Interesting oscillator potential

In summary, the conversation discusses the equations of motion for a particle in a one-dimensional potential and the solutions for the force equation. There is a discrepancy between the calculated solution and the expected oscillatory behavior, and suggestions are made to try the numerical calculation again. The conversation ends with a question about calculating the period.
  • #1
Mentz114
5,432
292
Hi Everyone,

I think I've solved the equations of motion for a particle in this one-dimensional potential -

V(x) = -k/(a-x) - k/(a+x) with |x| < a, a is real +ve, x is real.
K is a constant of suitable dimension.
It's a charge between two like charges with separation 2*a.

I start with the force equation

m*x'' = -4*a*k*x / (a^2 - x^2)^2
(Using ' to indicate differentiation wrt time)

I can integrate this to get

m*x' = sqrt(2*k*a)/sqrt(a^2 - x^2)

This ought to be textbook example, can anyone point me to an authoritative solution or come up with x ?
 
Last edited:
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  • #2
I get the separable ODE
[tex] \frac{dx}{dt}=\sqrt{\frac{4ak}{m}}\frac{1}{\sqrt{a^{2}-x^{2}}} [/tex]

Daniel.
 
  • #3
Thanks, Dextercioby.

We agree to a factor of sqrt(2). With some help I got

[tex]t + C =\frac{1}{2}\sqrt{ 2ak}( x\sqrt{a^2 - x^2} + arcsin( x) )[/tex]

for the relation between x and t. Doing some numerical work this looks
like a straight line, and not an oscillation.
 
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  • #4
I would suggest that you try your numerical calculation again. The arcsin(s) term would indicate that there should definitely be oscillations.
 
  • #5
I agree. Obviously, without the x*sqrt(a^2-x^2) term it's a sine wave.
I start with x=a/2 at t=0 which gives me k. The calculation gives the correct
x at t=0, then x falls linearly as t increases. Could be something wrong with some scaling.
 
  • #6
First post now Latexed

I think I've solved the equations of motion for a particle in this one-dimensional potential -

[tex]V(x)=\frac{-k}{a-x}-\frac{k}{a+x}[/tex]
with |x| < a, a is real +ve, x is real.
K is a constant of suitable dimension.
It's a charge between two like charges with separation 2a.

I start with the force equation

[tex]m\frac{d^2x}{dt^2} = -4akx / (a^2 - x^2)^2}[/tex]

I can integrate this to get

[tex]m\frac{dx}{dt} = \sqrt{2ka}/\sqrt{a^2 - x^2}[/tex]

This ought to be textbook example, can anyone point me to an authoritative solution or come up with x ?
 
Last edited:
  • #7
Sorry to necro-post, I made are errors in the above which I can now correct.

This equation is correct ( thanks Dan )

[tex] \frac{dx}{dt}=\sqrt{\frac{4ak}{m}}\frac{1}{\sqrt{a ^2-x^2}} [/tex]

and has solution

[tex]t + C = \frac{1}{2}\left(\frac{m}{4ak}\right)^\frac{1}{2}\left[x(a^2-x^2)^\frac{1}{2} - a^2cosh^{-1}(\frac{x}{a})\right][/tex]

If we choose coords so when x=0, t=0 then

[tex]C = -\frac{a^2}{2}\left(\frac{m}{4ak}\right)^\frac{1}{2}[/tex]

Not simple harmonic motion. How do I calculate ( guess ?) the period ?
 

1. What is an interesting oscillator potential?

An interesting oscillator potential is a mathematical function used to describe the potential energy of a particle in a physical system. It is typically used in the study of quantum mechanics and can take various forms, such as the harmonic oscillator potential or the anharmonic oscillator potential.

2. What makes an oscillator potential "interesting"?

An oscillator potential is considered interesting because it can exhibit unique behaviors and properties, such as quantized energy levels and wave-like behavior. It is also a fundamental concept in many areas of physics, making it a crucial topic for study and research.

3. How is an oscillator potential related to oscillation?

An oscillator potential is directly related to oscillation as it describes the potential energy of a particle that is undergoing oscillatory motion. It is a key component in understanding and predicting the behavior of systems that exhibit oscillatory motion, such as pendulums and springs.

4. Can an oscillator potential be applied to real-world systems?

Yes, an oscillator potential can be applied to real-world systems, particularly in the fields of physics, chemistry, and engineering. Many physical systems, such as atoms and molecules, can be modeled using oscillator potentials to gain insights into their behavior and properties.

5. What are some practical applications of an oscillator potential?

Oscillator potentials have a wide range of practical applications, including in the study of molecular vibrations, electronic energy levels, and atomic spectra. They are also used in the development of technologies such as lasers, sensors, and nanotechnology devices.

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