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.

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252 views

Projecting energy eigenstates of quantum harmonic oscillator into the coordinate basis

I am trying formally derive the projection of the energy eiegenstates of the 1D quantum harmonic oscillator into the $x$ basis $$ \phi_n(x) = \langle x | n \rangle = \langle x | \frac{{a^{\dagger}}^{n}...
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Do we ignore weight of the rod in vertical spring-block system?

Problem: The first spring is placed at the middle of the rod and the second one at the end as shown in the figure. If the end of the rod is slightly pulled up and released, determine the angular ...
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Experiments of quantum harmonic oscillator

I am new to quantum mechanics and this is probably a dumb question. What experiments can we do to produce a quantum harmonic oscillator? For example, for a classical harmonic oscillator, we can attach ...
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Hamiltonian of two coupled oscillators

Lets say I have this system: Two different masses with three different springs. It's not very nice to do, but I can find the eigenvalues of this system (It's not nice because the two masses are ...
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Modelling Forced Oscillations with a Single Trigonometric Function

In a physics book I came across the following solution to a differential equation modeling forced, damped, oscillating motion: $$x(t)=A\cos(\omega t+\phi)$$ where $$A=\frac{F_0}{\sqrt{m^2\left(\omega^...
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Coherent state and continuous basis of Harmonic oscillator

While deriving the coherent states of the harmonic oscillator, I used a heuristic argument : Since $\hat{a}$ and $\hat{a^\dagger}$ form a continuous basis, and $[\hat{a},\hat{a^\dagger}] = 1$, we can ...
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How to find the angular frequency of a simple pendulum using this method?

I am trying to derive an equation for angular frequency of a simple pendulum. Since the torque on the bob is only due to the horizontal component of mg, I can say that, $$ -mg(\sin\theta) l = \tau$$ $$...
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Angle in pendulum's motion

Let $\theta$ be the angle that a pendulum makes with the vertical. Is this $\theta$ the same as the $(\omega t+\delta)$ in $y=A\sin(\omega t+\delta)$?
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Eigenvalues and Eigenstates of Number operator [closed]

I have been working through a problem. It has asked me to determine the eigenstates and corresponding eigenvalues of the number operator in a quantum harmonic oscillator; $$\hat{n}=\hat{a}_+\hat{a}_-$$...
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Does changing phase constant also changes the mean position along with other things?

I am working on two different SHMs: $$P=a\sin\omega t \\ Q=a\sin(\omega t +\phi)$$ where $\omega$ = angular velocity, $\phi$ = phase constant, $P,Q$ = displacement at a instant, $a$ = amplitude Now ...
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Minimizing Coulomb potential of two electrons in non-degenerate state of a harmonic oscillator

Suppose you have a non degenerate state at harmonic oscillator. How will you distribute 2 electrons such that the coulomb potential will be minimum?
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SHM of a block attached to two springs and two light pulleys [closed]

The question I have is shown in the figure below: (Please note that all pulleys are ideal and massless). I have tried this question for a bit, and I'm attaching my work below. But the answer for the ...
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3D quantum harmonic oscillator with magnetic perturbation [closed]

Into how many distinct levels will the second excited state of a 3D quantum harmonic oscillator split, in the presence of a weak external magnetic field?
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Which equation to use for SHM?

Usually every simple harmonic question starts with the line: The block at equilibrium shifted to a position $X_0$ and released. I found this from the website: https://study.com/academy/lesson/simple-...
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Time Evolution of Coherent State (Gerry and Knight)

I'm stuck on some simple mathematics in finding the time evolved coherent state for a single-mode field from Gerry and Knight, Introductory Quantum Optics page 51. The Hamiltonian is given by $\hat{H}...
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Simple (classical, non-damped) 1D harmonic oscillator - how to represent phase space elliptical trajectory in polar form for an ellipse?

Considering a classical, non-damped 1D harmonic oscillator (e.g. mass $m$ oscillating along $x$-axis attached to spring with constant $k$) -- described by Hamiltonian (for constant energy $E$) $$E=p^2/...
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How to draw relation between Time Period and Amplitude of SHM?

