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
0 answers
16 views

Determining the wavelength of a virtual quantum fluctuation?

In the context of the Casimir effect, from what I've learned about it, there does seem to be a real effect (a force) on the 2 uncharged parallel plates due to quantum fluctuations/virtual particles ...
user avatar
  • 1,412
0 votes
0 answers
42 views

Transition probability for a time dependent perturbed quantum oscillator

I'm trying to solve a time-dependant perturbation theory type of problem which I've never seen before. Time-dependant perturbation theory applies for a Hamiltonian in the form: $\hat H = \hat H_0 + V(...
user avatar
0 votes
0 answers
9 views

Effective masses of spring-mass systems [duplicate]

It is known that the effective mass of the spring in this vertical spring-mass system(figure can be viewed by the following Wiki link) is 1/3 of the mass of the spring, for example, if the mass of ...
user avatar
0 votes
0 answers
18 views

Matrix element of $\sin{\phi/2}$ within the harmonic approximation

I'm reading Bogoliubov Quasiparticles in Superconducting Qubits by Glazman and Catelani and I'm stuck of one of their derivations. It's a long review article about many things not related to the ...
user avatar
  • 1,349
0 votes
0 answers
30 views

Coherent States of a Harmonic Oscillator [closed]

I have used the definitions of the annihilation and creation operators to determine the coherent state of a harmonic oscillator. I have derived the equation $$|\alpha \rangle=e^{\frac{-\alpha^{2}}{2}}\...
user avatar
1 vote
0 answers
25 views

Hilbert space of a diatomic molecule

In molecular quantum mechanics, it is very common to model a diatomic molecule as a two-level harmonic oscillator with vibrational levels lying within the electronic states: In most of the textbooks ...
user avatar
0 votes
1 answer
40 views

Solving Schrodinger equation with a harmonic oscillator potential

This is referenced from the textbook Introduction to Quantum Mechanics by Griffith. I am learning about the application of ladder operators to solve algebraically the Shrodinger equation for harmonic ...
user avatar
  • 23
1 vote
2 answers
72 views

Using Wick's Theorem in an example with the harmonic oscillator

I understand Wick's theorem to be, $$T(x)=\mathcal{N}(x)=\sum:\textbf{all contractions}:$$ And I'm researching combinatorics and quantum theory in general. How would one connect Wicks theorem to the ...
user avatar
1 vote
0 answers
41 views

Physical solutions in harmonic oscillations

https://www.youtube.com/watch?v=4ysFC9vd3GE&list=PLUl4u3cNGP61R5sPDPKVfcFlu95wSs2Kx&index=2&ab_channel=MITOpenCourseWare I was studying harmonic oscillators following this MIT class. ...
user avatar
  • 67
9 votes
1 answer
701 views

Intuition behind the differential equation for forced oscillations

The differential equation for forced oscillation is: $$m \ddot{x} + b\dot{x}+kx = F_{o}\sin(\omega''t)$$ I don't find this equation intuitively satisfying. My mind tends to think that as $F_{o}\sin(\...
user avatar
  • 225
2 votes
0 answers
21 views

What's the amplitude of the energy loosing oscillator as a function of time?

The problem comes from 'introduction to classical mechanics' by David Morin. It is as follows: A chain with mass density $\sigma$ kg/m hangs from a spring with spring constant $k$. In the equilibrium ...
user avatar
0 votes
1 answer
17 views

Prove that horizontal velocity component equation $u=Ue^{-ky}\cos(ky-\omega t)$ is valid (Stokes problem) [closed]

I was presented with this question in my fluid mechanics lecture, however I am a bit unsure how to solve it. The problem is as follows: Knowing that the bottom plate oscillates with velocity $u=U\cos(...
user avatar
0 votes
0 answers
40 views

A phase term problem in QM path integrals

Recently I was teaching myself Feynman's path integral part of QM and I ran into a problem when deriving the propagator of a 1D harmonic oscillator. First I got $$ K(x,t;x_0,t_0)=e^{\frac{i}{\hbar}S^c}...
user avatar
0 votes
1 answer
26 views

Change in Amplitude of motion in SHM when Constant Force is applied

The Question goes as follows: NOTE: Though the Question mentions $A_2$ to be Acceleration, please treat it to be the New Amplitude (since I have my conceptual doubt over that part only) Now, I ...
user avatar
  • 21
2 votes
1 answer
67 views

What is the mass of collective oscillations?

