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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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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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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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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Change in time period of a simple pendulum [duplicate]

There's a question in my book which goes as: A pendulum has a bob of a hollow sphere filled with a liquid and having a hole at the base. If the liquid is allowed to flow out, what would happen to the ...
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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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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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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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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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184 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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109 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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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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Deriving spring oscilation period [duplicate]

We have a spring attached to a wall and at the other part an object on a frictionless surface. I tried to calculate the spring oscilation period. I used the conservation of energy and kinematics ...
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922 views

Kinetic energy and potential energy variation over distance in SHM

When you compute the average potential energy of a horizontal spring mass system from the mean position to the positive amplitude A, the value comes out to be $\frac{1}{6}kA^2$. For the average ...
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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 = ...
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Dilation operator in CFT viewed as 'hamiltonian'?

From the commutation relations for the conformal Lie algebra, we may infer that the dilation operator plays the same role as the Hamiltonian in CFTs. The appropriate commutation relations are ...
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In what sense is a quantum field an infinite set of harmonic oscillators?

In what sense is a quantum field an infinite set of harmonic oscillators, one at each space-time point? When is it useful to think of a quantum field this way? The book I'm reading now, QFT by ...
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Finding the minimum radius of the pivoted disc

Here is a question based on Simple Harmonic Motion that I tackled just now. However I think I am having an approach to tackle this but I am not sure about it. Ouestion: A uniform disc of radius ...
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Finding the total space that an oscillating body has gone through via complex analysis

I was solving my homework and I got to an exercise that stated: An harmonic oscillating body has an equation of $$y(t) = A \sin(t)$$ Find the total space that the body has travelled during $t \in ...
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Help explain how direction change relates to acceleration [duplicate]

I was doing some simple harmonic motion problems and I came across this picture describing the position, velocity and acceleration of a linear oscillator. At the moment in time when v is 0 the ...
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210 views

Questions related to resonance/standing-waves and sound

I understand resonance for a simple harmonic oscillator but not for more complex systems like standing waves. How can I be in resonance with the normal mode in an organ pipe? I understand that the ...
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127 views

How can one calculate the phase difference between two quantum harmonic oscillator (Hermite-Gauss) states?

The analytic solutions of a quantum harmonic oscillator are given by Hermite-Gauss states, which differ in the order $n$ of the Hermite polynomials. If two such states are plotted, there will be a ...
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What is the relationship between Q-Factor and viscosity for a Newtonian fluid based damper?

I am working with a rotary damper (paddle hanging off a shaft attached to the bottom of a rotary table and into a vat of fluid). I know the viscosity of the current Newtonian fluid, I know the flow is ...
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Why isn't a pendulum clock considered a perpetual machine?

Why do people say that a perpetual machine is one that runs for eternity? Wouldn't a machine that runs taking no energy from outside but sometimes needs to be restarted, such as pendulum clock, be ...
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Energy of damped harmonic oscillator begins to increase with very large Q in numerical integration

I have numerically integrated the (reduced) homogeneous equation of a damped harmonic oscillator in order to see how the error propagates. $$\frac{d^2 X}{d\phi^2} + ...
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Angular momentum for 3D harmonic oscillator in two different bases

I know that the energy eigenstates of the 3D quantum harmonic oscillator can be characterized by three quantum numbers: $$ | n_1,n_2,n_3\rangle$$ or, if solved in the spherical coordinate system: ...
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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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Concrete example of a two-dimensional harmonic oscillator

I am a student of mathematics and some time ago I showed in general that for a two-dimensional harmonic oscillator one can apply the recurrence theorem. So far so good.. now I would like to have a ...
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271 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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Nuclear Physics: Eigenvalues of nucleus angular moment

In the shell model of nuclei, when we talk about collective motions, we describe any nucleus deformation, expanding its radius on spherical harmonics base like this $R(\theta,\phi) = ...
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Nodes in wave functions outside of the classical turning point

