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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Is symmetrization $xp-px$ required for commutation $[H,x]=0$?

Given a Quantum Hamiltonian: $$\hat{H}=ax^2+bp^2$$ It does not commute with either $x$ or $p$. Suppose we have a Hamiltonian :$$H = k \hat{p}\hat{x}$$ why do we need it to be: $$H = k (\hat{p}\hat{x} -...
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Generalised coordinates

I am working on a scientific project for my university and I am reading a german paper (Karas: "Platten unter seitlichem Stoß") which makes use of generalised coordinates. It's about an analytical ...
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Perturbation Theory Applied to the Quantum Harmonic Oscillator [closed]

I am trying to compare the wave function obtained by exact method and by approximated method. The potential is \begin{equation} V(x)=\frac{1}{2}m\omega^2+Ax\end{equation} I found a solution but I am ...
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Solving Liouville's Equation for the Harmonic Oscillator and Fluctuating energy

I am trying to solve the Liouville eqaution of the classic harmonic oscillator with fluctuating energy and arbitrary initial condition $\rho_0$. I want to approach the problem using the method of ...
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2D Harmonic Oscillator trajectory and Ergodicity [closed]

For 2D Harmonic Oscillator $H(p,q) = \frac12(p_x^2 + p_y^2 + x^2 + y^2)$ For a fixed energy, the motion of the system is uniquely determined by the initial conditions $(p(0), q(0)) = (p_0, q_0)$. I ...
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Why fields are solutions of waves equations?

This could be extremely trivial but I am having problems figuring it out. I think I understand properly the difference between waves and fields. A field is a function valued on space or spacetime ...
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Driven harmonic oscillator: driving constant force is only temporary

Example: Let $m$ be a point mass that hangs at the equilibrium point $y_0$ on a spring fixed at the end. No damping force acts on the particle. Let $k$ be the spring constant. If I want to calculate ...
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Quantitatively writing spring mass equation in accelerating and non -accelerating frames

I have two questions, the second one is related to the first one. 1.The first question is about direction of acceleration in SHM. Now, we all know that the acceleration in SHM is directed towards ...
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Bungee jumping: why does using the law of conservation of energy give me a different $k$ than solving for $k$ when the net force is 0? [closed]

I tried solving the following problem: "A 62.0kg bungee jumper jumps from a bridge. She is tied to a bungee cord whose unstretched length is 12.0m and falls a total of 31m. Calculate the spring ...
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Representation of the displacement-operator in number basis

According to the original paper of Glauber and Cahill Ordered Expansions in Boson Amplitude Operators. K. E. Cahill and R. J. Glauber. Phys. Rev. 177 no. 5, 1857-1881 (1969). the displacement ...
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Solving a differential equation [closed]

($x$ here refers to position) $\frac{d^{2} x}{dt^{2}} + ω^{2}x$ = 0 After solving the above differential equation, we get $x(t) = Ae^{iωt} + Be^{−iωt}$ where $A$ and $B$ are some constants. My ...
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Will a pendulum oscillate on the ISS?

A question I encountered is whether a pendulum undergoing SHM on earth will oscillate on the ISS. Apparently the answer is it won't due to no effective gravity. The argument is there will be no ...
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How do you randomly draw samples from the probability density function of the quantum harmonic oscillator in MATLAB?

The Quantum Harmonic Oscillator in the ground state is specified by the following Gaussian PDF in two dimensions: $$p(x,y)= \frac{M \omega_x}{\pi h}\sqrt{ \frac{\omega_y}{\omega_x}} e^{-\frac{M}{h}(\...
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Formula for period of pendulum using energy conservation

I'm trying the derive the period of a simple pendulum using energy conservation and without calculus. I'm doing something wrong which I can't figure out. I see a lot of other derivations online ...
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Vibrational spectrum of a mass disordered chain

Consider a linear spring-mass disordered chain with a large number of masses (say $10^6$ masses). The spring constant $k_i$ of each spring is set to 1. The chain consists of atoms of mass 1 and mass 2 ...
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Perturbation in 2D oscillator

2D oscillator $H_0=\frac{P_x^2}{2m}+\frac{P_y^2}{2m}+\frac{1}{2}m\omega^2\left(x^2+y^2\right)$ with perturbation $H_1=h\omega \left(\frac{L_z^2}{h}-2\right)$ How to write the perturbation in terms ...
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Why do systems keep oscillating after a small disturbance?

