Closed Orbit of Hydrogen Atoms. How to find the classical turning point

[rydberg157]

Hi.
I am trying to find the classical turning points in semi-parabolic coordinates for the hydrogen atom when an electric field is being applied to it in the y-axis. I am reading an article for those who are interested called Classical, semiclassical, and quantum dynamics in the lithium Stark System published in Physical Review A Volume 51, Number 5. It gives me the following three separated equations based on the Hamiltonian:

.5 * (p_u)^2 + .5*F*u^4 - E(u^2) = e_u
.5 * (p_v)^2 - .5*F*u^4 - E(u^2) = e_v

and e_u + e_v = 2

u and v are semi-parabolic position coordinates. eu and ev are separation constants.
The article also says that for the purposes of finding closed orbits, that's what I am looking for, E = scaled energy and F = 1. Scaled energy can be picked to be an arbitrary number. Scaled energy = E / (F^.5)

I know that at the classical turning point the kinetic energy is equal to 0 so the equations above simplify to

.5*u^4 - (scaled_energy)*u^2 = e_u
-.5*v^4 - (scaled_energy)*v^2 = e_v

Finding the classical turning point will hopefully help me determine closed orbits for hydrogen as they give me two integrals for determining when an orbit is closed if the electron is launched at a certain angle in a Field=1 and and Energy=Scaled Energy.

I have figured out that in the primitive orbit of hydrogen where the electron is launched vertically in the y-axis, parallel to the field, the electron will have u = 0 position and a velocity in the u direction = 0, so the quartic equation

0.5v^4 - (scaled_energy)*v^2 = e_v = 2

will determine the classical turning point for that orbit.

Thank you

 

[mark726]

for sharing your findings and the equations you have been working with. It seems like you are on the right track in your search for the classical turning points in semi-parabolic coordinates for the hydrogen atom.

To further assist you, here are a few suggestions:

1. Use a computer program or software to plot the equations and visualize the orbit. This will help you better understand the behavior of the electron in the applied electric field and identify the turning points.

2. Look into the concept of effective potential in classical mechanics. This can help you understand the behavior of the electron in the electric field and identify the points where the effective potential is equal to the electron's energy.

3. Consider using numerical methods, such as the shooting method or the Runge-Kutta method, to solve the equations and find the turning points. These methods can give you more accurate results and allow you to explore different initial conditions for the electron.

4. It might also be helpful to consult with other scientists or researchers who have studied similar systems or have expertise in classical mechanics. They may have valuable insights or suggestions for your research.

Best of luck in your study of the hydrogen atom in an applied electric field. I hope you are able to find the closed orbits and further advance our understanding of this fascinating system.