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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Using the Martin-Siggia-Rose (MSR) formalism for oscillator with general non-harmonicity
I am wondering if using the Martin-Siggia-Rose (MSR) formalism can be convenient/treatable for calculating correlation functions [or their spectral densities] of a linear [underdamped] oscillator with ...
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1answer
183 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
107 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
74 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
94 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}\...
4
votes
3answers
132 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
75 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
$$\phi(t) = \frac{z(t)}{1-\frac{\lambda}...
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3answers
179 views
What are the possible limits for when SHM is not applicable for a vertical mass-spring system?
I am not sure if this is more of a maths problem than a physics if so could admin place in the math stack.
So my question is as follows. I have recently been looking at SHM in a spring-mass, as shown ...
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2answers
46 views
How can a stiffness of a spring can be $k(x+x^3)$? Compute energy total of the system [closed]
Consider the motion of a particle of mass $m$ attached to a spring of stiffness $k(x+x^3)$ where $x$ is the displacement.
1) How can look such a spring? I thought that the stiffness of a spring ...
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1answer
84 views
Why should the $\phi^4$ term necessarily cause scattering while a $x^4$ term in anharmonic oscillator only causes correction of energy levels?
Consider an anharmonic oscillator in quantum mechanics, described by the Hamiltonian $$H=\frac{p^2}{2m}+\frac{1}{2}m\omega^2x^2+bx^4.$$ The $bx^4$ term doesn't cause scattering. The effect of this ...
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0answers
57 views
Asymptotic behavior of canonical perturbation theory for the classic anharmonic oscillator
What do we know about the asymptotic behavior of the perturbative expansion for the classical anharmonic oscillator?
The Hamiltonian is
$$
H = \frac{p^2}{2m}+\frac{1}{2}m\omega_0^2 q^2 +\mu q^4
$$
...
5
votes
2answers
831 views
Pendulum period [duplicate]
In plane pendulum problem, we can calculate its period using elliptic integration.
In SHO problem, we use approximation such that $\theta\ll 1$ and get the period, $2\pi\sqrt{l/g}$.
Is there another ...
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0answers
3k views
What is the difference between Non-harmonic oscillation, Anharmonic oscillation and Complex harmonic oscillation?
I am just wondering if the words Non-harmonic oscillation, Anharmonic oscillation and Complex harmonic oscillation mean the same thing. If not what exactly is the difference between them?
Since the ...
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1answer
22 views
Velocity changes due to crystal anharmonicity?
What is the effect of cubic and higher anharmonicities of a phonon hamiltonian on the velocity of phonons?
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1answer
259 views
Why is pendulum isochronic? [duplicate]
We all know that the pendulum is isochronic, i.e. that it takes the same time regardless of the amplitude is this is less than 20 degrees.
But how do we prove it mathematically?
What happens when the ...
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1answer
656 views
Why do large angular displacements affect the amplitude of a simple pendulum? [duplicate]
I was conducting an experiment involving the effect of large angular displacements (10°, 20°, ... 90°) on the amplitude of a simple pendulum. So far, I have found that the period of the pendulum ...
2
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1answer
832 views
Quantum pure quartic oscillator
It was recently brought to my attention, that there exist analytic solutions for the quantum pure-quartic oscillator with the hamiltonian
$$ \hat{H} = \frac{1}{2m} \hat{p}^2 + \frac{\lambda}{24} \hat{...
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1answer
440 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
431 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
107 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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votes
3answers
6k 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
65 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
92 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
309 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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1answer
84 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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votes
3answers
5k 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, ...
3
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1answer
481 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$$
and I want to find the free energy (so essentially $\log Z(\beta)...
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1answer
233 views
How to solve the 10000th eigenvalue of the anharmonic oscillator?
Given a certain Hamiltonian, for example,
$$ H = -\frac{1}{2}\frac{\partial^2}{\partial x^2 } + x^4 . $$
, what methods can we use to approximate the $n$th eigenvalue, for very large $n$?
For ...
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1answer
515 views
Can quartic oscillator's energy eigenvalue, i.e. $(0+1)$-dim $\phi^4$ theory, be solved exactly?
For a potential of anharmonic oscillator like this:
$$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
312 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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1answer
506 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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1answer
983 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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1answer
365 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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312 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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0answers
255 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$$
One could find the ...
