = \sqrt{V''(x_0) / m}$?

I know that in an equation such that $$\ddot{x} + \omega^2x = 0,$$ the angular frequency $ = \omega$. But why is that ever $ \sqrt{V''(x_0) / m}$? (where $x_0$ is the equilibrium point). I just saw ...
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110 views

An overdamped oscillator with natural frequency ω and damping coefficient γ starts out at position x0 > 0 [closed]

An overdamped oscillator with natural frequency ω and damping coefficient γ starts out at position x0 > 0. What is the maximum initial speed (directed toward the origin) it can have and not cross the ...
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Initial conditions for shm [closed]

This is the part of the question from the book that I am studying, "A mass of $0.75\:\mathrm{kg}$ is attached to one end of a horizontal spring of spring constant of $400\:\mathrm{N m^{−1}}$. The ...
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128 views

How to find frequency with only amplitude? [closed]

I came across the following problem earlier. A platform oscillates in the vertical direction with simple harmonic motion. It’s amplitude of oscillation is C. What is the range of frequency of ...
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61 views

Oscillating Ruler

I was bored in the office and stuck a ruler in my desk and started maing it oscillate. I noticed that when I looked at it from the top, there were some bands of color I observed (as in the pic below) ...
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Energy of driven dampened oscillator

Given the oscillator described by: $$m\ddot{x}+\gamma \dot{x}+kx=F_0\cos(\omega t)$$ And supposing the system is at it's stable state, I wish to calculate the following: 1) The system's energy at any ...
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Simple harmonic motion, maximum kinetic energy [closed]

Why kinetic energy is maximum at mean position?
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How to apply cutoff in path integral?

I am working on harmonic oscillator for quantum fluctuations (apart from clasical part), path may written as $$ S_q=\int_0^Tdt[(\partial_tq)^2+w^2q^2] $$ This may written as $$ S_q=\int dt(q\Delta q) ...
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Excitation source in 2D grid coupled harmonic oscillator

In A. Zee's Quantum field theory in a Nutshell, he describes the QFT analogy of a matress, a 2D grid of points $q_a$ connected by springs (first page of first chapter, $q_a$ is the vertical ...
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113 views

Motion of Thompson's jumping ring

Thompson's jumping ring experiment is set up as follows: There is a force acting on the ring $F(x)$ where $x$ is the vertical displacement. The force is due to the $90^\circ$ out of phase flux ...
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554 views

Pendulum's motion is simple harmonic motion

For a pendulum's motion to be simple harmonic motion (S.H.M.) is it necessary for a pendulum to have small amplitude or S.H.M. can be produced at large amplitudes as well? If it is really necessary ...
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Proportionality of states in quantum harmonic oscillator

What is the justification for $a_{\pm} \psi_{n}$ being proportional to $\psi_{n\pm1}$ in a quantum harmonic oscillator? Here $a_{\pm}$ is the raising/lowering ladder operator.
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Solution to Harmonic Oscillator

The damped oscillator is governed by the equation of motion: $\ddot{x} + 2\beta \dot{x}+\omega _{0}^2x=0$ where $\omega _{0}=\beta$ for an oscillator with critical damping. The solution is $x(t)= ...
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55 views

Driven harmonic oscillator problem : change of variables and expressing the extension of the spring

I'm struggling to understand the reasoning in question $b)$. Basically, I had to write Newton's Second Law and do a change of variables to put the equation of motion of a mass $m$ in a specific form, ...
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283 views

Spring-block system, simple harmonic motion, time period

In the case of simple harmonic motion of spring block system, why time period of the simple harmonic motion of the block is independent of acceleration of the system (spring-block system)?
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Calculating trajectory of particle moving in a potential (SHM)

I have been given the potential of a simple harmonic oscillator: $$V=\frac{1}{2}kx^{2}$$ I want to calculate the value $x(t)$ of a particle moving in this potential, with initial conditions ...
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Response functions for the quantum harmonic oscillator

I'm going through problems in Quantum Field Theory for the Gifted Amateur, and have been trying to understand a problem on the forced quantum oscillator [$L = ...
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56 views

Derive an equation related to magnetism [closed]

Solve the equations for $v_x$ and $v_y$ : $$m\frac{d({v_x)}}{dt} = qv_yB \qquad m\frac{d{(v_y)}}{dt} = -qv_xB$$ by differentiating them with respect to time to obtain two equations of the ...
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Simulation of oscillator with frequency dependent damping

What would be the equation for the frequency dependent damping of harmonic oscillator? Is there something like: $$ \ddot{x}+2\delta\dot{x}+\omega_0^2x = \frac{F}{m}f(t) $$ with frequency dependent ...