Hydrogen atom in plane wave electronic field

[Quantum River]

We just consider one dimensional case and the classical method.
Then the motion equation of the electron in Coulomb field and the plane wave electronic field is
d^2 x/ dt^2=-1/x^2+cos t. (x is the coordinate and t is the time. )
How to solve the equation exactly?
We don't consider such cases as the electron collision with the Hydrogen nucleus.

Quantum River

 

[HallsofIvy]

That's a badly non-linear equation. Do you have any reason to think that it has an exact solution?

 

[Quantum River]

The equation is a one-body problem. We can't solve the three-body problem, but maybe the one-body problem is always solvable, even if the answer could be highly complex.
The equation is very useful in physics. So I want to solve it exactly.
Quantum River

 

[Astronuc]

d^2 x/ dt^2=-1/x^2+cos t.

Are you sure about this equation?

The left hand side has dimensions of acceleration, while the right-hand side has a mixture of 1/L² and cos t, which is dimensionless, so the t would have to be mutiplied by 1 (1/T), where T is time.

if the equation was

d^2 x/ dt^2=-(cos t)/x^2, then it could be readily solvable, but I doubt it makes physical sense.

 

[Quantum River]

The part of -1/r^2 actually means the Coulomb force generated by the Hydrogen nucleus.
The part of cos t is the simplification of e*E0*cos (omega*t).
e is the electric charge of the particle;
E0 is the electric field intensity.
cos t is the change of the electric field.
Quantum River