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 $(V_0/ω)^2 + X_0^2 = A^2$ in SHM? $V_0$ is initial velocity and $A$ is amplitude [closed]
In the SHM chapter of the HCV book, after deriving the expression
$$ V = \omega \sqrt{\left(\frac{V_0}{\omega}\right)^2 + X_0^2 - X^2}, $$
the author simplifies it by substituting
[\left(\frac{V_0}{\...
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Reduced mass in a Harmonic Oscillator [closed]
I recently came across the harmonic oscillator and the concept of reduced mass, i.e
$$
\mu = \frac{m_1m_2}{m_1 + m_2}
$$
To begin, I understand the derivation from the point of view of sitting on one ...
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Fock Darwin wavefunction
The ground state of a 2D harmonic oscillator in magnetic field is a Gaussian wavepackets, and the spectrum of the Hamiltonian is solved by the Fock-Darwin states.
Are there textbooks (I want textbooks,...
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Tenth (even) harmonic of a open-closed tube
Can I say that a frequency (let's say f1) is the "tenth harmonic" of an open-closed tube?
I would say it does not because closed tubes only have odd harmonics, is that correct??I want to ...
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How to solve the 3 dimensional isotropic quantum harmonic oscillator with a non-zero natural length?
We are familiar with the Hamiltonian of a 3D isotropic harmonic oscillator
$$
H=\frac{1}{2m}\hat{p}^2+\frac{1}{2}m\omega^2r^2
$$
where we assume the equilibrium point of this harmonic oscillator is at ...
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Volume element in QFT
The Hamiltonian of a free scalar field in QFT is given by:
$$\hat{H} = \int{\frac{\mathrm{d}^3k}{8\pi^3}\hbar\omega_k\left(\hat{n}_k + 4\pi^3\delta^{(3)}(0)\right)}.$$
And $\hat{n}_k = \hat{a}^\...
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Confused in David Bohm's *Quantum Theory*
In discussing matrix mechanics Bohm says in chapter 16 of Quantum Theory that the $rm$ element of operator $A$, $a_{rm}$, is given by
$$a_{rm} = \int dx\ \psi^*_r(x)A\psi_m(x) \hspace{1in} (Eqn. ...
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Can all solutions from any possible dynamical systems be formulated as combinations of harmonic oscillators?
In my understanding, all physical oscillators are unit vectors of a Hilbert space that is represented from the group $SU(3)\times SU(2)\times U(1)$. Every dynamical system operates under this group (...
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Coherent state of a quantum harmonic oscillator [closed]
Let's consider a coherent state for a QHO which we denote as $\psi_{\lambda}(x,t)$.
While one can derive the appropriate expression for the coherent state, what I am interested in is a commentary made ...
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The eigenvalues of quantum harmonic oscillator [closed]
Can someone explain to me what is the green curve in the graphical rapresentation of energy levels for a quantum harmonic oscillator? I've always encounter this type of photo and nobody explains what ...
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Expectation value of position in first-order perturbation [closed]
Suppose I have a simple harmonic oscillator in ground state and a time-dependent perturbation $V(t)=f(t) \hat{x}$ that turns on at $t=0$. How do I find the expectation value of position in t goes to ...
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Proving the Plane Harmonic Oscillator move in an ellipse
My problems states that we have $r(t)$ satisfying $m\ddot{r}(t)=-kr(t)$
And in the first section we were asked to evaluate the derivative of $r(t)\times \dot{r}(t)$
And by cross product derivative law ...
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Gauge choice in showing Landau level degeneracy via the algebraic method
I'm trying to understand the algebraic method of formulating the Landau level problem better. I'm referring to David Tong's notes on the Quantum Hall effect for this (but not exactly following his ...
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It's more like a simple math question but how does this process to that? [closed]
I tried to adjust the equation multiple times to make it look like the second one, but I kept failing. Can someone please let me know how?
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Which symmetries lead to the ladder operators of the harmonic oscillator?
It seems like symmetries usually lead to ladder operators. For example in a central potential problems the conservation of angular momentum leads to angular momentum ladder operators being used in the ...
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Selection rule in three dimensional harmonic oscillator in representation $(n,l,m)$
Selection rule for transition probability of first order perturbation . I don't understand why the selection rule $$\langle l'm'n'|x|l,m,n \rangle =\delta_{m'm} \delta_{n'n} \langle l'|x|l \rangle$$ ...