I have seen many questions in the famous Indian "JEE exam" which involves a system being given a small disturbance and then oscillating in simple harmonic motion due to it. Examples: A horizontal ...
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Harmonic Oscillator: extract the ground state wave function from the propagator

I am currently studying the path integral formulation of quantum mechanics and have done a couple of problems (free particle and simple harmonic oscillator). Now, I am already done calculating the ...
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What are the necessary conditions on the energy of the Euler-Lagrange equation to have an oscillating solution? [closed]

Which condition(s) should the energy satisfy, such that the solution of the corresponding Euler-Lagrange equation is oscillating?
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Expression and explanation of quantum mechanical harmonic oscillator

I am currently studying the textbook Infrared and Raman Spectroscopy, 2nd edition, by Peter Larkin. In a section entitled Quantum Mechanical Harmonic Oscillator, the author says the following: Fig. ...
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Matrix form of 1D harmonic oscillator eigenfunctions [closed]

I've been asked to find the uncertainty in position for the harmonic oscillator where: $$\langle\hat x^2\rangle = \sum_{k}\langle\Psi_{0}|\hat x|\Psi_{k}\rangle\langle \Psi_{k}|\hat x|\Psi_{0}\rangle ...
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Fourier Transform in the Path Integral of a Harmonic Oscillator

My question comes directly from Section 7 of Srednicki's QFT textbook. I'm not able to reproduce Equation (7.5): $$\begin{aligned} [\cdots]=\frac{1}{2} \int_{-\infty}^{+\infty} \frac{d E}{2 \pi} \...
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Vibrational Spectra of a mass disordered chain

Consider a linear spring-mass disordered chain with a large number of masses (say $10^6$ masses). The spring constant $k_i$ of each spring is set to 1. The chain consists of atoms of mass 1 and mass 2 ...
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0answers
16 views

Counting number of Antinodes for Modes in 2D

In 1D cases for standing waves, if we have the 3rd harmonic on a string of length L, then there are 3 antinodes and 2 nodes in between. This means, for each wavelength, we can establish a relationship ...
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Giving some extra energy to a system of two coupled simple harmonic oscillators

As it is well known, about one simple harmonic oscillator if we give to it extra energy then there will be no change in it's frequency. About a system of two coupled SHOs the normal variables have ...
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What do Grassmann-valued terms in operators really mean?

I've read (for example, ch. 5 of Piers Coleman's book on Many Body Physics), that a simple general formulation of a fermionic driven harmonic oscillator problem is: $$H = H_0 + V(t)$$ $$H_0 = E_0 c^\...
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Reduced Green's function for a quantum harmonic oscillator

Is there a known formula for the reduced Green's function for the (1D) harmonic oscillator? As far as I am aware, the reduced Green's function could be used as a tool for working with time-...
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In what sense is the equation of motion of a damped oscillator not time-symmetric?

Consider the equation of motion of a damped oscillator $$\frac{d^2 x}{dt^2} + \gamma \frac{dx}{dt} + \omega_0^2 x = 0 \,. $$ Why does the equation of motion not satisfy time-symmetry? Is it related ...
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Aliasing of the oscillations of a one-dimensional harmonic chain

I am reading chapter 9 of the Oxford Solid State Basics and in this chapter, the author considers a chain of masses connected to each other via springs of constant $\kappa$. He solves the problem by ...
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1answer
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Why/Do Hamilton's equations Hold with Complex Variables?

I am investigating the problem of taking a hamiltonian of bilinear terms, and converting them into a bunch of uncoupled oscillators, such as in a periodic lattice. To do this, you have to introduce ...
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Are there fields, and therefore particles, which do not arise from the quantum harmonic oscillator?

BACKGROUND From what I understand of quantum optics, the creation and annihilation of photons is modeled by a quantum harmonic oscillator. The latter is obtained by applying the quantization "...
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More appropriate statement for the SHM

I know that the energy of SHM is given by $$E=\frac 12 kA^2$$ So which of these is more appropriate to say? Energy is increases because amplitude is increased. Amplitude is increased because ...
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Relative distance frame of reference partition function

I am preparing for an exam on Statistical Mechanics and came across the following integral: $$\int d^2q_1d^2q_2e^{-\frac{\beta}{2k}|q_1-q_2|^2} $$ where $q_i\in\mathbb{R}^2$ and $0\leq q_{i,x}\leq L$...
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“Natural Frequency” of A Quantum Simple Harmonic Oscillator

