Questions tagged [anharmonic-oscillators]

Problems concerning oscillators that are not harmonic (i.e. for forces that are not linearly proportional to position).

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Anharmonic oscilator without friction

Hello, everyone Me with my friend do a project on physics in the university. We want to describe the non-harmonic oscilator, so we built the system shown in the picture. The problem is there is a ...
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33 views

What is the next higher order correction to Hooke's Law for a typical steel spring?

Near equilibrium the potential of a typical steel spring is well approximated by a quadratic function. I'm having trouble finding a reference for the what is the next-to-leading order contribution, ...
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Anharmonic field theory: pedagogical simplifications

Thank you all for so much help as I work through Zee's QFT book. Here I finally have a question about physics instead of math. Zee makes several comments that the main thing left to do in QFT is find ...
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63 views

Landau-Lifshitz skips a step in anharmonic oscillations

In chapter 28 of Landau-Lifshitz Classical Mechanics textbook they try to explain how to get the motion of a particle with the Lagrangian: $L=\frac{1}{2}m\dot{x}^{2}-\frac{1}{2}m w_{0}^{2}x^{2}-\frac{...
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What are quantum anharmonic oscillators?

I have just started studying about quantum computers (hardware side) and I am really confused about what is a quantum anharmonic oscillator. I have read somewhere that a qubit is the physical ...
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186 views

Deriving a model of a point-driven Chladni plate

Please note — this question considers a point-driven Chladni plate, not Chladni's classical experiment. I'm aware of various other questions concerning the latter here on Physics.SE. As the title ...
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148 views

Divergent Energies and Analytical Continuation - Two questions on the inverted harmonic oscillator and the inverted double well

I have two questions on the general topic of energy potentials that diverge at infinity. First of all, the inverted harmonic oscillator. I found this post on Physics SE, Inverted Harmonic oscillator. ...
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84 views

Approximating the energy levels of the anharmonic oscillator using WKB

I got stuck trying to solve this problem: Given the potential $$V(x) = \frac{m\omega^2x^2}{2}-\beta x^4,\ \beta>0$$ I need to evaluate the deviation of the energy levels from the harmonic ...
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84 views

Ladder Operators for a nonlinear oscillator

I was wondering if there was a way to construct the ladder operators for a nonlinear oscillator given by the Hamiltonian $$H=x^2+p^2+\lambda x^4$$ If we were to just calculate scattering amplitudes, ...
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285 views

How to obtain large order perturbation series for cubic anharmonic oscillator?

Consider the potential $$V(x)= \frac{x^2}{2} + gx^3.\tag{1}$$ Then the time-independent Schrödinger equation becomes $$\left(-\frac{1}{2}\frac{d^2}{dx^2} + \frac{x^2}{2} + gx^3 \right)\psi = E(g) \...
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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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Why it is not possible to get exact solution for cubic potential perturbation for 1D SHO and we have to use perturbation theory? [duplicate]

Can anyone help me in providing the process of finding exact solution in case of cubic perturbation in 1D SHO, or any suitable resource?
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166 views

Why is the ground state energy of a linearly perturbed quantum oscillator always lower than its harmonic counterpart?

I am concerned with a QHO that is linearly perturbed in $x$, i.e. $$ H = \hbar \omega \left(\hat{n} + \frac{1}{2}\right) + \lambda \underbrace{\left(\hat{b}+ \hat{b}^\dagger \right)}_{\propto \hat{x}}....
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49 views

A question about derivation of the potential energy around the stable equilibrium point

I'm learning about harmonic oscillators. In the last lecture my teacher derived the potential energy of a system that has a stable equilibrium point but not a harmonic oscillator. At the beginning of ...
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466 views

How is a quartic oscillator solved in classical mechanics?

Quantum mechanically, a quartic anharmonic oscillator with potential $$V(x)=\frac{1}{2}m\omega^2x^2+\lambda x^4$$ is dealt with perturbation theory- the approximate energies $E_n$ and energy ...
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213 views

First-order correction to energy in perturbed harmonic oscillator [closed]

I know, from the perturbation theory, that, if I have the hamiltonian $$ \hat H = \hat H_0 + \lambda \hat W$$ where $\hat H_0$ is the unperturbed hamiltonian of which I know its eigenvectors and ...
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79 views

Time period of an oscillatory motion [closed]

The question: A particle of mass $m$ is executing oscillation on the $x$-axis. Its potential energy is $U(x)= K|x|^3$, where $K$ is a positive constant. If the amplitude of oscillations is $a$, ...
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162 views