Can we draw a relation between Time period and amplitude an object doing SHM? I came up with something but I’m not sure if it’s correct. $TA = k$, where k is a constant I came up with this just by ...
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How will the movement of cart create an effect on time period of oscillation of ball? [closed]

In this question for finding the time period of bob I considered using the formula $T=2\pi\sqrt{\frac{l}{g}}$. It gives (d) option as one of the answer. My question is, will the movement of cart to ...
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Why does resonance take place? [closed]

Resonance takes place when external driving frequency equals the natural frequency of an object. I know every objects have their natural frequency. But I can't see everything vibrating on its own, ...
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Solution of a polynomial boson problem

I was reading about anharmonicity that can be added to the harmonic oscillator model and I got myself wondering whether it is possible and whether there is a general method to solve (I mean ...
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Time spent by the bob of pendulum between two angles

Suppose we have a simple pendulum executing SHM about its mean position. We can easily notice that the bob of the pendulum swings back and forth making an angle, say $\theta$ about its central ...
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Two masses connected by a spring

I have a question about a problem I saw on a website A mass is attached to one end of a spring, and the other end of the spring is attached to an immovable wall. The system oscillates with period T. ...
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Is the drag coefficient the same thing as damping coefficient? Can I find the drag coefficient using the data of a damping oscillating sphere?

I am currently working on a lab experiment to find a relationship between the diameter of a sphere and its drag coefficient. I will be using a spring-mass system that oscillates vertically and then ...
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Complex notation in harmonic oscillator

For a simple harmonic oscillator, $$x(t) = A \cos(\omega t)$$ We can also write $x(t)$ as: $$x(t) = C_1 e^{i\omega t} + C_2 e^{-i\omega t}$$ Why is it necessary that the coefficients $C_1$ and $C_2$ ...
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Energy States of 2D Harmonic Oscillator with cross-terms in the potential

How can I find the energy of a particle in a 2D potential of form $V(x,y)= \frac{k}{2}(3x^2 + 3y^2 + xy + yx)$? It looks to have a close relation with Quantum Harmonic Oscillators, is it related to it?...
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Computing ${\rm tr}\left[\hat{a}^{\dagger}\hat{a}\hat{D}(\alpha)\right]$

I'm having some trouble computing ${\rm tr}\left[\hat{a}^{\dagger}\hat{a}\hat{D}(\alpha)\right]$ where $\hat{D}(\alpha)$ is the displacement operator. I tried doing this using the number state basis, ...
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Finding the drag coefficient of an object from damped simple harmonic motion (spring-mass system)

I am currently working on an experiment to determine the relationship between the diameter of a sphere and its coefficient of drag. I will be using a spring-mass system that oscillates vertically and ...
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44 views

Mass hanging from vertical spring

Why does a mass hanging from an ideal spring, under gravity, undergo periodic oscillations? Isn't gravity a damping force here? Equation for mass spring system is given by $m\ddot{y}+ky=0$ where ...
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Confusion in Quantum Harmonic Oscillator

I am confused with the meaning of the particle number of a quantum harmonic oscillator. Classically, the Hamiltonian of harmonic oscillator in phase space is defined as follows: $$H = \frac{p^{2}}{2m} ...
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Why is the velocity of a particle undergoing simple harmonic motion always positive according to this formula?

We know that the velocity of a particle undergoing simple harmonic motion, $v$, is given by- $$v=A\omega\cos(\omega t+\delta)...(i)$$ Now, depending on the phase $(\omega t+\delta)$, the value of $v$ ...
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How to find the initial phase and amplitude of a particle undergoing SHM when I know the initial position and velocity?

According to my book, we can find the amplitude $A$ and the initial phase $\delta$ if we know the initial displacement $x$ and the velocity of the particle at $x$, $v$. However, my book doesn't give ...
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Why does the 2D QHO live in a rectangle and not an ellipse?

If we have a potential of the form $V(x,y)=k_{x}x^{2}+k_{y}y^{2}$ then that potential looks like this: As you can see, this potential is clearly round. When dealing with simple harmonic oscillators (...
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What is wrong with considering $V=4(\frac{1}{2}m\omega^2x^2)$ as $V = \frac{1}{2}m\omega^2(2x)^2$? [duplicate]

To find the energy of the Harmonic Oscillator whose potential is given by $V=4(\frac{1}{2}m\omega^2x^2)$, we consider the following two Cases: CASE 1: We rewrite $V$ as $V = \frac{1}{2}m\omega^2(2x)^2$...
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Physical pendulum: moment of inertia vs center of mass

This has been confusing me for a while. Consider a solid, homogeneous rod of mass $m$ and length $l$, hanging from a fixed pivot. Its center of mass is located at $\frac{1}{2} l$, and its moment of ...
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What symmetry is responsible for the amplitude independence of the period of a simple harmonic oscillator?

