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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Why we taking $ a = A \sin \phi$ and $b = A \cos\phi$ in place of constants in the Linear Harmonic Oscillator eq.?

The General Physical Solution of motion of the linear harmonic oscillator, $d^2x/dt^2 + \omega^2 x(t)= 0 $ is $$ x = a \cos \omega t + b \sin \omega t$$ where $a, b$ are two arbitrary real constants. ...
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Can we think of spontaneous emission of a photon from an excited atom as a driven harmonic oscillator problem?

This is a kind of strange question, but I'm wondering, in the context of a fully quantum field theoretic treatment of spontaneous emission, if there is any model or way of calculating the process that ...
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Compound pendulum - understanding torque of elastic force [closed]

Suppose I have the following system: The red "line" is a bar with $m=2kg$, $l=2m$, the two springs both have $k=6.1 \frac{N}{m}$. Now, if we displace the bar by $x$, we have the ...
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Why is the commutator of ladder operators non-zero?

Griffiths states that the "ladder" of stationary states for a harmonic oscillator should be unique. That should mean that for one particular energy level, there exists only one energy state. ...
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Why is the occupation of harmonic oscillator the Bose function?

Is there an intuitive reason why the occupation for the harmonic oscillator is the Bose distribution? I know that a QM-oscillator with commutation relations is a bosonic system but I have no intuition ...
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Trouble determining initial phase in oscillations (two values for $\phi$)

I need to calculate the initial phase of a particle $m=4kg$ oscillating on the $x$ axis under the influence of $$F=\frac{- \pi x}{16}N,$$ if I know that when $t=2s$, the particle passes through the ...
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Linear Harmonic motion (simple oscillator)

We know that for a simple harmonic linear oscillator, the displacement is given by $x(t)=A\sin(\omega t + \phi)$, where $\phi$ denotes the phase angle. Now as per my understanding this $\phi$ is only ...
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Initial conditions in an infinite string of masses

Assume an infinite string of masses $m$ connected by springs with constant $\kappa$. The masses in equilibrium are evenly spaced by $a$. Then the equation of motion for the $j$-th mass is $$ \ddot{\...
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Frequency dependent spring

Consider a harmonic oscillator without damping: $$m x''+ F_s=0,$$ where $F_s$ is the force induced by the spring (usually $F_s=kx$). Now, consider that the spring's response for a harmonic excitation, ...
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Magnitude of centrifugal force [closed]

A pendulum is designed for use on a gravity-free spacecraft. The pendulum consists of a mass at the end of a rod of length $l$. The pivot at the other end of the rod is forced to move in a circle of ...
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Using variation principle on quantum oscillator with general potential

Consider a general bounding potential $V(x)$. The hamiltonian is $$H = \frac{p^2}{2m} + V(x).$$ We want to apply the variation principle in equation $$F\leq F_0+\langle H-H_0\rangle_0.$$ $\langle\...
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Simple Harmonic Motion: Relation between angular motion and linear to and from motion

What is angular frequency in simple Harmonic Motion? If simple harmonic motion is a linear to and fro motion then whose angular frequency are we talking about? A linearly moving body cannot have an ...
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Basis representation for isotropic 2D quantum harmonic oscillator

The basis functions of the 2D isotropic quantum harmonic oscillator are of the form $$ \psi_{n,\ell} (r,\varphi) = A_{n\ell}(r)e^{i\ell\varphi}$$ where $A_{n\ell}(r) = \frac{\sqrt{2 \times p!}}{\sqrt{(...
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Vibrational spectroscopy energy spectrum

I have a question regarding vibrational spectroscopy. In vibrational spectroscopy we are describing the vibration of molecules with the Morse potential which gives us stationary wavefunctions that ...
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Pendulum similarity with a ball attached to a uniform disk with constant [closed]

A particle of mass $m$ is supported by a frictionless horizontal disk which rotates about a vertical axis through its center with a constant angular velocity $\omega$ . The particle is connected by a ...
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Trying to get a magnetic field behave like a quantum harmonic oscillator

Background: Let’s take a magnetic vector potential $\vec{A} = (A_1(x_1,x_2),A_2(x_1,x_2),0)^T$, s.t. $\vec{B}=(0,0,B)^T$, the Hamiltonian in the position representation can be rewritten as: $$ H = -\...
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Solution for forced harmonic motion with non-constant frequency

Is there any integral form of the solution for the equation below? $$ \ddot{y}+\omega^2(t) y = f(t) $$ where it's basically the equation for forced harmonic motion with non-constant frequency. If $\...
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Calculation of normal frequencies of a normal mode

In the Classical mechanics book by Goldstein, it is stated that if one wants to find the normal frequencies of a system, $\omega$ then the following equation has to be solved: $\left|\hat{V}-\omega^2\...
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Converting Displacement-Time to Distance-Graph for Simple Harmonic Motion

An object undergoes simple harmonic motion with the position/displacement function $$Position=\text{sin } t$$ The distance function is: \begin{equation} Distance = d(t)= \left\{ \begin{array}{lr} ...
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Why does a larger radius of an oscillating sphere on a curved track result in a shorter period?