When looking at the solutions of the classical harmonic oscillator for instance from wikipedia one can observe that there are no nodes in the wavefunction outside the classical turning points. But I ...
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Calculation of energy eigenvalues of $\hat{x}^4$

I would appreciate help in calculating the energy eigenvalues associated with $\hat{x}^4$, with $\hat{x}$ expressed using the ladder operators for harmonic oscillators. $\hat{x} = ...
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quick question about degeneracy

For two non-interacting particles, with eigenfunctions $\phi_{n1}(x1)$ and $\phi_{n2}(x2)$ in a one-dimensional potential well $V_{(x)}$ with n = 1,2,.... Consider two spinless non-identical ...
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What happens between two harmonics?

I know that standing waves in a simple harmonic system occur when the "echo" of a wave overlaps completely with the original wave at a certain point in time, doubling the amplitude. And then, because ...
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Does the average momentum vanish for an eigenstate of the simple harmonic oscillator?

Suppose we have a simple harmonic oscillator, let's consider the ground state, $|0\rangle$ and the first excited state $|1\rangle$. $\langle 0|\hat p|0 \rangle$ is zero right? Since the particle can ...
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Hamiltonian Operator for Harmonic Oscillator

I have been solving the harmonic oscillator problem in quantum mechanics using Algebraic Method and since then I am consulting the books of Tannoudji and Griffiths for that matter. While studying both ...
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Ladder operator on momentum basis

Since in Quantum mechanics momentum operator can be written in terms of ladder operators $$\widehat{p}=-i\sqrt\frac{{\hbar m \omega}}{2}(\widehat{a}-\widehat{a}^\dagger)$$ these operators operate on ...
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Quantum oscillator, position mean value problem

A quantum harmonic oscillator of mass $m$ and frequency $\omega$ is at time $t=0$ in the state: $$ \left|\psi(t)\right> = \sum_{n=N-\Delta N}^{N+\Delta N}\left|n\right>\frac{1}{\sqrt{2\Delta N ...
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Meaning of “Simple” in Simple Pendulum and Simple Harmonic Motion?

I have gone through the Phys.SE question Why is simple harmonic motion called so?. From the 1st answer of this Question it seems to me that another type of "Harmonic motion" is "Damped Harmonic ...
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solution of pendulum equation [closed]

I have the pendulum expression $$\ddot{\theta}+\omega_{o}^{2}\sin(\theta)=0,$$ where I used a Taylor expansion for the sine term: ...
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damped and undamped oscillation graph comparison

I have been trying to solve a problem which compares the two motion. If the undamped free harmonic oscillator motion is $ x=A\sin\omega_{o}t$. So the corresponding motion in case of damped motion will ...
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Simple harmonic motion [closed]

A uniform straight rod of length $L$ is hinged at one end. It is free to oscillate in vertical plane. Time period of oscillation with small angular amplitude when a point mass of mass equal to that of ...
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Forced harmonic oscillator with two springs

Consider a vertical system of two springs in series, with a mass(50 g) between them. From below the system is driven by a vibration generator. The setup is shown here, but the picture is taken while ...
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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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Raising and lowering 3D harmonic oscillator state

A good solution to the 3D harmonic oscillator is shown here. This gives the basis states $|n,\ell\rangle$ My question is if there are some operators comparable to the 1D SHO that will raise either n ...
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Finding Tension in an Elastic String?

I know that this is a homework type question and I'm not asking a particular physics question, but I'm really desperate for help. Here's the question: I tried to divide the string to 2 parts with ...
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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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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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Trace as integral

Consider a system of two entangled harmonic oscillators. The normalised ground state is denoted by $\psi_0(x_1,x_2)$. I've been taught that a density matrix is constructed as $\rho = ...
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Uncoupling a coupled oscillator Hamiltonian by change of variables

I'm working on the problem of two entangled harmonic oscillators with Hamiltonian: $$H = \frac{1}{2} [p_1^2 + p_2^2 + k_0(x_1^2 + x_2^2) + k_1(x_1 - x_2)^2].$$ Introducing the variables $x_± = ...