When you step through the procedure of deriving a phonon dispersion relationship for a given crystal structure (i.e. small oscillations from equilibrium, harmonic approximation, collective coordinate ...
user avatar
10 votes
3 answers
1k views

Why is Dirac's Phase Operator Non-Hermitian?

I'm self-studying Gerry and Knight. To prove Dirac's phase operator is non-existent, the book makes the following argument. The conventions used are as follows: $\hat{n}$ is the number operator, $\hat{...
user avatar
0 votes
0 answers
61 views

Kinetic friction and damped harmonic oscillators

I was taught in school that the magnitude of the kinetic frictional force does not depend on the speed. Hence, the equation of motion of the harmonic oscillator in the presence of friction with the ...
user avatar
  • 15
0 votes
1 answer
22 views

What is the nature of motion formed by the superposition of two SHMs in the frequency ratio $1:\sqrt{2}$

When two simple harmonic motions (SHM) with frequencies in the ratio of two integers are superposed, the resulting motion is periodic. However, if SHMs with frequency ratio $1:\sqrt{2}$ are superposed,...
user avatar
3 votes
1 answer
584 views

Where is the energy going in this simple harmonic motion

Suppose i have a block of mass $M $ performing simple harmonic motion under a spring. Now suppose i gently place a particle of mass $m $ on top of it. Case 1 The mass $m$ is placed when block of mass $...
user avatar
1 vote
2 answers
55 views

Fermionic oscillator and reducible representation

Consider the fermionic oscillators $\{a, a^\dagger\} = 1$, $\{a, a\} = \{a^\dagger, a^\dagger\} = 0$. The commonly used irreducible representation is given by $|0\rangle$, $a^\dagger |0\rangle$ where $...
user avatar
  • 485
0 votes
0 answers
60 views

At resonance, there is infinite oscillation (new)

As per a previous question: Transient behavour For a driven harmonic oscillator: I was trying to show an exponential increase in amplitude using the transient solution, however I still got the sake ...
user avatar
  • 3,705
0 votes
1 answer
36 views

Existence of a unitary transform $(q,p) \rightarrow (-q, p)$

If $q$ and $p$ are the canonical position and momentum operators of a quantum harmonic oscillator, is there a unitary that transforms $(q,p)$ into $(-q, p)$? For instance, denoting the annihilation ...
user avatar
  • 357
2 votes
1 answer
55 views

Computing the partition funciton of 2 identical particles in a harmonic oscillator

Say I have two identical (fermionic) non-interacting particles in a 1D harmonic oscillator. I would like to compute the entropy of the system as the temperature $T$ varies, for which I need the ...
user avatar
  • 432
1 vote
1 answer
41 views

What happens to a quantum harmonic oscillator at $T = 0$?

Say I have two identical non-interacting particles of spin $\frac{1}{2}$ and masses $m$ in a harmonic potential with constant $k$. What would be the average kinetic energy of the gas at $T=0$? What is ...
user avatar
  • 432
5 votes
3 answers
212 views

Why does a critically damped oscillator undergo a quicker decay than an overdamped one? [duplicate]

Starting with the initial conditions, $x(0)=x_0$ and $v(0)=0$, the solution of a damped oscillator: $$\ddot{x}+2\gamma\dot{x}+\omega_0^2x=0,$$ for the overdamped case $(\gamma\gg \omega_0)$ is given ...
user avatar
1 vote
0 answers
18 views