Small oscillations in the given potential

The task is to find the period of small oscillations in the potential $$U=U_0\tan^2{\Big(\frac{x^2}{a^2}\Big)}.$$ I started with finding the stable equilibrium points: $\frac{dU}{dx}=0$ $2U_{0}\...
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239 views

Hooke's full unapproximated law

It is known that the Hooke's law relating the restoring force of a spring to the distance of retraction from the equilibrium position, is only an approximation. That is, the equation $F=-kx$ is only ...
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80 views

Solving ODE equation for classical field [closed]

I would like to solve the following homogeneous, ODE: $$\left[\frac{d^2}{dt^2} + m^2\right]\phi(t) + \frac{1}{6}\lambda \phi^3(t)=0.$$ I know the solution is $$\phi(t) = \frac{z(t)}{1-\frac{\lambda}...
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462 views

What are the possible limits for when SHM is not applicable for a vertical mass-spring system?

I am not sure if this is more of a maths problem than a physics if so could admin place in the math stack. So my question is as follows. I have recently been looking at SHM in a spring-mass, as shown ...
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How can a stiffness of a spring can be $k(x+x^3)$? Compute energy total of the system [closed]

Consider the motion of a particle of mass $m$ attached to a spring of stiffness $k(x+x^3)$ where $x$ is the displacement. 1) How can look such a spring? I thought that the stiffness of a spring ...
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104 views

Why should the $\phi^4$ term necessarily cause scattering while a $x^4$ term in anharmonic oscillator only causes correction of energy levels?

Consider an anharmonic oscillator in quantum mechanics, described by the Hamiltonian $$H=\frac{p^2}{2m}+\frac{1}{2}m\omega^2x^2+bx^4.$$ The $bx^4$ term doesn't cause scattering. The effect of this ...
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Asymptotic behavior of canonical perturbation theory for the classic anharmonic oscillator

What do we know about the asymptotic behavior of the perturbative expansion for the classical anharmonic oscillator? The Hamiltonian is $$ H = \frac{p^2}{2m}+\frac{1}{2}m\omega_0^2 q^2 +\mu q^4 $$ ...
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854 views

Pendulum period [duplicate]

In plane pendulum problem, we can calculate its period using elliptic integration. In SHO problem, we use approximation such that $\theta\ll 1$ and get the period, $2\pi\sqrt{l/g}$. Is there another ...
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4k views

What is the difference between Non-harmonic oscillation, Anharmonic oscillation and Complex harmonic oscillation?

I am just wondering if the words Non-harmonic oscillation, Anharmonic oscillation and Complex harmonic oscillation mean the same thing. If not what exactly is the difference between them? Since the ...
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22 views

Velocity changes due to crystal anharmonicity?

What is the effect of cubic and higher anharmonicities of a phonon hamiltonian on the velocity of phonons?
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338 views

Why is pendulum isochronic? [duplicate]

We all know that the pendulum is isochronic, i.e. that it takes the same time regardless of the amplitude is this is less than 20 degrees. But how do we prove it mathematically? What happens when the ...
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1answer
838 views

Why do large angular displacements affect the amplitude of a simple pendulum? [duplicate]

I was conducting an experiment involving the effect of large angular displacements (10°, 20°, ... 90°) on the amplitude of a simple pendulum. So far, I have found that the period of the pendulum ...
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975 views

Quantum pure quartic oscillator

It was recently brought to my attention, that there exist analytic solutions for the quantum pure-quartic oscillator with the hamiltonian $$ \hat{H} = \frac{1}{2m} \hat{p}^2 + \frac{\lambda}{24} \hat{...
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563 views

Simple pendulm motion for larger angular displacement? [duplicate]

What will be the nature of the motion of a simple pendulum for larger angular displacement? Will that be a periodic motion? If so, will the time period increase or decrease?
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607 views

A small error in Landau & Lifschitz “Mechanics” (3rd ed.)?

I think I found a small error in Landau & Lifschitz "Mechanics" (3rd ed.). In section 28 (Anharmonic oscillations), they are discussing how to solve the following anharmonic oscillator problem: ...
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1answer
144 views

Does increasing the magnitude of the pendulum angle cause its time period to be underestimated? [duplicate]

The pendulum equation states that the time period $T=2π\sqrt{l/g}$. This is based on the small angle approximation where we approximate l $$\frac{{\rm d}^2 θ}{{\rm d}t^2 }= -\frac{g}{l}\sin θ \...
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Does amplitude affect time period for spring mass system?

