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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A doubt regarding Quantum Harmonic Oscillator

Classically when we solve Newton's equation for $V=\frac{1}{2}m\omega^2x^2$ we get two linearly independent solutions (for $\omega\not=0$): $Ae^{\omega t}$ & $Be^{-\omega t}$, their linear ...
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Can we find the amplitude for small oscillations for the given system?

We have a uniformly distributed(with both mass and charge) rod, with mass $m$, positively charged with linear charge density,$-\lambda$, length $2l$, with a uniformly distributed charged ring at the ...
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Nuclear Physics: Shell Model [closed]

I am studying nuclear shell model with the assumption that potential is 3D harmonic potential $$V(r)= (1/2)m*w^2*r^2. $$ I got the energy dependence as $$E= (N+3/2)hw.$$ but in a question he asked to ...
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Rule swing with spring experiment: how can I modify it?

Basically I want to replicate this experiment (https://youtu.be/GqPGbHq2fxU). It's a ruler oscillating with one fixed end and one end attached to a spring. In my previous experiment, I used a short ...
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Length of a water column executing SHM in a U-shaped Tube

I was watching the lecture 30 of Walter Lewin ( 8.01 ). Here is a link: https://youtu.be/hAYeA3Wwb4U At 29:30, when he was describing the SHM of water in a U shaped tube. Lewin says that for the ...
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How can inverting a potential change the solutions? [closed]

Consider a one-dimensional parabolic potential of the form $V(z)=10z^2$, acting on a mass of 0.5kg number. How can inverting the potential on a system let's say where there is a mass and oscillation ...
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How to actually create a quantum harmonic oscillator potential well?

Classically, to put a particle in a harmonic oscillator potential I can just hook it up to a spring. But how is $V=\frac{1}{2}m \omega^2 x^2$ actually created for a particle like, say, an electron? Of ...
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Period of the propagator of quantum harmonic oscillators

I found something that I'm confused with when calculating the propagator of harmonic oscillator. Using the energy representation, the propagator of a quantum harmonic oscillator can be expressed as : $...
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Canonical Transform for a simple pendulum [closed]

Im have the following Hamiltonian, $$ H(q,p) = \frac{p^2}{2m}-\frac{m\omega^2 q^2}{2} \,\, .\tag{1} $$ I must show that the generating function defined by, $$ S(q, \theta) = \frac{1}{2}mq^2 \cot (\...
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Symmetry of infinite harmonic oscillators in quantum field theory

We know that the system (Hamiltonian) of $N$ harmonic oscillators possesses $SU(N)$ symmetry, where \begin{equation} H=\hbar \omega \sum_{i=1}^{N}\left( a_{i}^{\dagger } a_{i} +\frac{1}{2}\right) . \...
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Classical Harmonic Oscillator in a Magnetic Field

Consider a 3D charged Harmonic Oscillator which is placed in a homogenous magnetic field along the z direction. The equations of motion in plane are then given as: $$ m\ddot{x}=kx+a\dot{y},$$ $$ m\...
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Does the damping force oppose a spring's restorative force in damped oscillations?

If the damping force ($F_d=-bv)$ in damped simple harmonic motion opposes the spring's force $(F_s=-kx)$, why does the solution for the position as a function of time proceed from this equation?: $$-...
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Could one call eigenstates of $ \hat{a} = \hat{x} + i\hat{p}$ coherent states for other potentials than the harmonic oscillator?

Let's say I look at the quantum system of a particle in one dimension, subject to any other potential than the one of the harmonic oscillator, and I define $\hat{a}$ as stated above. I would find the ...
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Deriving $\langle H\rangle$ from average momentum and position for a LHO [closed]

Assume that we know the values of $\langle x\rangle$ and $\langle p\rangle$ for a LHO, that is in a random superposition of zeroth and first state. Derive $\langle H\rangle$. So I tried solving this ...
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2 answers
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Harmonic oscillator boundary condition issue for an impulse force

I am trying to solve an equation of an underdamped harmonic oscillator with a damping, and I get a weird boundary condition that perplexes me. Let me precise the issue, the equation is : \begin{...
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Zero frequency quantum oscillator rigorously

Problem description Consider quantum oscillator with $ \omega = 0 $. In other words, we have $$ \hat{H} = \hat{p}^2, $$ where $\hat{x}, \, \hat{p}$ are the usual coordinate and momentum operators with ...
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Harmonic Oscillator with imaginary frequency

What is an physical interpretation of these harmonic oscillators: $$\ddot{x}+i\cdot x=0$$ and $$\ddot{x}-1\cdot x=0.$$ I assume that the system satisfies this second order DE $$\ddot{x}+\omega^2\cdot ...
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In his July 1925 paper, how does Heisenberg solve the difference equation to get the first order amplitude as a function of $n$?

