Difficult geometries - Casimir Effect

Kruger
I really need your help.

Is there in the green field only place for one electromagnetic "oscillator" in ground state (the red one)? (Because these oscillators have to be at an angle of 90 degrees on the walls). I think this cannot be true.

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Kruger
nobody knows?

Kruger
Please, I wonder if you know.

marlon
What does this picture represent ? Is it a feynmann diagram that expresses a loop or is it referring to world lines ? Please, illustrate its context. Where did you get it from ? In what chapter has it been given?

It makes no sense to me

marlon

Kruger
Ok, first please visit this picture with the wave between the plates:

Now, you see there are standing waves (oscillations) between the plates. The thing what I try is to draw such oscillations between a circle and a plate.
And I don't really know how to draw this oscillations (between circle and plate), I mean in case of the Casimir-Effect they have to be at an angle of 90 degree at the wall, other oscillations are forbidden. Do you see?

Kruger
I thank the person or people who can help me.

marlon
Kruger said:
Ok, first please visit this picture with the wave between the plates:

Now, you see there are standing waves (oscillations) between the plates. The thing what I try is to draw such oscillations between a circle and a plate.
And I don't really know how to draw this oscillations (between circle and plate), I mean in case of the Casimir-Effect they have to be at an angle of 90 degree at the wall, other oscillations are forbidden. Do you see?

Err, what is the point using a circle and a plate ? What is the physical relevance ?

What you can do is this : the circle itself can be seen as an infinitesimal sequence of little lines. One can then treat each line and the other plate. Ofcourse, when both plates are not parallel then no standing waves are possible so basically there is only one contribution, ie the one i mentioned above with two parallel plates

marlon

Kruger
Err, what is the point using a circle and a plate ? What is the physical relevance ?

The relevance is to calculate the Casimir Force between sphere and plate that comes from the zero point energy of the electromagnetic field. The oscillations of the electromagnetic field are in their lowest possivle energy state, E=h(bar)omega/2, if they are in vacuum state. These oscillations can just build standing waves between two plates and between a sphere and a plate. You see?

Thanks marlon.

marlon
Kruger said:
The relevance is to calculate the Casimir Force between sphere and plate that comes from the zero point energy of the electromagnetic field. The oscillations of the electromagnetic field are in their lowest possivle energy state, E=h(bar)omega/2, if they are in vacuum state. These oscillations can just build standing waves between two plates and between a sphere and a plate. You see?

Thanks marlon.

But why a sphere and a plate ?
I mean, the reason we use the two plates is just to illustrate the existence of this Casimir force. The distance between the two plates selects only certain waves that can exist in between them...So why would you want to replace one plate by a sphere if you take this reasoning into account ?

marlon

Kruger
So why would you want to replace one plate by a sphere if you take this reasoning into account ?

Why not . I calculated the force between two plates, its quite easy. Now, I wanted to make some connections between various geometries of the Casimir-effect.

Another question is the following:
Between a plate there is one electromagnetic oscillation with wavenumber k, such that we have k=pi/L where L is the distance between the plates. This is one oscillation I picked out of the whole set of oscillations. The angular velocity omega of this oscillation has to bi: w=c*pi/L at time t1. Now, the plates will be pushed together (Casimir- Force). Can we then just say, that at time t2 where the new distance between the plates is smaller then L, the angular velocity of that oscillation has just changed or must we say this oscillation is "destroyed"?

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DaTario
In my opinion, your last question address the issue of adiabatic transfer of energy between the first set of modes to the second. If you have all of your energy concentrated, for exemple, in the third Harmonic of the first set of modes...if you start changing the configuration slowly enough to maintain the adiabatic regime, then you will end up with all the energy concetrated in the third harmonic of the sencond set of modes. If you make this transition out of adiabatic regime, energy from the first set of modes are arraged statistically among the frequencies of the sencond mode.

Regarding Casimir Effect, I wonder if is there anyone who had discovered a kind of topological logic in terms of which one can easily decide if a certain configuration of Casimir objects will yield attraction or repulsion.