The term "harmonic oscillator" is used to describe any system with a "linear" restoring force that tends to return the system to a 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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Pressure in Harmonic Oscillation

Classical Harmonic oscillator's energy depends on temperature as it equals $k_B$$T/2$. However, quantum harmonic oscillator energy is $(n+1/2)hf$. So, when T=0, quantum predicts motion. I have been ...
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Question about derivation of the Heisenberg Uncertainty Principle?

I am looking at the derivation presented here. The first thing I am unsure about is where the form of $\psi_0=Ae^{\frac{-m\omega x^2}{2\hbar}}$ came from. Also, is this form for all $\psi$, or just ...
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320 views

Is uniform circular motion an SHM?

I know the projection along a diameter is an SHM but is circular motion itself an SHM? If we consider the mean position to be the center of the circle then the centripetal acceleration is proportional ...
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272 views

Can the matrix element of the momentum operator be found in the momentum basis?

In Shankar's Quantum Mechanics example 7.3.4, the problem is to find $\langle n'\rvert P\lvert n\rangle$ for the harmonic oscillator. The answer contains imaginary parts; you can derive such an answer ...
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1answer
175 views

determining phase constants in SHM [closed]

A particle moves along the x axis. It is initially at the position $x$ of $0.300 m$, moving with velocity $v$ of $0.070 m/s$ and acceleration $a$ of $-0.330 m/s^2$. Suppose it moves with constant ...
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Simple harmonic motion frequency [closed]

I already did part a and b by using kinematics but I'm stuck for the next part A particle moves along the x axis. It is initially at the position 0.300 m, moving with velocity $0.070 m/s$ and ...
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469 views

Why do non-circular oscillations have angular frequency? [duplicate]

Why so the oscillations which are not circular also have angular frequency which is a quantity related to the circular motion? I have referred many articles related to simple harmonic motion where ...
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2answers
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Why does the acceleration $g$ due to gravity not affect the period of a vertically mounted spring?

For a vertically mounted spring, I was looking at the formula $ T= 2\pi \sqrt{m/k}$ for a period. Why doesn't the gravitational acceleration $g$ factor in?
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68 views

Integral over scalar product of eigenfunction of momentum operator and harmonic oscillator one

Recently I've met following expression: $$ \tag 2 \sum_{n}f(n)\int dp~ |\langle p | n\rangle|^{2} = 2\pi \sum_{n}f(n). $$ Here $|n>$ is eigenfunction of harmonic oscillator with energy $$E_{n} = ...
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89 views

Simple harmonic motion equation [duplicate]

I don't really understand this equation and was wondering if someone could help. The book says when the restoring force is directly proportional to the displacement the oscillation is called ...
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1answer
162 views

Analytical solution of Liouville's equation for classic harmonic oscillator - which book?

So the past five hours I've spend fruitlessly searching the web for any materials containing the analytical solution of the simple PDE: $$\frac{\partial f}{\partial t} - m\omega^2x\frac{\partial ...
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124 views

Damped Oscillations: Incoherence between a general solution and a specific one

In my 'Classical Dynamics of Particles and Systems, THORNTON/MARION, 5th Edition' book of classical mechanics it is given the following general solution for a damped oscillation solving ...
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What is the physical meaning of $a_{\vec{p}} \! \mid \! 0 \rangle$

$a^\dagger_{\vec{p}} \! \mid \! 0 \rangle = \mid \! p \rangle$ is interpreted as a creation of a particle with momentum $p$ from the vacuum. $a_{\vec{p}} \! \mid \! p \rangle = \mid \! 0 \rangle$ is ...
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1answer
709 views

The period of a mass-spring system including a pulley [closed]

A spring with spring constant $k$ is attached to a mass $m$ as illustrated in the following four set ups: Calculate the period of the motion for each of them. In cases (1) and (2) the period ...
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288 views

Why is $\pi$ used when calculating the value of $g$ in pendulum motion?

