Questions tagged [harmonic-oscillator]

The term "harmonic oscillator" is used to describe any system with a "linear" restoring force that tends to return the system to an 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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Quantum harmonic oscillators with momentum-position coupling

I have two coupled quantum harmonic oscillators given by the following Hamiltonian: $$H=\frac{p_{x}^{2}}{2}+\frac{\omega^{2} x^{2}}{2}+\frac{p_{y}^{2}}{2}+\frac{\Omega^{2} y^{2}}{2}+\frac{C p_{x} y}{2}...
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Pendulum with Oscillatory Support - A question on Lagrangian Mechanics [closed]

Recently I have been attempting Morin's Introduction to Classical Mechanics (2008) but I got rather stuck on question 6.3 on the topic of Langrangian Mechanics. Attached are the problem and the ...
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Creation and annihilation operators in coordinate space

I am trying to express the creation and annihilation operators of a single quantum harmonic oscillator in coordinate space. The problem is that, when I use $P \to -i\hbar d/dx$, I get $a=a^\dagger$: $$...
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Time evolution for the harmonic oscillator wave functions

The wave functions of the quantum harmonic oscillator are given by: $$\psi_n(x)=\frac{1}{\sqrt{2^nn!}} \left(\frac{m\omega}{\pi \hbar}\right)^{-1/4} e^{-m\omega x^2/2\hbar}H_n\left(\sqrt{m\omega/ \...
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Using Binomial and Taylor Expansions to Demonstrate Harmonic Motion [closed]

I'm doing a physics 2 self-study and I came across this question in my textbook: A ball of mass m and charge q is constrained to move along the y axis. At the origin is a stationary charge Q. The ...
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Derivation of the Displacement Operator [closed]

So I am trying to derive the displacement operator ($\hat{D}(\alpha)$). I have seen multiple sources that all seem to show that it comes out of nowhere, i.e. there is no derivation. So I have been ...
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Schroedinger cat states of the harmonic oscillator

I've found in an article that it is possible to prepare experimentally the superposition of two coherent (quasi-classical) states to obtain the Schroedinger cat state: $$ \left|\psi_{\pm}(t)\right\...
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How is harmonic motion (specific and not a special case) different from periodic motion? [duplicate]

I have seen written in many books that a motion that repeats itself after a specific time period follows periodic or harmonic motion. However I know for a fact that damped harmonic motion cannot ...
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Problem with spring oscillator measurement - inconsistent frequency

I measured the periodic movement of a spring with a mass and plotted it Looking at the first graph, I assumed the frequency is somewhere between 2-2.5 Hz. The mass hanged on spring was 100 g. Using ...
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How to properly find the spectrum of the Quantum Harmonic Oscillator? [duplicate]

We want to find the spectrum of the Harmonic Oscillator Hamiltonian: $$H=\frac{\hat{p}^2}{2m}+\frac{1}{2}m\omega ^2 \hat{x}^2$$ From what I have seen in many books the procedure is as follows: We can ...
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Using perturbation theory or small oscillation approximation in Harmonic oscillator

Let us assume, we are given the following potential, $$V(x)=\frac{1}{2}ax^2-2x+\epsilon x^3$$ We need to find the energy levels of a particle bound in this potential Let us think of the ground level ...
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Is there any phase delay between displacement and driving force in undamped driven oscillation when the driving frequency is below resonant frequency?

I know that there is phase delay in damped driven oscillation but I want to know is there any phase delay in undamped driven oscillation when we apply sinusoidal driving force. When driving force is ...
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Restoring force of electron-nucleus spring model (Lorentz oscillator model)

The restoring force should fulfill at least two criterion Experience repulsive force when it is compressed and attractive force when it is extended The restoring force always increases with distance ...
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Energy of a Quantum Harmonic Oscillator $(x-a)$ vs $x$

I am trying to compare the energy of quantum harmonic oscillators at different positions. Intuitively, I think they should be identical since the only difference is $(x)$ versus $(x-a)$. $(m=1, \hbar =...
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Why is the method I'm using incorrect to find out the time taken for the particle to complete 5/8 oscillations?

We use phasor diagram for SHM so when the phasor travels $2\pi$ radians it's projection on the y axis would have completed one oscillation. It says in the question that the particle completes $5/8$ of ...
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Simple Harmonic Motion - University Physics with Modern Physics problem 14.31 [closed]

I'm trying to solve problem 14.31 from University Physics with Modern Physics (13th E.) by Young & Freedman. Q: "You are watching an object moving in SHM. When the object is displaced $0.6\ ...
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Is there any effect that draws the oscillation frequencies of two particles together?