I understand the reason mathematically based on SHM of rolling spheres where T=2π√7(R-r)/5g But I don't know what is the theoretical explanation behind larger spheres being faster. I understand the ...
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How to find velocity at natural length when total mechanical energy is negative? [closed]

I'm attempting to answer the following questions: 'A particle of mass $1/2$ kg is attached to one end of a model spring which is hanging vertically from a fixed point A. The spring has stiffness $4 \, ...
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Hydrogen atom and two-dimensional harmonic oscillator

Starting from the radial one-dimensional equation for the hydrogen atom: $[-\frac{\hbar^2}{2\mu}\frac{d^2}{dr^2}+\frac{\hbar^2l(l+1)}{2\mu r^2}-\frac{e^2}{r}]\chi_{nl}(r)=E_{nl}\chi_{nl}(r)$ where $\...
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Is there a physical interpretation of the quadrature operators of the quantised EM field in a cavity?

I am considering a cavity setup using two mirrors perpendicular to the $z$-axis separated by a distance $L$, as seen here Assuming the electric field is polarised along the $x$-axis and uniform in ...
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Proof that the QHO annihilation operator has nullity of 1

In a traditional analysis of a quantum harmonic oscillator (QHO), operators $a$ and $a^\dagger$ are introduced and it is shown that $$ H a |{n}\rangle = (E_n - \hbar \omega_0)a|{n}\rangle, $$ $$ H a^\...
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Prevent Foucault's Pendulum from an Ellipticity

I am using a 15kg dumbbell as the bob of a Foucault's pendulum attached to a string of 4 meters. However, using the common method of burning a wire to set the bob in motion, it will begin to oscillate ...
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Energy of compound harmonic oscillator [closed]

Consider two harmonic oscillators of masses $m_1,m_2$ and spring constants $k_1,k_2$ respectively. Their motions are described by equations $$u_1=A_1\sin(\omega t+\varphi_1),\qquad u_2=A_2\sin(\omega ...
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Evolution of Quantum Harmonic Oscillator into coherent state

Why does a quantum harmonic oscillator that is driven by an electromagnetic wave in cosine form with its frequency equal to the resonance frequency of the oscillator evolve from its groundstate into a ...
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How can the quantisation of the energy of an oscillator be derived from the concept of entropy?

In quantum mechanics the energy of the harmonic oscillator is quantised, which means it can only take on discrete energy levels. In an equation: $$ E_n = nhf$$ Planck did a lot of research on entropy. ...
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Energy of polyatomic molecular vibrations

I understand that the energies of a simple diatomic molecular vibration are equal to $E_n=(n+\frac12)\hbar\omega$, and I also know the accompanying eigenfunctions for these energies. I have also heard ...
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Calculating $\hat{x}^2$ and $\hat{p}^2$ - harmonic oscillator matrix form [closed]

In harmonic oscillator, we can write $\hat{x}$ and $\hat{p}$ as (I obtained the $\hat{x}$ and $\hat{p}$ by using matrix form of the ladder operators) ; $$\hat{x} = \sqrt{\frac{\hbar}{2m\omega}}\begin{...
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Why here the time period of a simple pendulum is taken to be 50 seconds [closed]

The given question is The given answer is I want to know why here T is taken to be 50 s
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1 answer
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Is the average position for the ground state of a 1D simple harmonic oscillator zero? [closed]

My textbook claims the average position of the $n$-th state of 1D simple harmonic oscillator (SHO) is zero, which means $$ \def\bra #1{\langle #1 |} \def\ket #1{| #1 \rangle} \def\braket #1{\langle #1 ...
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Energy of molecular vibrations

I have just read that the energy of a molecular vibration with frequency $\omega$ has eigenvalues of $(n+\frac12)\hbar\omega$, where $n$ is the quantum number. However, this equation really surprises ...
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Why the first-order derivative is missing when composing a Hamiltonian of simple harmonic oscillator by the lowering and the raising operators? [closed]

Given the lowering operator ($a$) and the raising operator ($a^\dagger$) $$\begin{align*} a &= \frac{1}{\sqrt{2m \hbar \omega}}\left(-i \hbar \frac{\partial}{\partial x} - i m \omega x\right) \\ a^...
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Coherent state basis

I'm learning about coherent states in a more in depth lesson the the quantum harmonic oscillator. Coherent states are eigenstates of the lowering operator. In my head this is just saying: since any ...
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How can Two particles in perpendicular S.H.M. interfere?