Lateral Driven Oscillation of a bridge

I am working through a question on driven oscillations and was looking for a couple of pointers if possible. The question is regarding driven oscillations on a bridge, and we have established the ...
user avatar
1 vote
2 answers
88 views

Annihilation and creation operators in quantum harmonic oscillator

I'm new to quantum mechanics and I just have a doubt. If $\hat a$ is the annihilation operator of quantum harmonic oscillator and $ \hat a^{\dagger}$ is the creation operator, what is the value of $\...
user avatar
1 vote
1 answer
57 views

Time-dependent harmonic oscillator in classical mechanics

Consider a damped, driven harmonic oscillator: $$m\frac{d^2x}{dt^2} + \beta\frac{dx}{dt} + kx = F(t) $$ I want to write this equation for time-varying $m$, $\beta$ and $k$ which are the mass, viscous ...
user avatar
  • 149
1 vote
0 answers
61 views

What is the meaning of the time evolution of a product of coherent states of the QHO?

I am trying to analyze the dynamics of a coupled quantum harmonic oscillator (cQHO) system. The Hamiltonian of the system is given by: \begin{equation} \hat{H}_{Coupled}=\frac{1}{2m}\sum_{j}\hat{p}_{j}...
user avatar
2 votes
0 answers
46 views

Different solutions of the harmonic oscillator

This is a list question. How many different solution methods are there for the harmonic oscillator? I know three. The first is the algebraic method of the raise and lower operators, and the second is ...
user avatar
  • 1,008
2 votes
0 answers
40 views

Why doesn't angular frequency change in damped simple harmonic motion?

I recently carried out an experiment varying different factors affecting simple harmonic motion, namely friction and air resistance. Whilst carrying out research for this, I found the relationship ...
user avatar
0 votes
1 answer
49 views

Oscillation of heavy spring

I am trying to derive the solution for oscillation of a mass of a heavy spring. I have classical spring-pendulum (harmonic pendulum) in mind with one side of spring attached to ceiling and suspended ...
user avatar
  • 912
0 votes
1 answer
28 views

Initial phase of Simple Harmonic Oscillator

Starting with the following solution for SHO: $$x(t) = A\sin(\omega t+\varphi)\Rightarrow x_{0}=A\sin(\varphi)$$ $$\dot{x}(t) = v(t) = \omega A\cos(\omega t+\varphi) \Rightarrow v_{0}=\omega A\cos(\...
user avatar
0 votes
1 answer
45 views

$n$-number of creation operators on the ground state [closed]

I simply want to prove the following: for the given state, $|n\rangle = \frac{1}{\sqrt{n!}}(a^\dagger)^n|0\rangle$, show that this satisfies $\hat{N}|n\rangle = n|n\rangle$ given $\hat{N} = \hat{a}^\...
user avatar
  • 121
1 vote
0 answers
40 views

I'm getting the wrong Hamiltonian of the quantum oscillator

First, I generalised the oscillator's Hamilton's equations to complex variables: $$\frac{dz_1}{dt}=\frac{\partial (z_1^2+z_2^2)}{\partial z_1}=2z_1$$ $$\frac{dz_2}{dt}=2z_2$$ So the real world ...
user avatar
  • 917
0 votes
2 answers
53 views

Harmonic oscillator in pertubation theory-cosmology

I have a doubt at the following point, in the book "the primordial density pertubation-David H. Lith, Andrew R. Liddle" on page 383, it does that: $$\delta \ddot{\phi}_{k} + 3H\delta \dot{\...
user avatar
3 votes
2 answers
46 views

Units of damping terms for harmonic oscillator

I understand that in the equation of motion of a simple harmonic oscillator $\ddot{x} + \omega_0^2 x = 0$, $\omega_0$ has dimension inverse time. Since solutions are of the form $x(t) = A \cos (\...
user avatar
0 votes
2 answers
69 views

Finding angular frequency via integration of Newton's Second Law for a physical pendulum

For context: I am a student enrolled in AP Physics C with prior knowledge from AP Calculus AB and AP Physics 1. We just collected data for a lab to determine an experimental value for g. The setup ...
user avatar
4 votes
2 answers
1k views

Why does a spring mass system oscillate?