The question is about the paper "Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen." by Werner Heisenberg (1925) (See e.g. here). The PDF-file containing the (...
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Magnetic compound pendulum experimental setup [closed]

I plan to undertake a pendulum experiment involving a rectangular magnet swinging between two sheets of aluminium to investigate eddy current drag on the motion of the pendulum. I was however, ...
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What is the fractional frequency stability of a thermal damped harmonic oscillator?

Suppose I have a lightly driven (classical) damped harmonic oscillator at temperature $T$. Suppose $\omega$ and $Q$ are specified as well as the mean energy $\bar{E}$ in the oscillator due to the ...
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Dispersion equation with variable wavenumber

The wave equation $$u_{tt}=c^2 u_{xx}$$ is known to have a simple wave solution $u(x,t)=Ae^{i(kx-\omega t)}$ where the dispersion equation is simply $c=\omega/k$. Yet, let the wavenumber be a function ...
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Square root of number operator for quantum harmonic oscillator

Let $a$, $a^{\dagger}$ denote the standard annihilation and creation operators for the quantum harmonic oscillator, with $[a, a^{\dagger}] = \mathbb{I}$. The number operator is then defined as $a^{\...
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Critical damping oscilations - Equation in Fowles book

I'm not understanding a passage in the Fowles's book, seventh edition, equation 3.4.9. I understood that, considering: $x$ = position $\gamma$ = damping factor ${w_0}^2$ = k/m, where k is the ...
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Phases of vectors

This maybe because I am missing a whole concept itself, but how can vectors have a phase? I have been studying forced oscillations and I read that when multiplying a vector with i (sqrt(-1)) , the ...
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Quantum Harmonic Oscillator density matrix in coherent states base [closed]

I was trying to calculate matrix elements of the density operator for a 1D QHO (with Hamiltonian $\mathcal H = \hbar\omega a^\dagger a $) in the base of coherent states $\{\vert\alpha\rangle\}$ and ...
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Derivative of a real harmonic signal

simple question I can't figure out: $s(t) = A\cos(\omega t + \phi) = \mathfrak{R}[A e^{jwt}]$ is the temporal function of a real harmonic signal. I don't get how the derivative is still an imaginary ...
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Canonical transformation of the harmonic oscillator‘s Hamiltonian [closed]

I could deduce the Hamiltonian of the damped harmonic oscillator: $$ H=\frac{p^2}{2m}e^{-2 \gamma t}+\frac{m \omega_0^2 q^2}{2}e^{2 \gamma t} $$ Using the canonical transformation $Q=e^{\gamma t}q, P=...
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Generalized momentum in terms of wavefunction: Is it always $-i\hbar \partial/\partial q$?

I saw this kind of derivation several times in different notes/review/educational articles. (For example https://arxiv.org/abs/1904.06560 or http://wcchew.ece.illinois.edu/chew/course/QMALL20121005....
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Boundary conditions of anharmonic oscilation

I am reading a book talking about schrodinger's equation of diatomic molecules. $$[\frac{-\nabla_1^2}{2M}-\frac{-\nabla_2^2}{2M}+U(R_1+R_2)]\mathcal X=E_t\mathcal X$$ It says the wavefuncitons $\...
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What is the lagrangian for a collection (lattice) of harmonic oscilators?

I am a self-learner so I probably missed some exercises in classical mechanics. Anyhow, I'm learning QFT from A. Zee's book "Quantum Field Theory in a Nutshell", and on page 4, he wrote down ...
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How to show that $\psi(x)=-A\exp(-\alpha^2x^2)$ satisfies TISE for $V(x)=\frac 1 2 m\omega_0^2x^2$? [closed]

I'm struggling to approach this 'show that' question: Write down the time-independent Schrödinger differential equation for $\psi(x)$ in a one-dimensional and time-independent potential $V(x)$. In ...
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Minimum time required to reach the same phase

Let us consider three seperate springs with spring constants $k_1,k_2,k_3$. Suppose they are horizontal springs with simplicity. They are connected to a block of mass $m$ each. If they are oscillated ...
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$m$, $T$, $f$ and initial phase determines amplitude $A$ of a SHM?

