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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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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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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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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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
73 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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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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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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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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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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$Q$-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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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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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 ...
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Why is angular frequency $\omega = \sqrt(k/m)$ dimensionally correct?

So I'm learning about simple harmonic motion, and I came to the part where the differential equation $$\frac{\mathrm d^2x}{\mathrm dt^2} = -\frac{k}{m} x$$ is solved and simplified to $$x(t) = A\...
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Inverse of a matrix in a Path Integral

Good morning! I can't make sense of an inverse of a matrix appearing in a calculation for a Wiener Path Integral. In discretized form: $$\int \prod_{i=1}^N \frac{dx_i}{\sqrt{\pi \epsilon}} e^{-\frac{1}...
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Relation between destruction/creation operators of harmonic oscillator in QM and Second Quantization

It is well known that in elementary QM the so-called destruction/creation operators $$ a_i = \frac{Q_j + i P_j }{\sqrt{2}}, \quad a_i^* = \frac{Q_j - i P_j }{\sqrt{2}},$$ are introduced when ...
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Why is this he correct? [closed]

Here is the question: "A spring stretches 0.4m when a 2kg mass is hung from it. The spring is stretched an additional 0.2m from its equilibrium point and is released. Determine: The Max ...
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The simple harmonic oscillator model relating particles and fields in QFT

In all of the introductory Quantum Field Theory texts I gave read so far, (such as Zee, Srednicki, Luke), there is an introduction to the concept of fields as operators, following the simple harmonic ...
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The allowed energies of 3D harmonic oscillator [closed]

I'm trying to calculate the allowed energies of each state for 3D harmonic oscillator. $$ E_n = (n_x+\textstyle\frac{1}{2})\hbar \omega_x+ (n_y+\textstyle\frac{1}{2})\hbar\omega_y+ (n_z+\textstyle\...
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Proving the raising and lowering of the raising and lowering operator

I am given a written proof of $\hat A^{\dagger}[u_n] = \sqrt{n+1} \ u_{n+1}$, and from it, and told to similarly prove $\hat A[u_n] = \sqrt{n} \ u_{n-1}$. However, in the written proof for $\hat A^{\...
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Complex solutions to an Underdamped Oscillator

In many of the books talking about damped simple harmonic motion, the underdamped oscillator is treated as follows: Newton's second law says $$m\ddot{x} + r\dot{x} + sx = 0 $$where s is stiffness ...
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Spring constant and dispersion relation

In order to calculate the dispersion relation (i.e $w(k)$) for the electrons and protons, I used the following relations: $ E = ℏω$, $p = ℏk$, and I substituted them in this formula for energy: $E = ...
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Step-by-step guide to finding the phase constant in simple harmonic motion

Working on some simple harmonic motion problems involving an oscillating spring/mass system ... the usual. I never really understood exactly how to find the phase constant for the $$x(t)=Acos(wt+phase ...
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Harmonic oscillators in fluids and driven oscllations

If given a normal spring/mass system and letting the mass oscillate in a fluid say water, would it be possible for the motion of the fluid, if the fluid is moving to create a driven oscillation and ...
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Equation of coupled springs : where does this potential come from?

In this document, we try to derive the equation of two coupled springs as in this picture. At the bottom of the page 2, they say : it would be more efficient to introduce the potential energy ...
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When to use sine or cosine when computing simple harmonic motion

For simple harmonic motion (SHM), I am aware you can start of using either sine or cosine, but I am a bit confused as to when you would start off with sine rather than cosine. I know that a sine graph ...
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What would be the minimum velocity of a particle performing S.H.M.?

We were asked a simple question on a test: What is the maximum and minimum velocity of a particle performing an SHM? Note here that we're talking about a generic standard SHM here. If the maximum ...
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Energy of harmonic oscillators

I've calculated the energy of a classical harmonic oscillator (HO) as: \begin{align*} \overline E = \overline{E_K} + \overline{E_P} = \frac{\overline{p^2}}{2m} + \frac{k\overline{x^2}}{2} = \frac{...
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Why does the anisotropic harmonic oscillator has no coupling between different directions?

The hamiltonian of the anisotropic HO e.g. in 2d is typically written as $$H=\frac{1}{2m}\left(p_x^2+p_y^2\right)+\frac{1}{2}m(\omega_x^2 x^2+\omega_y^2y^2)$$ What I wonder is why there is no ...
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Oscillation experimetns [closed]

So currently I have trouble evaluating my experiment because I do not have literature values for my experiment. So the experiment was, I found the relationship between the surface area of the damping ...