This is perhaps a naive question, but I have just recently been introduced to QM so here it goes: we are studying the simple Q.M. Harmonic oscillator. I understand that in the classical picture, the ...
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Physical example of degenerate groundstate in harmonic oscillator

It says in my lecture notes, that it depends on the hilbertspace in question weather the ground state $|0\rangle$ of an harmonic oscillator is degenerate or not. The ground state fullfills $$N|0\...
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Physical meaning of Damped Exponential Sinusoidal [closed]

$$x(t)=A\sin(tf+p)e^{-dt}$$ The equation is used for damped exponential sinusoidal, if used for stress decomposition does it have any physical value, especially the exponential $d$?
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How does energy of a harmonic oscillator change when one shift the center of motion? [closed]

If one has a shifted harmonic oscillator, i.e $$x=asin(wt+\phi) + x_o$$ then, the differential equation is $$\ddot{x}= -w^2 (x-x_o)$$ Would this suggest that the maximum kinetic and potential ...
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Plotting phase diagram for the velocity vs position graph. Interpolation as a possible solution? [migrated]

I need to plot a few graphs. First is of the function \begin{equation} x(t)= -e^{ -0.1 t} \cos \left( 0.995 t \right) \end{equation} and of $\dot{x}$ (time derivative function) \begin{equation} ...
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Simple harmonic motion phase shift

If the simple harmonic motion equation, suppose the acceleration comes out to be $$a=-\omega^2(x-c)$$ then what is the significance of $c$?
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How to correct for finite step-size when simulating a harmonic oscillator?

I am a grad student in experimental physics and due to the Corona-lockdown, I am now tipping my toes in programming simulations. To start, I want to simulate a harmonic oscillator with frequency $\...
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When Is It Appropriate To Use The Ladder Operator Method in Quantum Mechanics?

I'm trying to understand when it is intuitively obvious that the ladder method would be best used to tackle a problem in quantum mechanics.
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Can a harmonic oscillator have different maximum potential and kinetic energy? [closed]

I was of the opinion that the maximum attainable kinetic energy was equal to the maximum attainable potential energy which is equal to the total mechanical energy of the system at at any point of time....
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How to compute expectation value $\langle e^{iH}\rangle$ for quadratic Hamiltonians?

I have a rather basic, but actually non-trivial question: We consider a bosonic system with creation operators $\hat{a}_i^\dagger$ and annihilation operators $\hat{a}_j$ and vacuum state $|0\rangle$ ...
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Kinetic energy distribution by mass in mass-spring harmonic oscillator system

Basically I want to find a minimum working example of a non dissipative system that is able to concentrate kinetic energy on some subparts of the system. My first guess was to use the coupled ...
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1answer
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Quantum fields in position space. Could it be considered as one simple harmonic oscillator at every point of space?

Quantum fields in almost every note that a have seen are considered in momentum space. I visualize this as one harmonic oscillator at every point momentum space. then Quantum field in position space ...
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Why doesn't the stored energy of a forced oscillator change if averaged over a long time?

In Feynman Lectures Vol.1 on transients, when calculating the power of a forced oscillation it is written that the stored energy of the oscillation (the spring's kinetic energy and potential energy) ...
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Rotating wave approximation and anharmonic oscillator

Given an Hamiltonian of this form $$ \hat H = \alpha \; \hat b^\dagger \hat b + \beta \; (\hat b^\dagger + \hat b)^4 $$ where $[\hat b^\dagger, \hat b]=1$. In this video the host gets rid of all ...
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1answer
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Constantly damped pendulum

The drag force in a damped pendulum is often assumed to be either linear (viscous drag) or quadratic (air drag). However, there is another case where I have failed to find any analysis. If we have a ...
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1answer
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On the prefactor in the path integral formulation

The propagator $K$ from ($x_a,t_a$) to ($x_b,t_b$), as defined by Gottfried, can be written as $$ K(b,a) = F(t_b-t_a)\exp\left(\frac{i}{\hbar}S_{c}(b,a)\right) $$ where $S_c$ is the classical action ...
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Predict the behaviour of this system involving magnets oscillating on springs through connected coils

I came across an interesting problem recently and started to think about the situation in more detail. Here is the setup: Identical bar magnets are suspended from identical springs, with the North ...

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