I'm looking for any sort of coupling that draws oscillation amplitudes together if one couples two (nearly) harmonic oscillators, basically the opposite of avoided crossing or level repulsion. Is ...
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Pendulum Tuned Mass Damper - Mathematical Relationship between Mass and Damping Ratio

I am doing an experiment where I built a test tower with a pendulum to act as a tuned mass damper, similar to this picture below: I want my independent variable to be the mass of the pendulum (which ...
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Why is a phase shift through a time delay not used for damping in vibration dampers?

Why do oscillation dampers use signal conversion through a sufficiently massive electrical circuit (with resistors, capacitors, diodes) to create antiphase, instead of simply shifting the signal in ...
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Matter wave coherence length

Coherence length is defined as (see here) The propagation distance over which a coherent wave (e.g. an electromagnetic wave) maintains a specified degree of coherence. Other questions see here, and ...
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Deriving the bosonic action of annihilation and creation operators in QFT

The definition of bosonic annihilation and creation operators are given as: $$a^{\dagger}_{\vec{k}}\lvert n_{\vec{k}}\rangle =\sqrt{n+1}\lvert (n+1)_{\vec{k}}\rangle$$ $$a_{\vec{k}}\lvert n_{\vec{k}}\...
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Can we use the equations $-AωSin(ωt+φ)$ and $AωSin(ωt+φ)$ interchangably?

My book says that, For a simple harmonic motion, velocity of the particle = -AωSin(ωt+φ) Now, assuming φ to be π/2, in the next sentence, they say that, maximum velocity = Aω. Why have they simply ...
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Decreased period of pendulum

I'm doing an experiment with a physical pendulum and as time passes, the time taken for n cycles is shorter than would be predicted from the period (i.e. the time taken for the first n cycles is less ...
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Does the moment of inertia of a tidally locked revolving body differ from that of a non-tidally locked revolving body?

This is a question I encountered(for context) : A solid sphere of (radius =R) rolls without slipping in a cylindrical vessel (radius =5R). Find the time period of small of oscillations of the sphere I ...
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Why is there only partial energy exchange for coupled oscillators with different masses?

So I was playing around with this widget and I noticed that when the spring masses are the same, the energy is completely exchanged (note one has to set the "show graph" option to "...
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Thermodynamics of harmonic oscillators [closed]

I have N harmonic oscillators in thermal equilibrium at temperature $T$, $$ E(n_{1},...,n_{N})= \hbar \omega \sum_{i=1}^{N}(n_{i} + \frac{1}{2})$$ And I calculated partition function like this $$Z = \...
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How can you approximate number of bound states in a harmonic oscillator potential $V$ and also for a Dirac delta function using uncertainty principle?

How can you approximate the number of bound states in a harmonic oscillator potential $V$ and also for a Dirac delta function using the uncertainty principle?
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What is my misunderstanding in Wick's theorem?

Trying to understand Wick's theorem, I took most of my knowledge from the corresponding Wikipedia article. The statement is that given the definition of normal ordering of operators $A,B,C,\ldots$ any ...
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An algebra step in the Quantum Partition Function for the Harmonic Oscillator

On page 183 of Altland Simons, we are told: $$ \prod_{n = 1}^{\infty} \Big[ \Big( \frac{2\pi n}{\beta} \Big)^2 + \omega^2 \Big]^{-1} \sim \prod_{n = 1}^{\infty} \Big[ 1 + \Big( \frac{\beta \omega}{2\...
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How to find maximum velocity of an image of an object excecuting a SHM in front of a concave mirror?

I was doing a problem in which a block of mass of 1 kg attached to a spring of spring constant 100 N/m excecutes a SHM with the blocks equilibrium position at the centre of curvature of a concave ...
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Forced oscillation resonance frequencies [closed]

Given this forced oscillator: $$ \ddot{x}+\gamma\dot{x}+\omega_0^2x=\frac{F(t)}{m} $$ Where $F(t)$ is: $$ F(t)=\sum_{n=1}^{\infty}\frac{4F_0}{n\pi}\sin\left(\frac{2n\pi t}{T}\right) \hspace{0.4cm} \...
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Confusion about modes and quantum field theory

I'm learning quantum field theory from P&S and Srednicki. I'm having a lot of difficulties understanding the concept of a momentum state. In particular, I'm confused about how to interpret the ...
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Transformation between two different sets of molecular vibrational normal coordinate systems [migrated]

Lets assume we have $N$ Atoms and we treat them within the Born-Oppenheimer Approximation. We can calculate the adiabatic electronic groundstate potential. Lets assume we observe two local minima, ...
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Why do you add independent solutions when finding a general equation for SHM?

