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.

Filter by
Sorted by
Tagged with
0
votes
0answers
52 views

Driving a quantum system

I don't understand the mathematical form of an external driving force on a quantum system. For example, say we were driving a quantum harmonic oscillator. You would have the Hamiltonian for the ...
0
votes
0answers
36 views

Linear harmonic oscillator in quantum mechanics, why we expand the potential into Taylor series?

We have started our first problem in QM by solving Schrödinger equation for the linear harmonic oscillator potential. I noticed that the first thing we did was to expand the potential near the stable ...
3
votes
2answers
49 views

What are the maximum spring lengths of a double spring pendulum?

NOT a duplicate of Maximum length stretch of vertical spring with a mass?, I am asking about a system with two connected springs, as shown in this diagram For a single spring, you can simply equate ...
1
vote
0answers
42 views

Power Spectral Density of vacuum fluctuations in harmonic oscillator

I was thinking about the question whether vacuum fluctuations in the non relativistic quantum harmonic oscillator actually have a power spectral density. The power spectral density is (to my knowledge)...
1
vote
2answers
80 views

Effect of shortening length of pendulum on angular amplitude

A simple pendulum has a point like bob of mass $m$ and a light inextensible string. Suppose, by means of some arrangement, we shorten the string extremely slowly, so that at any instant, the ratio of ...
6
votes
1answer
184 views

Orbital angular momentum as sum of harmonic oscillators

On section 7.3 of Ballentine's "Quantum Mechanics: A Modern Development" there is a really nice argument on why the eigenvalues of the total angular momentum operator must be integer, cf. e....
0
votes
1answer
33 views

Why is max power below natural frequency

I can't understand why intuitively the max power is at the natural frequency. We were told in our notes that the power absorbed is exactly equal to the rate at which energy is dissipated. Yet, if we ...
0
votes
0answers
48 views

Why does the Klein-Gordon harmonic oscillator have non-normalizable states?

The equation given here is the vector Klein-Gordon equation for the Harmonic oscillator potential. I have read about it in some papers. As far as I know, it gives the correct energy eigenvalues but ...
1
vote
1answer
118 views

An approximation for path integral kernels [closed]

The kernel in the path integral formulation for a one dimensional system is given by \begin{equation} \label{eq:kernel1} K(x_b, t_b; x_a, t_a) = \lim_{N \to \infty} \sqrt{\frac{m}{2\pi i \...
0
votes
0answers
29 views

For what configuration of oscillators on a ring the ring will not move?

Once upon a time, on a two-dimensional plane. Imagine we place pairs of oscillators on an incompressible ring. The ring has mass $M$ and radius $R$ (though I don't think this is of any importance), ...
1
vote
1answer
48 views

Key on a string: double pendulum & consequences of not using a bob on a string?

While trying to do an at home experiment about a single pendulum, I used a key on a rope and let it swing. I came to the conclusion that this could not be a single pendulum, because of the extra '...
0
votes
0answers
36 views

Difference between reactive (coherent) and dissipative coupling in open quantum systems?

I am unsure about the physical interpretation of the different types of coupling, I understand that reactive coupling is manifested by a term within the Hamiltonian and dissipative is via a term in ...
1
vote
1answer
62 views

Is the amplitude of vibration of all particles equal in a sound wave?

[Assume ideal conditions and simple harmonic wave] My book showed resemblance in a lot of equations related to both transverse and longitudinal waves although they work very differently. the ...
0
votes
2answers
46 views

Mass-spring damper system subjected to harmonic excitation [closed]

The following graphs represent the steady state responses shown in red of a mass-spring-damper system subject to harmonic forcing shown in blue in the graphs below. How can you determine which of the ...
1
vote
2answers
103 views

Is the period of a harmonic oscillator really independent of amplitude?

Say we have a harmonic oscillator that obeys the force rule: $$F=-kx$$ Hence, the equation of motion is: $$\ddot{x}+\frac{k}{m}x=0$$ which may be solved analytically as: $$x(t)=x(0)\cos\left(\sqrt{\...
1
vote
3answers
50 views

What is the significance of the sign of the velocity for a particle executing SHM?

So while deriving equation for the velocity of particle executing SHM at any point, I noticed a difference in the result depending on what wave (sine or cosine) you chose. For $x=A\cos\omega t$: $\...
0
votes
3answers
78 views

A system of two quantum harmonic oscillators

I have two quantum mechanical harmonic oscillators with the same frequency. The Hamiltonian of the combined system is: $$ H= \hbar \omega (2a^\dagger a+b^\dagger b+2)$$ In attempting to find the ...
0
votes
1answer
40 views

In a sine wave, what is the minimum possible distance between two particles which always have the same speed?