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How Does Frequency Change With Damping (Underdamped Harmonic Oscillators) [closed]
I'm studying harmonic oscillators and I'm trying to model a system where both the frequency and amplitude decay over time. This is throwing me off because frequency decay is much less intuitive than ...
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How can I interpret the normal modes of this mechanical system?
How can I interpret the normal modes of this mechanical system?
The equations of motion for the system are as follows:
$$\left[\begin{array}{ccc}
m_{1}\\
& m_{2}\\
& & 0
\end{array}\...
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The period of simple harmonic motion [closed]
Am i understanding this correctly?
The harmonic oscillation of an object can be seen as the movement in the y direction along a circular path. So the time for one revolution around the circle will be ...
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What happens to the amplitude when a spring is compressed?
Say there's a spring lying on a horizontal table, with one end attached to a wall (say the left end) and it is in it's natural length. Now I compress the spring from the right end, and leave it. So ...
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Why Is There No Oscillator Representation for Operators in Planar ${\cal N}=4$ SYM Theory?
I'm studying the planar ${\cal N}=4$ Super Yang-Mills (SYM) theory and I'm curious about the representations of its operators, specifically the Hamiltonian and the dilatation operator. In many quantum ...
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Action-angle variables for three-dimensional harmonic oscillator using cylindrical coordinates
I am solving problem 19 of ch 10 of Goldstein mechanics. The problem is:
A three-dimensional harmonic oscillator has the force constant k1 in the x- and y- directions and k3 in the z-direction. Using ...
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When is minimum potential energy in simple harmonic motion not zero?
We know that in simple harmonic motion, potential energy is minimum at the mean position and it is zero since displacement is zero. So what are some cases in which minimum potential energy is not zero?...
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Article on 1D deformed quantum harmonic oscillator
Few years ago I was reading an article which I'm trying to find for quite some time but with no success so far. It was a paper about deformation of 1D quantum harmonic oscillator with continuous ...
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How is the quantum harmonic oscillator related to Fock states?
The question is basically in the title.
From what I understand, in the Fock state there is a certain number of particles in each energy level. The creation/annihilation operators create or destroy a ...
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If friction is not proportional to velocity, why do we model it as such when considering damped oscillations? [duplicate]
Early in our study of mechanics, we learn that friction is usually proportional only to normal force, without dependence on velocity. However, during our studies of damped oscillations, we often model ...
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Why am I getting this derivation of time period of pendulum in an accelerated frame wrong? [closed]
We are working in the frame of the cart and we are trying to obtain the $\tau=k\theta$ form.
So, let's write the $\tau=I_{axis}\alpha$ first for a small deviation $\theta$ from the vartical.
(The ...
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Potentials increasing faster than harmonic oscillator
I'm reading a book which says: (HO stands for harmonic oscillator):
The spectrum of the HO has equidistant energy eigenvalues. A potential that increases quicker than the HO has states which become ...
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Spherical quantum oscillator: Is energy smaller than the potential?
A particle with mass $m$ is inside the spherical quantum well $V(r)$:
\begin{equation}
V(r)=
\begin{cases}
-V_0, & \text{if}\ r<a \\
0, & \text{otherwise}
\end{cases} \...
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Understanding the dynamics of a perturbed quantum harmonic oscillator system
I'm trying to understand how quantum systems behave when they are perturbed, and I'm using the quantum harmonic oscillator as a model.
I start by implementing a symmetric gaussian shaped bump in the ...
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Investigation Results of Damping of A Spring Showing Changing Phase Angle? Why?
In an experiment I've recorded the displacement of the spring over time, investigating underdamped simple harmonic motion.
Using pre-existing formulae the data should conform to a curve of the form
$$...
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Infrared regularizing the harmonic oscillator path integral
This is from Laine and Vuorinen’s Basics of Thermal Field Theory. I do not understand why the fact that the integral over $x(\tau)$ implies the following regularization scheme. That is, I don’t ...
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Oscillating body and Doppler effect
Say we have a body attached to a spring, oscillating with some frequency $\nu$. This is one of the simplest problems studied in elementary Physics, and yet I've noticed we always study it positioning ...