I am trying to intuitively understand why $\pi$ is used when calculating the value of $g$ using the harmonic motion of a pendulum: $$g ~=~\frac{4\pi^2L}{T^2}.$$ Does it have something to do with the ...
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Driven-damped oscillator: deduce the phase and/or resonant freq from amplitudes at varying freqs

Suppose that we have a fairly standard driven-damped harmonic oscillator (i.e. linear spring restoring force, linear damping force, sinusoidal driving force, etc). The catch is: we don't know the ...
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1answer
964 views

What could be the applications of Damped Oscillation? [closed]

I've been researching on Damped Oscillation for a few days for a research paper, however I couldn't find any of its applications on the web, though there are few examples of it, but they couldn't be ...
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106 views

A driven quantum harmonic oscillator (DQHO) [closed]

I'm trying to find the dispersion relation for the DQHO with Lagrangian $$ L(q,\dot q,t)=\frac{1}{2}\dot{q}^2-\frac{1}{2}\omega q^2+F(t)q $$ with $F(t)$ being non-zero for $0<t<T$. The ...
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What is the significance of clamping the center of the spring?

7. A block is hung on a spring, and the frequency $f$ of the oscillation of the system is measured. The block, a second identical block, and the spring are carried in the Space Shuttle to space. ...
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3answers
163 views

Damped simple harmonic oscillator problem

I'm supposed to calculate and draw the phase space trajectory for this: for the two different cases when and . I've never done this sort of question before, how are they done? I've tried ...
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1answer
507 views

Quantum harmonic oscillator solved by analytic method using Schrödinger equation and wave function

I'm having trouble understanding the recursion formula. Using $\xi \equiv \sqrt{m\omega/\hbar}x$ and $K = 2E/\hbar\omega$, the time-independent Schrödinger equation becomes $$\frac{d^2\psi}{d\xi ^2} ...
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Decoupling the Hamiltonian by a Discrete Fourier transform

For $N$ coupled oscillators(periodic BC) whose Hamiltonian is given as $H=\sum\limits_{i=1}^N (\frac{p_i}{2m} + \lambda(x_{i+1} - x_i)^2)$ decoupling can be achieved by change of variables by using ...
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1answer
50 views

What is the dormant common ground between harmonic mechanical oscilators and electromagnetical ones?

When I learnt electromagnetic oscillators I couldn't help but notice that it has many common stuff with mechanical ones. I know that it had to have sinusoidal equations. I (firstly, without ...
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Unstable equilibrium in a pendulum

Consider a pendulum with a bob and a massless, rigid, hinged rod attached to the bob. The bob is at rest at the bottom most position. Neglecting friction, is it possible to impart such a velocity ...
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300 views

Simple harmonic oscillator, calculate the trajectory in real space

Potential of a simple harmonic oscillator: $$U=\frac{1}{2}k x^2$$ I'm asked to calculate the trajectory of a particle moving in this potential, with initial conditions $x(t=0) = 0$ and $v(t=0)=v_0$. ...
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659 views

Green function for simple harmonic oscillator

I'm interested in examples on how to use Green function (GF)for simple harmonic oscillator (SHO)? I am from undergrad physics, so I need a fundamental math and quantum mechanical application of GF ...
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172 views

Forced Quantum Harmonic Oscillator

I'm just starting my journey to QFT and Particles physics and I have a question about the problem of QHO witch we hit with a force $F(t)$ for $ t< t' $, for which the force is zero for $t>t'$. ...
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1answer
92 views

Harmonic oscillator differential equation solution [closed]

My math book explains how to solve second order equations like : $$\ddot{x} + \omega^2x = 0$$ but I end up with the general solution : $$A\cos(\omega t) + iB\sin(\omega t).$$ Now my physics book ...
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57 views

Under-damped oscillator - how to determine speed?

At t=0 a mass M is stretched from equilibrium by x with spring constant k and damping coefficient $\gamma$. How do I determine how fast a weight is moving when it passes through equilibrium on an ...
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37 views

Confusion regarding the trial solution taken in the mathematical treatment of forced oscillations, at steady state

In the text-book that I am currently using, it is given that in case of forced oscillations, the periodic external driving force is a complex-driving force, and is generally of the form $F_0e^{jwt}$. ...
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131 views

Evolution of harmonic oscillator in path integral formulation

The unnormalized ground state of the harmonic oscillator (choosing units such that $m = \hbar = \omega = 1)$ is $$\tag{1}\psi(q,t) = \exp(-q^2/2-it/2).$$ The transition function is ...
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693 views

Is angular frequency dependent on time in damped harmonic motion?

I have a doubt regarding the angular frequency of a harmonic oscillator when there is damping involved. The frequency of the oscillation changes with time in the case of damping, but I haven't seen ...
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1answer
163 views

Why do we use sine/cosines in Simple Harmonic Motion? [duplicate]

For example, to calculate the displacement of the particle in an harmonic oscillator we do: $$x(t) = x_{\max} \cos(ωt+φ)$$ What do we find out taking the cosine of (ωt+φ)? Example Graph:
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How to derive the time period equation for a spring mass system taking into account the mass of the spring without involving energy analysis?