For simply harmonic motion, acceleration $= -\omega^2 x$, where $\omega$ is the angular frequency. Within limits of Hooke's law, the restoring force on the spring is given by $$F= -k \cdot x$$ This ...
user avatar
  • 45
1 vote
0 answers
48 views

Can $A_{nm} = |n\rangle \langle m|$ be written in terms of boson operators $b, b^\dagger$? [duplicate]

I am curious whether all operators can be written as a linear combination of product of boson operators $b, b^\dagger$. More precisely, consider the single harmonic oscillator Hilbert space $H$, whose ...
user avatar
0 votes
0 answers
65 views

Definition of Quality Factor $Q$

The stored energy definition of the quality factor $Q$from wiki Q-factor from wiki is given by $$ Q = 2\pi \frac{\text{energy stored}}{\text{energy dissipated per cycle}}= 2\pi f_0 \frac{\text{...
user avatar
0 votes
1 answer
68 views

Simple harmonic oscillator in a rocket which accelerates upward [closed]

I'm working my way through a textbook that deals with differential equations. Here's a problem that I need some help to solve: Suppose a spring with a constant 4.5 kg/sec² is attached to a body with ...
user avatar
1 vote
1 answer
81 views

Qubit system coupled to a bath of quantum harmonic oscillators

It is well known that when we consider a probe harmonic oscillators (called system) that is coupled to a reservoir of N harmonic oscillators, i.e. the Hamiltonian is written as the following, the ...
user avatar
5 votes
3 answers
108 views

How to express $|m\rangle\langle n|$ in terms of ladder operators?

Let us consider the Hamiltonian of a single harmonic oscillator, which is expressed in terms of creation/annihilation operators as $H=\hbar \omega (a^{\dagger}a+1/2)$. The eigenstates of this ...
user avatar
0 votes
1 answer
61 views

What does the total energy of a simple harmonic oscillator depend on?

For an oscillating system that undergoes simple harmonic motion, the total energy remains constant while the kinetic and potential energy constantly varies. From what is taught at school, the total ...
user avatar
  • 3
0 votes
1 answer
112 views

Relationship between spring constant and amplitude in vertical spring-mass system [closed]

Two objects of equal mass hang from independent springs of unequal spring constant and oscillate up and down. The spring of greater spring constant must have the (A) smaller amplitude of oscillation (...
user avatar
1 vote
1 answer
30 views

Expression for Steady state of Forced vibration [duplicate]

In my book under the topic Steady state of the forced oscillator, they started with the equation: $$\frac{d^2x}{dt^2}+γ\frac{dx}{dt}+ω_0^2x=fe^{jωt}$$ I know the equation for damped oscillation but it ...
user avatar
5 votes
1 answer
203 views

Entanglement in coupled harmonic oscillator system

I am considering the problem of two coupled harmonic oscillators. Ignoring factors of two, the Hamiltonian for this system is $$H=p_1^2+p_2^2 +k x_1^2+kx_2^2 +k(x_2-x_1)^2$$ One can do a nice ...
user avatar
  • 1,226
1 vote
1 answer
101 views

Harmonic oscillator propagator in Euclidean time

I'm following Nastase's book on Quantum Field Theory but this question is just about quantum mechanics in the path integral formalism. In chapter 8 he considers the propagator equation for a harmonic ...
user avatar
  • 146
1 vote
1 answer
100 views

Solving perturbed quantum harmonic oscillator [closed]

I am trying to solve the following perturbed Hamiltonian $$ H = H_0 + H' = \frac{\omega}{2}\Big(a^{\dagger}a + 1\Big) + \lambda \frac{\omega}{2}\Big(a^{\dagger}a^{\dagger}+aa\Big), $$ namely, I am ...
user avatar
  • 432

1
2 3 4 5
43