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7answers
3k views
Why is the simple harmonic motion idealization inaccurate?
While in my physics classes, I've always heard that the simple harmonic motion formulas are inaccurate e.g. In a pendulum, we should use them only when the angles are small; in springs, only when the ...
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1answer
110 views
Pendulum motion equation issue
The differential equation that gives the equation of motion of a pendulum where:
$m$ is the mass
$L$ is the distance between the pivot and the body's centre of mass
$g$ is the acceleration due to ...
8
votes
2answers
586 views
The role of anharmonic oscillator(s) in Heisenberg's 1925 paper
I am talking about the most famous paper of Heisenberg, which I know from the translation of van der Waerden (Sources in Quantum mechanics, North Holland, 1967).
After introducing matrix mechanics ...
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0answers
622 views
Solving the quantum an-harmonic oscillator pertubatively?
Background
Generally while solving the quantum an-harmonic oscillator:
$$ -\frac{d^2 y}{dx^2} + k_1 x^4 y + k_2 x^2 y= E y $$
Most people (I've googled) on the internet always solve this using:
...
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0answers
745 views
What is the mechanism of subharmonic oscillations?
It's clear to me from linear systems theory that energy manifested within a fundamental mode of resonance can saturate with the excess energy spilling over into harmonic frequencies greater than the ...
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1answer
148 views
Equations of motion for a system of $n$ particles given the potetial [closed]
I am having difficulties on the following question:
The equations of motion for a system of n particles are: $$m \ddot{x}_i = - \dfrac{\partial U(x_1,...,x_n)}{\partial x_i}$$ $$\ddot{x}_i = \...
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1answer
186 views
How to solve highly oscillating differential equation [closed]
The equation looks like: $$x''(t)+bx'(t)+c x(t)+dx^3(t)=0.$$ This is the motion of a particle in a potential $cx^2/2+dx^4/4$ with friction force $bx'$. In my case, the friction term is very small and ...
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1answer
115 views
Quantum anharmonic ocscillator $E_0(\lambda)$ curve or table
I am looking for the exact data on $E_0(\lambda)$ for the anharmonicity $\lambda x^4$. The perturbative expansion is the following: $E_0(\lambda)\approx 0.5(1+1.5\lambda -5.25\lambda ^2+41.625\lambda^...
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0answers
171 views
What can I expect to see in a oscillator exhibiting bifurcation?
I have a program which aims to simulate a Josephson Bifurcation Amplifier. I am currently trying to obtain a plot of the probability of bifurcation as a function of the ratio between the driving and ...
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2answers
352 views
How can an inverted anharmonic potential $V(x)=-x^4$ have discrete bound states?
I've been watching the lectures on mathematical physics by Carl Bender on youtube where he uses the non-Hermitian Hamiltonian methods to prove that the inverted anharmonic potential $V(x)=-x^4$ has a ...
4
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4answers
1k views
Show bigger amplitude of physical pendulum means bigger period
Suppose you have a physical pendulum. It is true that as amplitude increases, the period increases. Can we demonstrate this fact without explicitly finding the period (which is pretty involved and ...
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1answer
181 views
When is this integral zero?
I have a particle with total energy $E$ confined in a potential
$$U(x) = -\frac{\cos^4x}{2} - m \cos x - f \sin x. $$
The constants $f$ and $m$ are both in the range (-2,2). The energy is such that ...
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1answer
842 views
Does sound absorption depends upon the amplitude of sound wave?
I can understand the mechanism of frequency dependant sound absorption by most materials but does the sound attenuation also depends upon the AMPLITUDE(sound pressure or rather loudness/sound ...
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0answers
883 views
Stationary Perturbation Theory : Estimating higher order corrections for anharmonic oscillator
Note $\hbar = 1$.
$$H = H_0 + \lambda V =\frac{p^2}{2m} + m\omega^2x^2 + \lambda m^2\omega^3 x^4$$
Supposedly the perturbation expansion diverges. We are supposed to estimate for what order we have a ...
8
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1answer
2k views
Anharmonic oscillators: why is $F=-k x-k' x^3$, with no quadratic terms?
The equation of motion of a general anharmonic oscillator includes a position-dependent force that can be expanded in a Taylor series as
$$m\ddot{x}+2\mu\dot{x}+k_0+k_1x+k_2x^2+k_3x^3\ldots=F.$$
I ...