While studying the simple harmonic motion of a particle, I came across this concept of the interference of two particles in S.H.M. Everything made sense until it was about two mutually perpendicular ...
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Dimensional Analysis Does Not Check Out [closed]

I'm looking over the lecture notes found here, and if you scroll down to the end of page 2, the writers make the statement $ A(\omega _d) = \dfrac{f_0}{\sqrt{\omega_0^2 - \omega_d^2 + \omega_d^2\Gamma^...
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Oscillations vs Simple Harmonic Motion

From the formulas I am seeing, these are two different physical concepts. Is oscillation the motion backwards and forwards motion, such as a spring moving backwards and forwards? And then, if a mass ...
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On Quantum Harmonic Oscillator in Canonical Ensemble [duplicate]

I'm a bit confused with some results from the interpretation of the QHO in a canonical ensemble. The partition function is given by the expression $$Z = \sum_{s} e^{-\beta E_s},$$ where $s$ represents ...
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Solution for Phol's pendulum (forced and dampened harmonic oscillator)

The equation of motion for the forced and dampened harmonic oscillator for Phol's pendulum is the following: $$I_{zz}\ddot{\phi}+b\dot{\phi}+k\phi=M_0\cos{(\omega t +\phi_0)}$$ or $$\ddot{\phi} + \...
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Near Resonant Behaviour

I'm reading Landau's Mechanics and on Chapter 5 on small oscillations he says in a footnote 'The "constant" term in the phase of the oscillation also varies". I am a bit confused as to ...
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Stability of Euler-Cromer method

Euler method doesn't perform well in the context of oscillatory problems like the harmonic oscillator; the amplitude of the oscillation gets bigger with time, which clearly contradicts theory as no ...
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Can Gaussian states be entangled without two-mode squeezing?

The Duan Criterion, when written as a function of creation and annihilation operators $b$ and $b^\dagger$ depends only on $\langle b_1^\dagger b_1\rangle$, $\langle b_2^\dagger b_2\rangle$ and $\...
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What is the "spring" and what is the "mass" in a loudspeaker system? [closed]

I've been reading about how speakers work and keep seeing that it is analogous to the mass on a spring system. I'm trying to identify what is the "mass" and what is the "spring" in ...
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Definition of elongation in SHM?

If we have a weight oscillating in a vertical spring, is the elongation defined as a position (a coordinate $\vec{r}$) or as the distance (i.e $|\vec{r}-\vec{r_0}|$) between the position and the ...
2 votes
2 answers
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Interesting relationship between the 2D Harmonic Oscillator and Pauli Spin matrices

I have an isotropic 2D Harmonic Oscillator in cartesian coordinates \begin{equation} H = \frac{p_x^2}{2m} + \frac{p_y^2}{2m} + \frac{1}{2} m\omega^2 (x^2 + y^2) \end{equation} In terms of the usual ...
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Is the angular frequency in damped oscillations constant?

I've seen thousands of graphs of $A$ (amplitude) changing per cycle, which for me intuitively made me think that $\omega$ will change as well. However, that thought clashes when I think that $$\omega^...
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Scenario on the sign of hooke's law

Doubt is on the concept of the sign, not on the problem itself The main doubt is focused around the sign of $\vec{F}=-k\vec{x}$, and yes I've read a lot of posts on this page but it never ceases to be ...
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2 votes
2 answers
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Oscillations of a cylinder inside a cylinder [closed]

Please read the whole thing I'm asking for a concept not the problem itself, but I have to show the problem to explain myself Find the period of the small oscillations of a cylinder of radius r that ...
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1 answer
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Variational Method for A Symmetric double well Potential

I am given a set of trial wave functions of the form $$ Φ_n^{\pm}(x)=Ψ_{n}(x-α)\pm Ψ_{n}(x+a) $$ Where $Ψ_n$ are the $n$th Harmonic oscillator wavefunctions. in order to approximate the energy levels ...

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