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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questions
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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
votes
1answer
71 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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4answers
235 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
$$\left(-\frac{1}{2}\frac{d^2}{dx^2} + \frac{x^2}{2} + gx^3 \right)\psi = E(g) \...
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0answers
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Rotating wave approximation and anharmonic oscillator
Given an Hamiltonian of this form
$$ \hat H = \alpha \; \hat b^\dagger \hat b + \beta \; (\hat b^\dagger + \hat b)^4 $$
where $[\hat b^\dagger, \hat b]=1$. In this video the host gets rid of all ...
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Why it is not possible to get exact solution for cubic potential perturbation for 1D SHO and we have to use perturbation theory? [duplicate]
Can anyone help me in providing the process of finding exact solution in case of cubic perturbation in 1D SHO, or any suitable resource?
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3answers
118 views
Why is the ground state energy of a linearly perturbed quantum oscillator always lower than its harmonic counterpart?
I am concerned with a QHO that is linearly perturbed in $x$, i.e.
$$ 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
45 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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52 views
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
317 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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1answer
170 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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1answer
77 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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1answer
134 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
182 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
79 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
$$\phi(t) = \frac{z(t)}{1-\frac{\lambda}...
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3answers
328 views
What are the possible limits for when SHM is not applicable for a vertical mass-spring system?
I am not sure if this is more of a maths problem than a physics if so could admin place in the math stack.
So my question is as follows. I have recently been looking at SHM in a spring-mass, as shown ...
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2answers
50 views
How can a stiffness of a spring can be $k(x+x^3)$? Compute energy total of the system [closed]
Consider the motion of a particle of mass $m$ attached to a spring of stiffness $k(x+x^3)$ where $x$ is the displacement.
1) How can look such a spring? I thought that the stiffness of a spring ...
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1answer
93 views
Why should the $\phi^4$ term necessarily cause scattering while a $x^4$ term in anharmonic oscillator only causes correction of energy levels?
Consider an anharmonic oscillator in quantum mechanics, described by the Hamiltonian $$H=\frac{p^2}{2m}+\frac{1}{2}m\omega^2x^2+bx^4.$$ The $bx^4$ term doesn't cause scattering. The effect of this ...
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0answers
63 views
Asymptotic behavior of canonical perturbation theory for the classic anharmonic oscillator
What do we know about the asymptotic behavior of the perturbative expansion for the classical anharmonic oscillator?
The Hamiltonian is
$$
H = \frac{p^2}{2m}+\frac{1}{2}m\omega_0^2 q^2 +\mu q^4
$$
...
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votes
2answers
847 views
Pendulum period [duplicate]
In plane pendulum problem, we can calculate its period using elliptic integration.
In SHO problem, we use approximation such that $\theta\ll 1$ and get the period, $2\pi\sqrt{l/g}$.
Is there another ...
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4k views
What is the difference between Non-harmonic oscillation, Anharmonic oscillation and Complex harmonic oscillation?
I am just wondering if the words Non-harmonic oscillation, Anharmonic oscillation and Complex harmonic oscillation mean the same thing. If not what exactly is the difference between them?
Since the ...
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1answer
22 views
Velocity changes due to crystal anharmonicity?
What is the effect of cubic and higher anharmonicities of a phonon hamiltonian on the velocity of phonons?
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1answer
304 views
Why is pendulum isochronic? [duplicate]
We all know that the pendulum is isochronic, i.e. that it takes the same time regardless of the amplitude is this is less than 20 degrees.
But how do we prove it mathematically?
What happens when the ...
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1answer
777 views
Why do large angular displacements affect the amplitude of a simple pendulum? [duplicate]
I was conducting an experiment involving the effect of large angular displacements (10°, 20°, ... 90°) on the amplitude of a simple pendulum. So far, I have found that the period of the pendulum ...
2
votes
1answer
926 views
Quantum pure quartic oscillator
It was recently brought to my attention, that there exist analytic solutions for the quantum pure-quartic oscillator with the hamiltonian
$$ \hat{H} = \frac{1}{2m} \hat{p}^2 + \frac{\lambda}{24} \hat{...
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votes
1answer
530 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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votes
1answer
504 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
129 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
7k 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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66 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
95 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
351 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
85 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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3answers
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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, ...
3
votes
1answer
506 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$$
and I want to find the free energy (so essentially $\log Z(\beta)...
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1answer
243 views
How to solve the 10000th eigenvalue of the anharmonic oscillator?
Given a certain Hamiltonian, for example,
$$ H = -\frac{1}{2}\frac{\partial^2}{\partial x^2 } + x^4 . $$
, what methods can we use to approximate the $n$th eigenvalue, for very large $n$?
For ...
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vote
1answer
571 views
Can quartic oscillator's energy eigenvalue, i.e. $(0+1)$-dim $\phi^4$ theory, be solved exactly?
For a potential of anharmonic oscillator like this:
$$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
335 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
578 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
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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1answer
372 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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322 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
265 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$$
One could find the ...
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votes
7answers
3k views
Why is the simple harmonic motion idealization inaccurate?
While in my physics classes, I've always heard that the simple harmonic motion formulas are inaccurate e.g. In a pendulum, we should use them only when the angles are small; in springs, only when the ...
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1answer
116 views
Pendulum motion equation issue
The differential equation that gives the equation of motion of a pendulum where:
$m$ is the mass
$L$ is the distance between the pivot and the body's centre of mass
$g$ is the acceleration due to ...
8
votes
2answers
633 views
The role of anharmonic oscillator(s) in Heisenberg's 1925 paper
I am talking about the most famous paper of Heisenberg, which I know from the translation of van der Waerden (Sources in Quantum mechanics, North Holland, 1967).
After introducing matrix mechanics ...
3
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0answers
641 views
Solving the quantum an-harmonic oscillator pertubatively?
Background
Generally while solving the quantum an-harmonic oscillator:
$$ -\frac{d^2 y}{dx^2} + k_1 x^4 y + k_2 x^2 y= E y $$
Most people (I've googled) on the internet always solve this using:
...
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0answers
797 views
What is the mechanism of subharmonic oscillations?
It's clear to me from linear systems theory that energy manifested within a fundamental mode of resonance can saturate with the excess energy spilling over into harmonic frequencies greater than the ...
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1answer
154 views
Equations of motion for a system of $n$ particles given the potetial [closed]
I am having difficulties on the following question:
The equations of motion for a system of n particles are: $$m \ddot{x}_i = - \dfrac{\partial U(x_1,...,x_n)}{\partial x_i}$$ $$\ddot{x}_i = \...
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votes
1answer
202 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
118 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^...
