Why do waves need to match object size for interaction?

[Mr rabbit]

I have always been told that waves interact with objects with a size similar to their wavelength. For example: if you look at your microwave you notice that there is a grid with a size much smaller than the wavelength (so that they can not pass over), but visible light can pass very easy because its wavelength is much smaller than the size of the grid. It's also known that atoms can not be seen with visible light because their sizes are much smaller than the visible wavelengths (hundreds and hundreds of atoms can be placed in a blue wavelength).

I also thought of the hydrogen atom: at n = 1, l = 0 state, the electron does not rotate around the proton (because l = 0) and describes a rectilinear trajectory (crossing the proton!) until it is re-oriented. I asked a professor a few months ago "Why does not the electron collide with the proton?" and he replied me "because you can calculate the associated wavelength for 13.6 eV and you see that it is much larger than the size of the proton (0.84 fm)."

And my question is: why do the waves need to have a wavelength similar to the size of objects to interact? Is there a mathematical justification?

 

[Orodruin]

the electron does not rotate around the proton

This is correct. However, it is misleading you to think classically and using nomenclature that suggests the electron is a little point. It is not.

and describes a rectilinear trajectory (crossing the proton!) until it is re-oriented.

This is not correct. The electron is just in a stationary state that does not carry angular momentum. On the subatomic level, you simply cannot think of the particles as little balls.

 

[sophiecentaur]

On the subatomic level, you simply cannot think of the particles as little balls.

Or even bigger than that. Electron diffraction can be seen for larger structures than an atom.

 

[Orodruin]

Well, yes, but the OP was explicitly discussing the hydrogen atom.

 

[Drakkith]

And my question is: why do the waves need to have a wavelength similar to the size of objects to interact? Is there a mathematical justification?

In the case of the grid on your microwave door, the microwaves interact with the grid in such a way that most of the energy is reflected back into the microwave, with a small amount escaping through to the outside. Describing this interaction in detail is rather complex, but it is fully supported by mathematical and physical justification. To give you an idea of how complicated this can be, note that a typical radio antenna acts as if it has a much larger surface area than it actually does. If the antenna is roughly 1/4 to 1/2 the size of the wavelength the EM wave "sees" the antenna as being larger than it actually is and efficiently couples with it.

As for why, I don't think I know enough to explain why in any detail. Just be aware that this is only one particular type of interaction and other types don't necessarily follow this. For example, an electron is perfectly happy responding to low-frequency EM waves whose wavelengths are far larger than the electron. If it didn't, we wouldn't be able to listen to the radio or transmit information wirelessly. It is most likely the bulk interaction of great numbers of electrons in conductors that gives rise to this property that EM waves interact best with objects approximately the size of their wavelengths.

 

[Lord Jestocost]

And my question is: why do the waves need to have a wavelength similar to the size of objects to interact? Is there a mathematical justification?

This is a somehow infelicitous wording. The ratio between the wavelength and the size of an "object" determines the extent of diffraction of an incoming wave that is disturbed by an "object": a smaller "object" gives rise to a wider diffraction pattern. As "objects" you can consider obstacles which are placed somewhere in the path of the incoming wave or a holes in a plate which is somewhere placed in the path of the incoming wave. The "wavelength/size" ratio is thus a measure of the extent of the bending of a wave around the edges of an opening or an obstacle. For quantitative descriptions (intesity of the diffracted waves), one can rely on the Huygens-Fresnel Principle.

As for the microwave oven: The small amount of the microwave field's energy that leaks out through the small grid openings doesn't thus leak out in a "directional" way but is - considering the "wavelength/opening size" ratio - additionally "spread out" over space due to diffraction.

 

[sophiecentaur]

I have always been told that waves interact with objects with a size similar to their wavelength.

You would need to quote where you actually got that information from. We all get "told" stuff that ranges from solid gold Science to absolute nonsense so you should really state the source of that information so that it's quality can be assessed.
I would say that the relevant factor is not so much size as resonant frequency. A Long Wave radio signal (1500m wavelength) can interact significantly enough with a portable transistor radio antenna which may be only 10cm in length but it resonates at 200kHz. Many molecules will interact with EM waves with wavelengths many times their dimension.
However, a simple wire dipole will resonate when it is around a half wavelength long and make a good transmitting or receiving antenna. That could be along the lines of what you were told.

 

[Mr rabbit]

This is not correct. The electron is just in a stationary state that does not carry angular momentum. On the subatomic level, you simply cannot think of the particles as little balls.

Well, this is true. Sometimes I get carried away by the intuition of classical physics. However, the interpretation is the same: both particles don't 'collide' because of the difference of the wavelengths. He told me he have been in an experiment with high energy electrons and at that conditions is true that there are many interactions.

You would need to quote where you actually got that information from

Mmm... from professors (optics, quantum mechanics, electrodynamics...) and science communicators. It's, for example, the typical argument of why we can't see atoms using light.

A Long Wave radio signal (1500m wavelength) can interact significantly enough with a portable transistor radio antenna which may be only 10cm in length but it resonates at 200kHz. Many molecules will interact with EM waves with wavelengths many times their dimension.

This is true, I had not thought about it.

 

[sophiecentaur]

This is true, I had not thought about it.

So what is it that you have "always been told"? Who has been telling you that and ignoring the two examples I gave you? This is why I asked you for an actual reference that doesn't just rely on your memory.

 

[nasu]

Mmm... from professors (optics, quantum mechanics, electrodynamics...) and science communicators. It's, for example, the typical argument of why we can't see atoms using light.

Do you think that we cannot see atoms because they do not interact with light? What did you learn in QM about atomic energy levels and absorption/emission of light by atoms? Absorption spectrum of a gas?