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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Harmonic oscillator - wavefunctions

I understand now how I can derive the lowest energy state $W_0 = \tfrac{1}{2}\hbar \omega$ of the quantum harmonic oscillator (HO) using the ladder operators. What is the easiest way to now derive ...
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351 views

Accessible microstates of harmonic oscillator in microcanonical enemble

While reading up on statistical physics, I am going through the calculation of the partition function of the harmonic oscillator in the microcanonical ensemble. The result for the partition function ...
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Why uncertainity is minimum for coherent states?

While reading for quantum damped harmonic oscillator, I came across coherent states, and I asked my prof about them and he said me it is the state at which $\Delta x\Delta y$ is minimum. I didn't ...
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179 views

Coordinate representation of quantum ladder operator?

I can't seem to figure out how to derive the coordinate representation of the $a_+$ ladder operator in quantum mechanics. I know that $a_-$ is $\sqrt{\frac{1}{2mwh}} (mwx + i\dot{p}) $ in which where ...
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98 views

Sitting on the bob of a pendulum

Walter Lewin's best performance was the pendulum demonstration, and I copy the transcript now: Would the period come out to be the same or not? [students respond] Some of you think it's ...
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Non-Degeneracy of Eigenvalues of Number Operator for Simple Harmonic Oscillator [duplicate]

Possible Duplicate: Proof that the One-Dimensional Simple Harmonic Oscillator is Non-Degenerate? I'm trying to convince myself that the eigenvalues $n$ of the number operator ...
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175 views

Force to use in harmonic oscillation through the inside of a planet

I am to find an equation for the time it takes when one falls through a planet to the other side and returns to the starting point. I have seven different sets of values - mass of object falling, mass ...
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When are Maximum Velocity and Acceleration acheived in Simple Harmonic Motion?

Im trying to get my head around SMH out of curiosity because it seems simple yet I'm not getting the concept behind some ideas. For a SMH equation : $$ x=a \sin(\omega t+\phi) $$ Under what ...
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690 views

Using Fourier Transforms to Solve Systems with springs of high frequency

I'm trying to numerically solve the differential equations of motion in a system with multiple springs of very high frequency. Because the solution is often a combination of rapidly-oscillating sine ...
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How to calculate viscous damping coefficient?

The damping of a spring is calculated with: $$[\zeta] = \frac{[c]}{\sqrt{[m][k]}}$$ Where c is the 'viscous damping coefficient' of the spring, according to Wikipedia. m is the mass, k is the spring ...
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59 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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71 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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94 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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124 views

Nonzero ground state energy of the quantum harmonic oscillator [duplicate]

Since $\frac{1}{2}\hbar \omega$ is the zero point energy of the ground state of the harmonic oscillator, then there is no way to extract this energy. Therefore, in what way is this value different ...
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91 views

Eigenstates of a density matrix of continuous variables

Consider a system of two entangled harmonic oscillators. The normalised ground state is denoted by $\psi_0(x_1,x_2)$. The reduced density matrix of the second oscillator is given by: $$\rho_2 = ...
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166 views

Harmonic Oscillator driven by a Dirac delta-like force

Consider that there is no damping for simplicity. As we know, a driving force of the form $\sin(\omega t)$ will make the oscillator at steady state vibrates at the external frequency $\omega$. What ...
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173 views

Harmonic Oscillator - Energy quantisation

The one-dimensional quantum HO can be solved in Schrodinger representation by getting Hermite Differential Equation $$ \frac{d^2y}{dx^2} - 2x \frac{dy}{dx} + \lambda y = 0 $$ with solutions $$ y(x) = ...
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578 views

Equations of motion for a pendulum in 3D?

I am trying to solve for the equations of motion to simulate a pendulum. I decided to use the spherical coordinates. The Lagrange equation is: where L = length of the rope ϕ= angle of the ...
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180 views

Harmonic Oscillator Expectation Value

In Calculating the expectation value of the quantum harmonic oscillator, I've come across a problem for finding $\left \langle x \right \rangle$ for the coherent state $\left| \alpha \right \rangle$ ...
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113 views

One body harmonic oscillator states expressed in terms of creation operators

I am reading trough chapter one of Moshinsky's "The harmonic Oscillator in Modern Physics". However i am having some trouble with the mathematics in section 8 of chapter 1. I will sketch a summary of ...
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400 views

Simple harmonic oscillator system and changes in its total energy

Suppose I have a body of mass $M$ connected to a spring (which is connected to a vertical wall) with a stiffness coefficient of $k$ on some frictionless surface. The body oscillates from point $C$ to ...
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Using eigenvalues to determine the stability/behaviour of the system

first time I've been on physics.se but have used the math and cs before... Anyway, here's my question: If we have a damped pendulum described by the equation $$y'' + ay' + b = 0 , a>0$$ Using the ...
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WKB approximation in two dimensions

Does anybody know how to implement the WKB approximation for the two-dimensional Schrodinger equation with a harmonic oscillator potential: $\frac{1}{2}\Biggl[-\biggl(\frac{\partial^2}{\partial ...
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What are you studying when you study a Harmonic Oscillator in QM?

This probably is a naive question - so please forgive a self-studier. In the text I am studying, one builds a HO by placing a particle in a potential that increases quadratically from the origin. The ...
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96 views

Different hamiltonians for quantum harmonic oscillator?