I want to know the way to derive the time period equation of a spring mass system accounting for the mass of the spring but not using the energy analysis method but by proceeding in the same way as we ...
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890 views

What do they mean by a pendulum losing seconds?

In many pendulum related question, a pendulum is taken do a different place where it loses seconds. For example: A second's pendulum is taken to a mountain and it loses 20 seconds per day. What ...
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356 views

3DAnisotropic oscillator in Spherical Harmonic basis-States with $L_z=0$

I've been trying to prove a rather simple looking concept. I have a code that calculates states of a 3D anisotropic oscillator in spherical coordinates. The spherical harmonics basis used to expand ...
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498 views

Simplest explanation of pendulum having a constant time period at low angles

What is the simplest explanation for the pendulum having a constant time period at low angles?
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1answer
110 views

Quantum Harmonic Oscillator - Normalizability of Annihilated Ground State

The common line of deductions in the operator analysis of the quantum harmonic oscillator goes something like this: It is derived that the action of the annihilation operator $a$ on an eigenfunction ...
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Coupled quantum harmonic oscillator

Given the following Hamiltonian for two identical linear oscillators with spring constant $k$ and interaction potential $\alpha x_1x_2$; I was asked to find the expectation value $\langle ...
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910 views

Bound states, scattering states and infinite potentials

I am doing my first semester of Quantum Mechanics and we're using Griffith's Introduction to Quantum Mechanics. As he is introducing the Dirac delta function potential he explains bound and scattering ...
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1answer
248 views

Creation and Annihilation Operators

Let $\widehat{a}^{+}_{i}$ and $\widehat{a}_{i}$ be the usual bosonic creation and annihilation operators. Consider $$\widehat{q}_{i} = \sqrt{\frac{\hbar}{2m_{i}w_{i}}}(\widehat{a}_{i}+ ...
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63 views

Sums of operators in practice

Consider a one dimensional harmonic oscillator. We have: $$\hat{n} = \hat{a}^{\dagger} \hat{a} = \frac{m \omega}{2 \hbar} \hat{x}^2 + \frac{1}{2 \hbar m \omega} \hat{p}^2 - \frac{1}{2}$$ And: ...
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1answer
425 views

Quantum field theory: field operators in terms of creation/annihilation operators

I am learning Quantum Field Theory and there is a step in my notes that I do not really understand. It starts with the classical definitions of position $q$ and momentum $p$: $$ q = ...
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455 views

Can we represent Simple Harmonic function as triangular waves?

Having studied the topic recently I found out that simple harmonic motion can represented well with sine and cosine functions.Take for example a pendulum swing which could look like : and the ...
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1answer
389 views

Given the spring constant & maximum kinetic energy; length of spring extension? [closed]

I need to understand the following question before i right my exam tomorrow. A body attached to a spring with spring constant 100 N/m executes simple harmonic motion. The maximum kinetiv energy of ...
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676 views

How can I derive the Hamiltonian of simple harmonic oscillator from this Lagrangian?

I'm working through Leonard Susskind's Theoretical Minimum: Classical Mechanics and I can't seem to understand how the Hamiltonian of a simple harmonic oscillator is derived from the following ...
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299 views

Why is $\omega = \sqrt{K/m}$ valid for a quantum oscillator?

I'm working in the 3rd edition of Modern Physics by Serway, Moses, and Moyer. In 6.6, it talks about a quantum oscillator. I don't fully understand how the definition of frequency works. Now, we ...
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1answer
188 views

Rational ratio of frequencies leads to isolating integral of motion

Padmanabhan's discussion of dynamics mentions that in general the two dimensional harmonic oscillator fills the surface of a two torus. He further notes that there will be an extra isolating integral ...
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Eigenstates of a shifted harmonic oscillator

Let's say I have a quantum harmonic oscillator $H = \omega a^\dagger a$, where $a^\dagger$ is the raising operator and $a$ is the lowering operator and $H |n\rangle = \omega n |n\rangle$. Now assume ...
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1answer
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Derivation of $a_{j}$ coefficients in the quantum harmonic oscillator

In Griffiths' book page 53, when we derive the solution of the quantum harmonic oscillator by using the power series way, we have: $$a_{j+2} = \frac{2j+1-K}{(j+1)(j+2)}\, a_{j} .$$ And for large $j$, ...