I was going through my book when I came across this statement which was stated as a fact. 'That the minimum distance between two particles always having the same speed is $\frac{\lambda}{2}$ where $\...
3
votes
1answer
82 views

Behaviour of the path integral kernel $K(x_b,t_b;x_a,t_a)$ for the harmonic oscillator when $t_b \to t_a$

The exact propagator for the harmonic oscillator in atomic units in the path integral formulation is given by $$K(x_b,t_b;x_a,t_a) = \left( \frac{m\omega}{2 \pi i \sin(\omega t)} \right)^{1/2} \exp({...
3
votes
0answers
47 views

Temporal stability of multimode coherent states

For the standard quantum harmonic oscillator, the coherent states $\{|\alpha\rangle, \alpha\in \mathbb{C}\}$ are temporally stable. That is, $$ e^{-iH t}|\alpha\rangle = |e^{-i t} \alpha\rangle, $$ ...
1
vote
1answer
43 views

Stopping time of a damped pendulum

I'm interested in how long it would take for a damped pendulum to stop moving. I've found that the general form of a function $x(t)$ which returns the angular displacement of the pendulum given some ...
1
vote
3answers
104 views

Order of terms & harmonic approximation

I have a question regarding some basics about determining the order of a term in the context of the harmonic approximation. I am regarding the equation \begin{align} 2ml\dot{l}\dot{\phi} + ml^2\ddot{\...
0
votes
1answer
132 views

What does 'seconds squared per meter' mean?

I was wondering about the meaning of seconds^2 / meters (more specifically, period^2 / length - of a pendulum, after the square root graph of period vs length is linearized). I know that acceleration ...
2
votes
0answers
20 views

How to perform the multi-scale analysis beyond harmonic oscillations?

I occasionally see this interesting method called multi-scale analysis. From what I understood, it is used to perturbatively solve a perturbed harmonic oscillator, meaning that the equation of motion ...
3
votes
2answers
65 views

Normalisation in Harmonic Oscillators

For a harmonic oscillator, I can write $$ |\alpha \rangle = e^{-\frac{1}{2}|\alpha|^2} \Sigma_n \frac{\alpha^n}{\sqrt{n!}}|n\rangle = \sum_n\langle n|\alpha\rangle|n\rangle $$ I can also write: $$ |x \...
1
vote
1answer
67 views

Wavefunctions as Inner Product

In the following expression, n and m belong to the number basis and x is the position: $$ \langle n|m \rangle = \int_x n^*(x) m(x) dx = \int_x \langle n|x \rangle \langle x|m \rangle dx $$ I ...
0
votes
1answer
45 views

Do one dimensional SHMs follow a specific pattern when it comes to their displacements with respect to time?

I was looking at different kinds of SHMs on $x$-axis and I was wondering if the situations of positions of the particle at different times are similar for different SHMs as the most general SHM (the ...
0
votes
1answer
41 views

How can we determine the equation of motion of a specific SHM with initial displacement and velocity specified? [closed]

To explain my query, let's consider the following situation: A particle is performing SHM on the $x$-axis with the origin as its mean position. At $t=0 $, the particle is on the positive side of $x$ ...
0
votes
0answers
16 views

How to model guitar string behavior at a certain frequency

I am trying to model the behaviour of a string, when it's forced by a certain frequency. I know that when the frequency is near the resonant frequency or it's higher harmonics, a standing wave will be ...
3
votes
1answer
79 views

What is the experimental evidence for the hydrogen atom having a Coulomb potential?

It is famously impossible to deduce the shape of a drum from its spectrum, in general. In the case of the hydrogen atom, there are non-Coulomb potentials that produce the same spectral series! (See ...
0
votes
1answer
70 views

A pendulum is formed by pivoting a long thin rod of length $L$ and mass $m$ about point P on rod which is distance $d$ above the centre [closed]

How will the Time period be affected as d changes from $\frac{L}{2}$ to zero? I tried obtaining a relation between time period(T) and d: assuming the angular oscillations to be small, we can balance ...
0
votes
1answer
32 views

Vibration of a string at a frquency other than the resonance frequency

We know that if we attach an oscillator/vibrator to a string with either the natural frequency or one of its harmonics, it exhibits resonance and a pure standing wave is produced. But here, i am ...
1
vote
2answers
51 views

Can you intuitively explain the decreasing time-period of oscillation with increasing pendulum length in some cases?

