A jump rope in the ground state gets excited.

[Spinnor]

Say two girls spin a jump rope, of length ∏ meters, so that the only nodes are where the girls hold the rope, ψ= sin(x)exp(iω_1t) (actually there are no nodes at the girls hands as where they hold the rope goes in a smallish circle). Now instruct the girls how to shake the rope so as to most easily form the "first excited state", ψ= sin(2x)exp(iω_2t), from the ground state (the first excited state now with one more node in the middle of the rope). Do they just spin the ends of the rope faster and faster with one girl's phase lagging or leading the other girls. Can the ground state smoothly evolve into the first excited state with proper inputs from the girls? I would guess this would be easy to try with a rope?

Is this similar to a ground state quantum particle in a one dimensional infinite square well and turning on a perturbing potential of the proper frequency, say

V_p = a[x sin(ωt)]

How do I take ψ= sin(x)exp(iω_1t) and smoothly evolve it into
ψ= sin(2x)exp(iω_2t) ? What "forces" will do this?

Thanks for any help!

 

[Spinnor]

...

How do I take ψ= sin(x)exp(iω_1t) and smoothly evolve it into
ψ= sin(2x)exp(iω_2t) ? What "forces" will do this?

Thanks for any help!

Let ψ = exp(-δt)sin(x)exp(iω_1t) + [1 - exp(-δt)]sin(2x)exp(iω_2t)

Is this a possible smooth transition (waves on a string)? If so should I be able to figure out the required "force"?

Thanks for any help!

 

[haruspex]

If the rope is uniform, the shape won't be a sine function, nor a parabola nor catenary. Looks rather nasty.
You can play around with this as a solo effort, using gravity as your partner. Dangle a rope and rotate the wrist to get it spinning. Of course, you'll get a loose end beyond the bottom node, but it's basically valid. To get it to ratchet up to the next mode you'll need to rotate the wrist a lot faster. The transition will look like a a rather complicated helical wave traveling down the rope.