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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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Is a quantum harmonic oscillator always infinite dimensional?

Let us assume we have a quantum particle in a harmonic potential with the Hamiltonian $$H = \sum_n n \omega |n\rangle\langle n|$$ If I am not mistaken. Now when talking about harmonic oscillators ...
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Conceptual understanding of the Quantum Harmonic oscillator

First: When we consider a quantum particle in a harmonic (quadratic) potential we say that this particle is a harmonic oscillator, because it behaves like one. Is this correct? Now let us assume our ...
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Why can all solutions to the simple harmonic motion equation be written in terms of sines and cosines?

The defining property of SHM (simple harmonic motion) is that the force experienced at any value of displacement from the mean position is directly proportional to it and is directed towards the mean ...
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Vertical Oscillation of A Spring - Effective Spring Constant [on hold]

Good night. I am doing a physics homework and I need some small assistance. It is about the vertical oscillation of a spring. I have done the experiment, found the times, ect of the spring in series ...
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Invariant density of Harmonic oscillator

In general, dynamical systems described by a pure Hamiltonian can have an infinite number of invariant densities. In fact, each initial state determines exactly a closed path in phase-space and the ...
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harmonic oscillator position expectation value

I'm trying to get the expected value as a function of time for the position, of a harmonic oscillator hamiltonian and a state vector $|\psi\rangle=a|0\rangle+b|2\rangle$. I have $$|\psi(t)\rangle=...
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Energy of a Harmonic Progressive Wave

In a harmonic Progressive wave all the particles of the medium perform Simple Harmonic Motion. In case of SHM, when the kinetic energy is maximum the potential energy is minimum and vice versa. Is the ...
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A mass is subject to a resistive force, but no restoring force [closed]

Homework question that is set up as a damped harmonic oscillator problem. There is one issue, no restoring force, $kx$. A mass $m$ is subject to a resistive force $- b v$ but no springlike restoring ...
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Determining stiffness of spring by dynamic method [closed]

I have been measuring stiffness of spring in physics lab by dynamic method. I made graph in MS Excel $\omega^2 =f (1/m)$ and than linear regression. I need to know, if in equation $y = kx + q$ the $k$ ...
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Physical understanding of the 'box' in quantum mechanics models [closed]

Few classic quantum mechanic models will be particle in a box (infinite depth), particle in a box (finite depth), harmonic oscillator (HO) and mechanical rotation (MR). The wave functions of a QM-HO ...
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Parity of Harmonic oscillator in 2 and 3 dimensions: the case of $l_z$

From doing exercises and trying to understand their solutions, i figured in two dimensions, not all values of $l_z$ can be taken by the particles (this is to conserve parity). For example, for n=0, ...
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Computation of $e^{i \hbar \omega a^{\dagger} a} a e^{-i \hbar \omega a^{\dagger} a}$

I need to compute terms like : $$e^{i \omega t a^{\dagger} a} a e^{-i \omega t a^{\dagger} a}$$ Where $[a,a^{\dagger}]=1$ (they are the bosonic annihilation/creation operators). I wonder if there ...
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What's the relation between acceleration, position and angular velocity?

I just encountered a problem involving lift and oscillations where I found the following differential equation: $$\ddot y = -\frac{ \rho gA}{m}y = -\omega^2 y$$ What's the relation between $\ddot y$, ...
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How to find amplitude of oscillation of SHM in a non inertial frame of reference? [closed]

I'm trying to solve this problem which asks to find the amplitude of oscillation of a spring block system, which was initially hung in a stationary lift and was at rest, when the lift begins to ...
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Is the quantum harmonic oscillator energy $E = n\hbar\omega$ or $E = (n + 1/2)\hbar\omega$? (Feynman Lectures)

Please, read the whole question. I've discussed a few contradictions and so far have not found an explanation for them. I was reading The Feynman Lectures on Physics (vol. 1), the part where he talks ...
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Approximating ground-state energy without using variational principle

Given the Hamiltonian for one dimension harmonic oscillator: $$H=-\frac{\hbar^2}{2\mu}\frac{d^2}{dx^2}+\frac{\mu\omega}{2}x^2 ,$$ I need to calculate the approximate ground state energy using the ...
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Several spring coupled: can such a movement happen or is it only theoretical?