Consider a rigid body suspended about an axis of rotation which, in general does not pass through it's center of mass (COM) and has a moment of inertia (MOI) $0 < I_{axis}$ about that axis. Let $...
0
votes
3answers
283 views

Where must one use $x=A \cos \omega t$ and $x=A \sin \omega t$? [duplicate]

Consider the following example: A particle oscillates with simple harmonic motion in a straight line. In the first $t$ seconds, it travels a distance $a$. In the next $t$ seconds, it travels an ...
0
votes
3answers
62 views

Is oscillatory motion possible at constant speed?

If I do pull-ups without any gradual acceleration like the following graph would it be oscillatory motion? I read in a book that the equation of oscillatory motion is $$F=- k x^n$$ where, $n$ = an ...
0
votes
1answer
70 views

Solving simple harmonic oscillator equation using $x(t)=Ce^{pt}$ solution proposal [closed]

The simple harmonic oscillator equation is, $$\frac{d^2x(t)}{dt^2}+w^2x(t)=0$$ I'm trying to solve this equation using $x(t)=Ce^{pt}$ solution proposal. $$\frac{dx(t)}{dt}=pCe^{pt}, \frac{d^{2}x(t)}{...
0
votes
1answer
36 views

Pendulum on an accelerating train with changing length

Based on my own research, I found a general solution that can model a pendulum found on an accelerating train. The following solutions are based on small angle approximations. F=-mgsin θ F≈-mgθ, ...
0
votes
1answer
21 views

Does a tuning fork's resonance frequency increase with a repulsive force on it in AFM?

I have a tuning fork sensor with a probe tip that would be used in atomic force microscopy. Am I correct that the quartz tuning fork's resonance frequency increases with a repulsive force on it and ...
0
votes
0answers
37 views

Simple Harmonic Motion of a particle attached to a spring all inside a circle

I want to understand the following problem: The way I approached this question was using Hooke' Law. I managed to solve for $\lambda$ and later made a differential equation through which I got a ...
1
vote
1answer
19 views

Overlap between relative motion eigenbasis and single-particle eigenbasis in harmonic oscillator

The energy eigenstates of two particles in a 2d isotropic harmonic oscillator can be described in terms of a product basis of one particle states (I'm using the angular momentum basis with angular ...
1
vote
1answer
30 views

Along which axis is the moment of inertia of a harmonically oscillating body calculated?

I have been learning about oscillating bodies and recently stumbled upon physical pendulums. Now the problem is i don't understand about which axis is the MOI calculated while finding the TIME PERIOD(...
0
votes
1answer
110 views

How to calculate the damping ratio of a structure with a pendulum tuned mass damper?

I'm a highschool student investigating the damping of an oscillating structure by a pendulum mass damper. The structure has an accelerometer at the top to measure the acceleration. Although I know it ...
0
votes
0answers
13 views

Sidebands in spin-oscillator coupled devices

I am researching spin-oscillator hybrid devices (particularly those coupled to NV centers), and I have come across this paper investigating the strain field of a cantilever and its effect on the NV ...
1
vote
0answers
26 views

What is the velocity of the surface used to project the motion in this experiment for SHM? [closed]

What is the velocity with which the projection surface moves in this experiment?
0
votes
1answer
47 views

Showing a system is in Simple Harmonic Motion [closed]

A force of $F=(8-2x)$ is applied to a $2kg$ object in the $X$-direction. It is released from $6m$ away from the $x=0$ point. Show that the object is in a Simple Harmonic Motion. And derive a formula ...
0
votes
0answers
26 views

Double Pendulum and newtonian mechanics [duplicate]

I am searching for a way by which I can find the displacement vector of a double pendulum as functions of time.( without using lagrangian mechanics)
0
votes
5answers
99 views

Why does SHM happen in vacuum?

Very simply put, I don't understand why SHM happens. If a spring or a pendulum can attain lowest potential energy by stopping at the mean position, why don't both of them stop at the mean position ...
0
votes
0answers
58 views

Differential equations of a forced coupled spring-pendulum system

Currently working on a problem and I can really figure out how to write the differential equations for it. Here's the situation: So we have a mass $m$ tied to the wall with a spring of constant $k$. ...
0
votes
0answers
130 views

Density of state of a 2D harmonic oscillator

I tried to find the DOS of a 2D harmonic oscillator using $2$ different methods but the results aren't the same. The energy spectrum is: $$E_n=\hbar\omega(n+1)\tag{1}$$ and the degeneracy of the $n$-...
0
votes
0answers
18 views

How do I deduce the sign for the restoring forces in a system of N masses connected by N+1 springs?

Suppose we have a system of $N$ masses connected by $N+1$ springs, where the stiffness of the springs alternates between $k$ and $2k$. We assume N to be an even number. Determine the forms of the ...

1 2
3
4 5
38