I know that with the formula $T=2\pi\sqrt{\frac{m}{k}}$ the time period is not related to the amplitude. However, would amplitude matter if i do this experiment in real life. Would a greater amplitude ...
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69 views

How to write classical Hamiltonian $H = \frac{I^{2}}{2}$ in $(p,q)$ variables?

Suppose I have a completely integrable $1$ degree of freedom Hamiltonian $H(I, \varphi) = \frac{I^{2}}{2}$ written in action-angle variables $(I, \varphi) \in \mathbb{R} \times \mathbb{S}$. What ...
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Papers/books with theoretical data on anharmonic oscillators

I am trying to solve an-harmonic oscillator (with $x^4$ terms) and need theoretical data to compare with my numerical data. I have searched research papers on anharmonic oscillators and few ...
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367 views

Can there exist harmonic oscillator with asymmetric coupling?

In Classical Mechanics textbooks usually, for a coupled harmonic oscillator with two masses, coupling is taken to be same in both directions (i.e coupling constant w.r.t to m1 is same as that with ...
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87 views

Why does harmonic oscillation propagate better?

I read that EM radiation can propagate forever only if it follows the harmonic pattern. If that is true, can you explain why? Why doesn't a different oscillation propagate forever? What happens: is ...
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Why does a simple pendulum or a spring-mass system show simple harmonic motion only for small amplitudes?

I've been taught that in a simple pendulum, for small $x$, $\sin x \approx x$. We then derive the formula for the time period of the pendulum. But I still don't understand the Physics behind it. Also, ...
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531 views

QM anharmonic oscillator - Feynman diagrams, calculating free energy

I have the quantum anharmonic oscillator: $$H = \frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega^2 \hat{x}^2 + \frac{\lambda}{4!}\hat{x}^4$$ and I want to find the free energy (so essentially $\log Z(\beta)...
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245 views

How to solve the 10000th eigenvalue of the anharmonic oscillator?

Given a certain Hamiltonian, for example, $$ H = -\frac{1}{2}\frac{\partial^2}{\partial x^2 } + x^4 . $$ , what methods can we use to approximate the $n$th eigenvalue, for very large $n$? For ...
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588 views

Can quartic oscillator's energy eigenvalue, i.e. $(0+1)$-dim $\phi^4$ theory, be solved exactly?

For a potential of anharmonic oscillator like this: $$V= \frac{1}{2}m \omega^2 x^2+\frac{\lambda}{4}x^4$$ Can eigenfunction and eigenvalue of Hamiltonian be solved non-perturbatively that is ...
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349 views

Using perturbation theory to solve classical anharmonic oscillations

A point mass $m$ is allowed to move in the $x-y$ plane. Given $k$, $a$ to be the spring constant and springs' natural length respectively. The springs are parallel to the $x$ axis. $\hskip2in$ ...
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627 views

Scaling the Time Independent Schrodinger Equation

What seems like a rather simple question is causing me a lot of difficulty as my base in mathematics is weak. I want to know how I would scale the Schrodinger equation to find dependence on mass, $m$,...
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1k views

Finding Energy Eigenvalues of Simple Harmonic Oscillator for Higher Order Potentials

I am trying to find energy eigenvalues for a particle in a potential $\ V(x) = Bx^\gamma$ where $\gamma$ is a positive, even integer (2,4,6,8....). Considering boundary conditions, V(x) will go to ...
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384 views

The actual period of a pendulum at 90°. Looking for the correct formula

Do you have access to any scientific experiment which gives the period of a pendulum when the angle is $90^\circ$: this article says $T$ varies to about $18\%$ up to $90^\circ,$ so for a seconds ...
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324 views

Anharmonic oscillator on quantum mechanics

I'm studying the following Hamiltonian for an anharmonic oscillator in quantum mechanics: \begin{equation} \hat{H} = \frac{1}{2 m} \left( \hat{\vec{p}} - \frac{e}{c} \hat{\vec{A}} \right) + \frac{ m \...
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270 views

Anharmonic quantum oscillator with momentum perturbation

Given the following quantum oscillator for a particle with mass $m$, and perturbation $-\gamma P$ ($\gamma$ is a constant): $$H=\frac{P^2}{2m}+\frac{1}{2}m\omega^2X^2-\gamma P$$ One could find the ...
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Why is the simple harmonic motion idealization inaccurate?

While in my physics classes, I've always heard that the simple harmonic motion formulas are inaccurate e.g. In a pendulum, we should use them only when the angles are small; in springs, only when the ...
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118 views

Pendulum motion equation issue

The differential equation that gives the equation of motion of a pendulum where: $m$ is the mass $L$ is the distance between the pivot and the body's centre of mass $g$ is the acceleration due to ...