I was trying to calculate all the stuffs of simple harmonic motion knowing the mass, frequency and initial phase. with $\omega$ and $m$ I can calculate $k$, $\omega^2m=k$, with $f$, I can calculate $T$...
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Why is there only one eigenket per eigenvalue of the number operator? [duplicate]

Let's "define" (I put quotes since it's not a definition, but just requiring a property) the operator $a$ such that: $$[a,a^\dagger]=1$$ then $$n=a^\dagger a$$ No other assumptions are made ...
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Frequency of vibration of a spring

Hi, actually I'm confused about the velocity formula (In blue boundary) why the velocity of that small element taken in that way.
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Choosing a Complex Trial Solution for a Forced Vibration Problem and Expressing The Driving Force in Complex Terms

When solving forced vibration problems I would always choose a trial solution for the particular (stead-state) case in the form $x(t) = A \cos{\omega t} + B \sin{\omega t} $, but reading some books I ...
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Swap entanglement from harmonic oscillator chain

I am currently analyzing a system of coupled harmonic oscillators in a thermal state and have already asked a question related to this. I have calculated the entanglement entropy of a chain of two ...
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How to theoretically raise the temperature of a thermal state?

I am currently studying thermal states of coupled harmonic oscillators and was wondering, how can I theoretically raise the temperature of a thermal state? I.e. how does the unitary transformation ...
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Entropy of harmonic oscillator in 1d, 3d and anisotropic 3d

I'm curious about the entropy of a simple harmonic oscillator in a few different scenarios: 1d: particle with mass m moving in one dimension, potential $U = \frac{1}{2} k x^2$ 3d isotropic: particle ...
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Deriving an equation for the SHM of a cantilever beam with a mass attached at the free end

I'm a high school student doing a research paper on the vibration of cantilever beams. Specifically, I am investigating the relationship between the mass on a cantilever beam and the time period of ...
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Raising and Lowering operators acting on a tensor product of oscillators [closed]

I refer to this paper: "Quantum source of entropy for black holes" by L Bombelli et al. On the fourth page, there is an expression for the state of two coupled oscillators, $a$ and $b$ (...
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What is the justification of imposing the commutation relation between $a$ and $a^\dagger$ in the quantized electromagnetic field?

From this commutation relation, many things are proved such that $a$ and $a^\dagger$ are creation and annhilation operators in some basis (the eigenstates of $n$). But is there a justification to this ...
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Bound states -> Schrodinger equation

I am having some problems to understand the meaning of bound states on quantum mechanics... In classical mechanics, since we adopt V as zero on infinity, generally bound states means $E<0$. But, in ...
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Problem dealing with creation and annihilation operators of two uncoupled oscillators [closed]

I encountered a expression of the form while computing the Lindbladian of two uncoupled harmonic oscillators $$2 b^{\dagger}a a^{\dagger}a^{\dagger}b-a^{\dagger}a^{\dagger}b b^{\dagger}a-a^{\dagger}b ...
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When will the velocity of oscillating particle become zero in simple harmonic motion?

This question came in the Jagannath University admission exam 13-14 Q) In simple harmonic oscillation, the velocity of an oscillating particle becomes zero- (a) when acceleration is maximum (b) when ...
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Natural units and $\Delta E_n$ for an harmonic potential

For context, I am currently studying perturbation theory, as well as variational methods in quantum physics. My professor uses natural units when solving problems, and he states in every problem that $...
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Non-conservative force, but equation of harmonic motion

A small disc is projected on a horizontal floor with speed $u$. Coefficient of friction between the disc and floor varies as $\mu = \mu_0+ kx$, where $x$ is the distance covered. Find distance slid by ...
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How to know if the vibration system requires one degree of freedom or two? and how to pick the right coordinate to describe the movement?

I want to know a trick that helps me understand oscillatory systems and how to pick the correct general coordinates that describe the movement, I tried everything but I still can't get the solution ...
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(Classical) Probability distribution of momentum for a harmonic oscillator (Griffiths problem 1.12)

I am trying to solve problem 1.12 in Griffith's Introduction to Quantum Mechanics, but when I compare my answer to the solutions online it is wrong. We want to find the probability distribution of ...
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Einstein solid multiplicity

I have a question about the multiplicity of an Einstein solid and the probability. I have the book An introduction to thermal physics by Schroeder. It says all accessible microstates are equally ...
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