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Regarding to the asymptotic solution of quantum harmonic oscillator
In quantum mechanics, the radial equation of the SHO takes the form
\begin{align}
\frac{d^2 u}{dx^2}+\left(\epsilon-x^2-\frac{l(l+1)}{x^2}\right)u=0,
\end{align}
where $x=\sqrt{\frac{m\omega}{\hbar}}r$...
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Two Simple Harmonic Motion (S.H.M.) in Perpendicular Direction
Suppose a particle is moving under the superposition of two S.H.M in the perpendicular direction... The general equation for the trajectory for the resultant motion arising due to the two component S....
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Can a harmonic oscillator never be Raman active?
Assuming we have some harmonic oscillator
\begin{equation}
H = \omega_0 (a^\dagger a + \frac{1}{2}) = \frac{p^2}{2m} + k x^2
\end{equation}
for which the excitations have even wavefunctions $\Psi_n(x)=...
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Closed expression for expected values of $\hat{p}\,\,^{2j}$ for the vacuum state
I am wondering if there is a closed expression for the expected value $\left<0\lvert \hat{p}\,\,^{2j}\lvert 0\right>$ with $j\in\mathbb{N}$, where $\left|0\right>$ is the vacuum state of the ...
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Quantum harmonic oscillator meaning
Imagine we want to solve the equations
$$
i \hbar \frac{\partial}{\partial t} \left| \Psi \right> = \hat{H}\left| \Psi \right>
$$
where $$\hat{H} = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial ...
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How can maximum kinetic energy not equal to total energy in SHM$?$ [closed]
A linear harmonic oscillator of force constant $2×10^6$$ \,\text{N}\,\text{m}^{-1}$ and amplitude $0.01 \,\text{m}$ has a total mechanical energy of $160 \,\text{J}$. Find ratio of maximum potential ...
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Generalizing Wave Equation to two strings connected at a point
Hi physics noob here with a question about strings.
I saw that you can derive the wave equation assuming an increasing density of masses and increasing spring constants in a 1-dimensional system of ...
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Quantum Harmonic Oscillator With a Linear "Perturbation"
It is well known that the energy solutions for the unidimensional quantum harmonic oscillator $V(x) = \frac{1}{2}m\omega^2x^2$ are $E_n = (n + \frac{1}{2})\hbar\omega, n \in \mathbb{N}$. In particular,...
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What are the different types of resonances in forced oscillation systems?
I'm currently studying resonances in systems subjected to forced oscillations and have come across various terms and cases that I'd like to understand more clearly. Specifically, I am analyzing a ...
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Physical meaning of Zero-Point Energy
I know that a quantum system can never have 0 energy due the Uncertainty Principle, and its lowest energy is called the Zero point Energy. However, Energy is a relative quantity (atleast in classical ...
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Derivation of Differential Equation of a Simple Pendulum [closed]
This pretty much a simple question and i seem to be making a dumb error here, but nonetheless I can't get the correct answer for the general equation of a pendulum which is :$$\ddot\theta=-\frac{g}{L}...
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Question regarding the half Harmonic Oscillator
In the normal Quantum Harmonic Oscillator (QHO), we normally use the operator method (because it's to elegant), but I recently discovered the problem in Griffiths (prob 2.42) where they ask the same ...
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Velocity Formula in SHM
In Simple Harmonic Motion in one dimension, if we assume
$$\text{Displacement}=x=A \text{sin} (\omega t+\phi)\implies \text{velocity}=v=A \omega \text{cos} (\omega t+\phi)$$
From here by substitution ...
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How can you have position basis and energy basis? [duplicate]
In Quantum Mechanics, my understanding is that we have a Hilbert space.
If we to model a particle in space we consider the space defined by the basis
$$|x\rangle$$
for each $x \in \mathbb{R}$
We then ...
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Interpretation of perpendendicularity of paths
Two particles are oscillating along two close parallel straight lines
side by side, with the same frequency and amplitudes. They pass each
other, moving in opposite directions when their ...
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How can one "encode" momentum into the wave-equation of a QM harmonic oscillator? [duplicate]
I am learning about Quantum Mechanics using Griffiths book and after reading the section about the quantum harmonic oscillator, I was left wondering how one can construct a solution to the Schrodinger ...
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Calculating the resonance frequency of a spring based on adding additional mass
I have a following problem. I have a spring of unknown spring constant and resonance frequency. I can measure only the force on the spring and the change in length of the spring. I can add mass and ...