In the ICTP lectures of Y. Grossman: Standard Model 1, in about minute 54:00, he leaves an informal homework for the students. He ask to find the symmetry related to the conservation of the amplitude ...
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How does a simple pendulum gain its acceleration?

The following diagram shows the direction of the acceleration of a pendulum at different states. Also, consider the below diagram. It shows the state before the pendulum starts simple harmonic motion....
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How to use the translation operator in order to find the eigenstates in a perturbed QM system?

Given a quantum mechanical system with Hamiltonian $\hat{H_0}$, introduce a perturbation $\lambda \hat{H_1}$ with $\lambda$ sufficiently small. Define now the spacial translation operator to be $\hat{...
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Damped quantum harmonic oscillator - evolution of coherent state

I am trying to solve the following Master equation (also similar to damped quantum harmonic oscillator): $$\frac{d\hat{\rho}}{dt} = \frac{\Gamma}{2}\left(2\hat{a}\hat{\rho}\hat{a}^{\dagger} - \hat{a}^...
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How did Feynman derive the potential energy of the ballistic galvanometer?

In the Feynman Lectures Vol 1. Chapter 41 - The Brownian movement, the following statement is present under Section 1 - We know the formula for the kinetic energy of rotation—it is given by Eq. (19.8)...
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Average energy of an SHM

Why do we usually calculate the average potential or kinetic energy of a simple harmonic motion with respect to time, why not with respect to position? Why even calculate average energy for an SHM? ...
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Deriving the wavefunctions of coupled QHO without diagonalizing the Hamiltonian

I have the following Hamiltonian for two coupled quantum harmonic oscillators $$H=\frac{p_{x}^{2}}{2}+\frac{\omega^{2} x^{2}}{2}+\frac{p_{y}^{2}}{2}+\frac{\Omega^{2} y^{2}}{2}+\frac{C p_{x} y}{2}.$$ ...
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Trouble following a chapter on harmonic oscillators (classical mechanics 5th edition)

I'm following Classical Mechanics, 5th Edition by Tom W.B. Kibble and Frank H. Berkshire. I'm following it since I'm interested in studying physics (although, am doing it at home myself). I've worked ...
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Time period of a physical pendulum

Considering a physical pendulum consists of a thin homogeneous rod, is it possible to find the time period without calculating the moment of inertia of the rod?
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Confusion on kinetic energy quadratic forms and eigenfrequencies

I am new to the idea of expressing kinetic energy in terms of the quadratic form. I noticed that online, people often express the kinetic energy as: $$T = \frac{1}{2} \dot q^T M \dot q \tag{1}$$ ...
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Entanglement in 2D Harmonic Oscillator system

Let's assume a 2 dimensional harmonic oscillator system with the Hamiltonian $\hat{H} = \frac{1}{2} p_x^2 + \frac{1}{2} p_y^2 + \frac{1}{2} \omega_x^2 x^2 + \frac{1}{2} \omega_y^2 y^2$ with the ...
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Quantum harmonic oscillators with momentum-position coupling

I have two coupled quantum harmonic oscillators given by the following Hamiltonian: $$H=\frac{p_{x}^{2}}{2}+\frac{\omega^{2} x^{2}}{2}+\frac{p_{y}^{2}}{2}+\frac{\Omega^{2} y^{2}}{2}+\frac{C p_{x} y}{2}...
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Velocity of a body undergoing Simple harmonic motion

Consider the equation for velocity of a body undergoing SHM- $$ v(t)=-\omega A\sin(\omega t+\phi)$$ What does this negative sign mean? Does it mean that velocity can be positive or negative at the ...
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Does the brightness of day follow simple harmonic 'motion'?

Is it really true that the brightness during the day on earth follows simple harmonic motion? My teacher mentioned this as an example but it doesn't feel obvious to me by any stretch of the ...
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Pendulum Tuned Mass Damper - Mathematical Relationship between Mass and Damping Ratio

I am doing an experiment where I built a test tower with a pendulum to act as a tuned mass damper, similar to this picture below: I want my independent variable to be the mass of the pendulum (which ...
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Double weighted pendulum of a metronome

Recently I have been studying about the pendulum and had an investigation of the double weighted pendulum of the metronome. Referring to the diagram in the following site, I have some parts that I don'...

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