We have 6 particles. We couple them 2 by 2 with a spring of strength $K$ (as in the picture below). We then have 3 harmonic oscillators. Then we couple each oscillator by a spring of strength $S\ll K$ ...
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Hamiltonian of quantum harmonic oscillator with $\psi(x)=\delta(x)$: comparison to classical mechanics

I was just reading the question Why can't $\psi(x)=\delta(x)$ in the case of a harmonic oscillator? The accepted answer says that $\psi(x)=\delta(x)$ is a mathematically valid state, though it's not ...
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Why can't $\psi(x) = \delta(x)$ in the case of Harmonic oscillator?

In the analysis of Harmonic Oscillator, it is claimed that $\langle\hat H\rangle$ cannot be zero, why is it so? I mean $\hat H = \frac{ \hat p^2 }{2m } + \frac12 k \hat x^2$, and $$\left<x^2\...
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Why does angular frequency of a particle in SHM does not change when it's velocity is changed

$V = A \omega \sin(\omega t + \theta)$ gives velocity of a particle in SHM at time $t$. But, why does the value of $\omega$ doesn't change when $V$ is changed?
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Volume within equi-energetic surface of a classical harmonic oscillator in microcanonical ensemble

$$ V(E) = \int_{H\leq E} d\mu = \int_\Gamma d\mu\, \Theta\bigl(E-H(q,p)\bigr) . $$ To compute the volume within the equinergetic surface in the microcanonical ensamble, we use the formula above, ...
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An equation for the simple harmonic motion

I know that this seems to be a pretty easy question but for some reason I can’t find any explanation for the following equation in any high school text book or on the internet. So according to my ...
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Damped drive oscillating systems

I am currently looking at the theory of find the viscosity of and object through damped harmonic motion, and tho it can be done there is obviously a limitation with regrades to the medium. If the ...
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Simple harmonic motion phase difference problem

Two particles executing SHM of same amplitude of 20cm with same period along the same line about same equilibrium position. The maximum distance between the two is 20cm. The trouble for me is what ...
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Photon number representation of a position eigenstate

I wanted to calculate the photon number representation of a position eigenstate, so I developed as follows. \begin{align} \vert x\rangle =\sum_{n}\vert n\rangle\langle n\vert x\rangle =\sum_{n}\...
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Why we neglect the $\hbar ω/2$ in the Hamiltonian of the the Electromagnetic Field?

After the quantization of the electric and the magnetic field, we get the Hamiltonian of the electromagnetic field: $$H= \hbar ω(a^{\dagger}a +1/2) .$$ with $\hbar$ the planck constant and $a^{\...
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Understanding a reference (Cummins on 2d order ODE)

In the first page of The Impulse Response Function and Ship Motions (Cummins, 1962), it is written that: We can now write an equation, which has the appearance of a differential equation, relating ...
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1answer
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The origin of quantization

I will present a question which already is buzzing in my head for quite a time. Actually quantum physics developed as a interplay of empirical results and theoretical developments where it is ...
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1answer
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Different action-angle variables for a 2D harmonic oscillator

Consider a bidimensional harmonic oscillator. Ref. 1 says that, when the frequencies are commensurable, separating the variables in cartesian or polar coordinates leads to different action-angle ...
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Harmonic motion with friction [closed]

Consider an undamped mass and spring system with friction. The friction is not proportional to the velocity of the mass. If we solve the corresponding differential equation,the spring will oscillate ...
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1answer
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Calculating the Variance of the Thermal state

We have a Harmonic Oscillator in the Thermal state $\tau(\beta)$ which is defined $$\tau(\beta) = \frac{e^{-\beta H}}{\mathrm{Tr}(e^{-\beta H})}$$ where $Z = \mathrm{Tr}(e^{-\beta H})$ is known as ...
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Will the motion in both systems be same?

