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7
votes
2answers
184 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
votes
0answers
513 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: ...
1
vote
0answers
107 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 ...
1
vote
1answer
91 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 = ...
0
votes
1answer
92 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 ...
1
vote
1answer
100 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 ...
3
votes
0answers
97 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 ...
5
votes
2answers
176 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 ...
3
votes
3answers
287 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 ...
1
vote
1answer
166 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 ...
0
votes
1answer
254 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 ...
2
votes
0answers
489 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
votes
1answer
788 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 ...