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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4answers
229 views

Asymptotic frequency of nonlinear oscillator $\ddot x = -x-{\dot x}^3$ (speed cubed)

A particle oscillates according to the equation $\ddot x = -x-{\dot x}^3.$ The positive positions of the particle when it changes direction, $\dot x = 0$, are $x_1,x_2,\ldots$. I want to show that $$\...
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1answer
42 views

Equations of the spherical pendulum in different coordinates [closed]

I am trying to derive the equations of motion of a spherical pendulum, but instead of using the angles of the spherical coordinate system $\theta$ and $\varphi$, I want to use the angles $\alpha$ and $...
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0answers
82 views

Solve nonlinear, forced and damped Duffing oscillator

I am trying to solve a Duffing type equation by using Van Der Paul's method: \begin{align} \ddot{x} + \omega^2 x + 2 \gamma \dot{x} + \beta x^3 = f \cos(\Omega t) \end{align} with $$x(t) = Re[A(t) \...
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1answer
52 views

Differential equation for the anharmonic oscillator

In my project me and my partner used the engine to constrain the system so we can see the anharmonic oscillations. In our first analysis we get only odd powers in differential equation, so there ...
4
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1answer
338 views

Why is a kalimba note anharmonic?

I play a kalimba, and have also recently written a toy program that helps me tune it using a "naively applied FFT" without any sophisticated DSP. The Nyquist frequency is 24 kHz. The kalimba ...
4
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0answers
69 views

Compute probability current from WKB approximation

I struggle to reproduce a calculation from the Appendix of the paper "Anharmonic Oscillator: A Study of Perturbation Theory in Large Order", Physical Review D, 7 (6) 1973, link to abstract. ...
2
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1answer
59 views

Anharmonic oscillator without friction

My friend and I are doing a project on physics at the university. We want to describe the non-harmonic oscillator, so we built the system shown in the picture. The problem is there is friction between ...
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0answers
36 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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1answer
45 views

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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1answer
74 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{...
2
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2answers
623 views

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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0answers
204 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 ...
5
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1answer
160 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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1answer
111 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 ...
4
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1answer
99 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, ...
4
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4answers
489 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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26 views

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?
4
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3answers
284 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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1answer
54 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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1answer
693 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 ...
2
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1answer
311 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 ...
2
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1answer
81 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 $a$, ...
2
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1answer
250 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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3answers
346 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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3answers
86 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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3answers
751 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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2answers
56 views

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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1answer
110 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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0answers
68 views

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 $$ H = \frac{p^2}{2m}+\frac{1}{2}m\omega_0^2 q^2 +\mu q^4 $$ ...
5
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2answers
859 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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0answers
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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1answer
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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1answer
397 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
993 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 ...
2
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1answer
1k 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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1answer
724 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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1answer
694 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
166 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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3answers
8k views

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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0answers
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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0answers
98 views

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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2answers
400 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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2answers
6k views

Nonlinear spring $F=-kx^3$

A nonlinear spring whose restoring force is given by $F=-kx^3$ where $x$ is the displacement from equilibrium , is stretched a distance $A$. Attached to its end is a mass $m$. Calculate....(I can do ...
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1answer
88 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 ...
26
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3answers
6k views

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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1answer
565 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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1answer
246 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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1answer
638 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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1answer
373 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. $\hskip2in$ ...
2
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1answer
686 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, $m$,...