In my book, the discussion in SHM is mainly about a mass hanging vertically from a spring. But there are some exercises which contain questions which involves a mass attached to a spring and the whole ...
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Calculating initial average energy of a thermal state

We are given a system with the Hamiltionian $$H = \sum_i \omega_i a^{\dagger}_ia_i \tag{1}$$ where $a^{\dagger}_i, a_i$ are creation and anihilation operators. I did the calculations and got the ...
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The role of the harmonic oscillator eigenfunctions in quantum optics

In quantum optics we quantize the electromagnetic field and describe it using the harmonic oscillator model and the formalism of annihilation and creation operators. For the electric field operator we ...
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Conceptual understanding of Harmonic oscillators in Thermal states

my professor in the lecture defined a general thermal 'state' (which actually is a density matrix) in the following way $$\tau(\beta) = \frac{e^{-\beta H}}{Z}\tag{1}$$ where Z is the partition ...
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Conceptual understanding of quantum harmonic oscillators

The way I understand it is that we have the time-independent Schrödinger equation for a particle described by a wave function $\psi$ in a potential V(x) $$-\frac{\hbar}{2m}\frac{d^2}{dx^2} \psi + V(x)...
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Are Thermal states Harmonic oscillators?

Excuse me if I use somewhat wrong terminology. But I've always been confused about this. So firstly when we talk about a 2-state system, like a qubit, it has dimension d=2, no? But what if we ...
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Error propagation with a pendulum

The following is a 2018 F=ma exam question. I know that this isn't a homework site, but I think that my question is conceptually relevant. Here's the problem: A group of students wish to measure ...
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1answer
67 views

Is the given system going to perform a simple harmonic motion? [closed]

The system shown in the picture consists of a spring of constant $k$, a pulley (disk) of mass $M$ and radius $R$ and a block of mass $m$ is let free from rest. There is no slipping between the rope ...
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1answer
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Simple harmonic question [closed]

consider a spring with a block of mass $m$ and spring constant $k$ that is inside a lift. the cable breaks and the lift falls freely. Show that the block now executes a simple harmonic motion of ...
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1answer
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If oscillatory motion is not simple (or chaotic), is it then by definition complex?

I'm trying to logically deduce or show that a specific type of motion is complex. It is two-dimensional oscillatory motion that can be expressed by coupled second order non-linear differential ...
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1answer
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Scalar product of squeezed coherent states

Consider two states of the type $|\alpha,\xi \rangle = \hat{D}(\alpha) \hat{S}(\xi) |0\rangle$, where $D$ and $S$ are the displacement and squeeze operators, respectively, and $|0\rangle$ is a 1D ...
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Calculate weight of a barbell via measurement of period of oscillations

I have seen some videos on YouTube about some people that use "fake weights" in the gym, declaring to be able to lift much more weight than what is actually on the barbell. However, I think it should ...
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Damped harmonic oscillator with different initial amplitudes

If a string under tension is plucked, and that string goes into underdamped harmonic oscillations, the graph of the exponential decay of the amplitude looks something like this: If I were to pluck ...
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Proof $\exp(-\beta H)$ trace-class operator

Let $H=\frac{p^2}{2}+\frac{x^2}{2}\, : D(H) \to L^2(\mathbb{R})$ be the Hamiltonian of the harmonic oscillator with $m=\hbar=\omega=1$. Prove that $\exp(-\beta H)$ is a trace-class operator if $\beta&...
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How to find equation of simple harmonic motion from positional information at 3 different times?

Given a particle at three distinct position $x_1, x_2 \ and \ x_3 $from equilibrium position at different times $ t_1, t_2 \ and \ t_3 $ how can we find the amplitude, frequency and initial phase? It ...
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1answer
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Green's Function Method for a Spring and mass system [closed]

I think I've done part a) correctly and I have a general solution. However, I now have two unknown constants in my general solution and, as far as I can see, only one condition ($x(0)=-1$) with which ...
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3answers
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$Q$-factor for damped oscillator (not driven)?

How would this be defined? Some of the Q-factor definitions I have encountered include: $$Q=2\pi\frac{Energy \space stored}{Mean \space power \space per \space cycle}\\Q=2\pi\frac{Energy \space ...
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1answer
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Density of states for 3D simple harmonic oscillator

I have the thermal partition function and the density of states for the 3D simple harmonic oscillator, which are given below $$ Z(\beta) = \frac { 1 } { \left( 2 \sinh \left( \frac { \beta \omega } { ...
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1answer
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Harmonic oscillator energy difference between $(n+\frac{1}{2})h \omega$ and $(n+\frac{1}{2})\hbar \omega$

When I was studying the Harmonic Oscillator using the Schrödinger equation, I was told in lectures to pay attention to the units. There were 2 different equations given for the Energy of a Harmonic ...