The Hamiltonian for a classical simple harmonic oscillator is $$ H = \frac{p^2}{2m} + \frac{1}{2}m\omega^2x^2$$ With the usual choice of the ladder operators $$a = ...
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Mean energy harmonic oscillator

I know that for a particle under the potential $$V(x,y,z)=\frac{k}{2}(x^2+y^2+z^2)$$ the equipartition theorem says that it contributes to the mean energy to $\frac{3k_BT}{2} $ (one half for each ...
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Conservation of energy in a quantum harmonic oscillator after a sudden change in spring constant

At a given instant of time, a harmonic oscillator undergoes a sudden change in spring constant from $k$ to $k'$. Show that for energy to be conserved in the accompanying transition, $\sqrt{k/k'}$ must ...
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Is there an equation that tells you more about the amplitude of an object which is in resonance?

I'm a high school senior and I have to write a paper about resonance and differential equations. I've been searching the Internet for a long time, but I haven't found an equation that is properly ...
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110 views

Number theoretical function applied in physics? [closed]

I have a series of number theoretic phenomena (mathematics) that I can describe exactly by the superpositions or linear combination of the below function (I know it is an inverse Fourier type). Does ...
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106 views

2nd order pertubation theory for harmonic oscillator

I'm having some trouble calculating the 2nd order energy shift in a problem. I am given the pertubation: $\hat{H}'=\alpha \hat{p}$, where $\alpha$ is a constant, and $\hat{p}$ is given by: ...
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499 views

Amplitude of a Forced Harmonic Oscillator

For an assignment in one of my maths units at uni, I've been asked to derive and solve the differential equation of motion for a forced harmonic oscillator, with the forcing function having the form ...
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251 views

Quantum harmonic Oscillator analytic method

I'm using a book from Griffiths, I got really stuck about how he arrived at the approximate solution, is it just by trying( trial solution method?), I really appreciate any help on this. ...
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970 views

Does the Fundamental Frequency in a Vibrating String NOT Necessarily Have the Strongest Amplitude?

I am doing some experiments on musical strings (guitar, piano, etc.). After performing a Fourier Transform on the sound recorded from those string vibrations, I find that the fundamental frequency is ...
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Partition function for quantum harmonic oscillator

Hi guys I'm currently trying to solve a mock exam for an exam in a few days and am a bit confused by the solutions they gave us for this exercise: Exercise: A solid is composed of N atoms which ...
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450 views

What is the correct Hamiltonian for a system of coupled quantum oscillators?

The Hamiltonian (see Eqn. 1 in Appendix 2 of this paper) for a system of coupled quantum oscillators is given as $$H=\frac{1}{2}∑_{i}p^{2}_{i}+\frac{1}{2}∑_{j,k}A_{jk}q_{i}q_{k}$$ Yet, in my QM ...
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Time period of torsion oscillation

For the oscillation of a torsion pendulum (a mechanical motion), the time period is given by $T=2\pi\sqrt{\frac{I}{C}}$ which is a result of the angular acceleration ...
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192 views

Spring oscillations and waves

Consider a block of mass $m$ attached to a spring. Let it oscillate at a frequency $f$. Now each part of the spring is in SHM. so this means a wave is propagating through this spring.bCan this wave be ...
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5answers
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Is $\langle\psi_1|p\psi_1\rangle$ necessarily 0 for eigenstates? [closed]

Is $\langle\psi_1|p\psi_1\rangle$ necessarily 0 for harmonic oscillator eigenstates? If $\Psi(x,t)= c_0\psi_0(x)e^{-iE_0t/\hbar}+c_1\psi_1(x)e^{-iE_1t/\hbar}$, is the following true? Where $p$ is ...
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How to prove that a motion is Simple Harmonic Motion (SHM)?

I would like to know how one could show and prove that a given motion is simple harmonic motion. Once given an answer, I'll apply that technique to an example I am trying to figure out. Thank you ...
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59 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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81 views

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 $j$, ...
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70 views

How can I find the motion equations of the 2-dim harmonic oscillator?

First of all: I am no physicist, so I am rather helpless. I need to find the moving equations of the 2-dim. harmonic oscillator. If it is possible it should be rather elementary, because, as I said, ...
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136 views

Energy eigenvalues of a Q.H.Oscillator with $[\hat{H},\hat{a}] = -\hbar \omega \hat{a}$ and $[\hat{H},\hat{a}^\dagger] = \hbar \omega \hat{a}^\dagger$

I just finished deriving the commutators: \begin{align} [\hat{H}, \hat{a}] &= -\hbar \omega \hat{a}\\ [\hat{H}, \hat{a}^\dagger] &= \hbar \omega \hat{a}^\dagger\\ \end{align} On the ...
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125 views

How does one subtract two light beams?

From what I understand, it seems like you can only "add" beams together. You can use a beam combiner, basically using a beam splitter in reverse, to combine two beams. In homodyne detection, you use a ...
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142 views

Strange Behavior in Spring Computer Model

To learn more about oscillatory motion which I am learning about in my high school physics class, I have created a computer model of a damped spring where the damping force is proportional to ...
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219 views

Simple step in time evolution of position operator in simple harmonic motion

When considering the 'Heisenberg' picture of the harmonic oscillator, I've come across the step: $$\begin{align} \left\langle n\left|(\hat{q_H}\hat{H}-\hat{H}\hat{q_H})\right|k\right\rangle &= ...
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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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60 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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107 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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Phase space derivation of quantum harmonic oscillator partition function

I would like to derive the partition function for the quantum Harmonic oscillator from scratch: $$\tag{1} Z = \int dp \, dx\, e^{-\beta H}.$$ The free particle appears in many textbooks. $H = ...