In my physics class, we have an assignment based on simple harmonic motion with the differential equation: $$ \frac{d^2x}{dt^2} + a\frac{dx}{dt} + a^2x = 0 $$ Different parts of the question help us ...
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Identical particles placed on the energy levels of a 1D harmonic oscillator

If the particles are 6 spinless bosons, would they tend to occupy the ground state together and make the lowest total energy of the system $E=6E_0=3 \hbar \omega$? And If the particles are 6 spin 1/2 ...
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Constructing the density of states of multiple independent harmonic oscillators

I have a system of $N$ uncoupled 1D quantum harmonic oscillators, each with its own frequency $\omega_i$. The density of states for a single quantum harmonic oscillator shall be defined as $$ \rho(E) =...
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Axis of oscillation

what would be the effective length of the pendulum if the block start oscillating in the plane perpendicular to the screen? according to me the block would oscillate about AB and with the green line ...
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Evenly spaced energy level systems [duplicate]

I'm looking for examples of quantum mechanical systems which have evenly spaced energy levels. A couple of them are - Quantum Harmonic Oscillator; Inverted Quantum Harmonic Oscillator. It appears to ...
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Is it possible to get the SHO approximation of a pendulum without using energy conservation?

I tried to get the approximation for small angle of a simple pendulum using only $\sum \mathbf F = m\mathbf a$ and cartesian coordinates (that means only $x$'s and $y$'s, without $\theta$). After some ...
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Time period of a simple pendulum at the centre of earth

So I thought of this hypothetical experiment and wondered whether it be possible for it to even happen at center of earth even though gravity is zero. So considering a pendulum with length much much ...
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Equivalent length of a simple pendulum

i was solving some question based on harmonic oscillations and a question popped up: If the angle between the the wires and the surface is 45 and the mass of the bob is $m$ calculate the time period ...
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Damped oscillator retarded Green function

The Green function for the damped oscillator in Fourier space is $\tilde{G}(\omega)=\frac1{\omega_0^2 -\omega^2 + 2i\gamma\omega}$, where $\gamma$ is the damping parameter and $\omega_0$ the ...
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Energy analysis of standing waves

Why is the kinetic energy of antinodes for a standing wave maximum when all the particals pass through their mean position? The elastic potential energy of particals near the nodes are maximum when ...
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Steady state solution to a coupled oscillator equation where one of the oscillators are driven?

I am analysing the dynamics of quantum van der Pol oscillators in the classical limit and I found the following equation of motions for the complex amplitudes of oscillator A and B: $$ \dot \alpha = (-...
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The infinite-fold degeneracy of an oscillator when becoming a free particle

Considering a one-dimensional case, and if it has the following relation: $$(1) : [H,a]= \pm \omega a$$ then there are evenly spaced spectrum lines. If we sink $\omega \rightarrow 0$ , or, ...
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Can different quantum field oscillators have different frequencies $ω_k$? [closed]

Can different quantum field oscillators have different frequencies $ω_k$? If so, does it follow from that the energy gaps between two neighboring eigenvalues $ℏω_k$ could be different for different ...
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Statistical weight for $N$ harmonic oscillators in microcanonical ensemble

I would like to compute the statistical weight for the microcanonical ensemble for $N$ harmonic oscillators. To do that i use the hamiltonian of the harmonic oszillator: $$H(q,p)=\sum\limits_{i=1}^N \...
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Transmon: why do we need unharmonic hamiltonian to isolate energy levels? [duplicate]

With a quantum harmonic oscillator, we cannot isolate energy levels, e.g. to create a qubit. We need to embed anharmonicity, to get unevenly-spaced energy levels and so making them distinguishable. ...
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Proof that body performs SHM

At core, SHM is the shadow of a particle revolving with omega on a circle of radius equal to amplitude. Now we say that body performs SHM either if the equation of its position makes a sinusoidal ...
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Using operators to solve SHO with $x^2$ perturbation

I have simple harmonic oscillator potential:$V_0 =\frac{1}{2} m\omega^2x^2$. So the Hamiltonian is: $H_0=\frac{p^2}{2m}+\frac{1}{2} m\omega^2x^2$. We now add a perturbation which is $V_1 =\frac{1}{2} \...

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