This question may be stupid, but I would be very grateful for an answer. As far as I understand, a matter wave has a wavelength and an amplitude. The amplitude does NOT extend in space, it simply indicates the probability of the respective particle being within a certain region. The wavelength IS a length in space and extends in the direction of the (relative) velocity of the particle. (Correct me if I am wrong.) However, in my limited understanding, a matter wave must also have a lateral extension (which is NOT its amplitude) since it is "broad" enough to cover, e.g., the two slits in a double slit experiment. If the matter wave was only extended along the direction of its movement, it should hit any horizontal or vertical obstacle at exactly one point, even though it's probability of location is extended on a line (since this line runs at right angles to the obstacle). To use an analogy: if you observe a water wave from the beach, coming in your direction, it's amplitude is its hight, and its wavelenght extends in the frontal direction (relative to you), at right angles to the shoreline. But the water wave also stretches out to the sides, parallel to the shoreline. This extension may be longer or shorter, but, in the case of a water wave, it evidently cannot be zero. Why isn't it zero in the case of a matter wave? (Or is it? But if so, how can it pass two slits which are laterally distant, relative to it?) How far does a matter wave extend laterally? Does it only extend in one direction or in all (spatial) directions? - If these questions do not make sense: could you explain why?
1 Answer
It's a bit incorrect to use term "wave of matter", it's wave of possibility and nothing more, another interpretations can cause wrong understating and paradoxes, matter looks exactly like in classical understanding and has sizes exactly like in classical predictions, electron is spherical and has certain radius and doesn't look like wave.
You can use Fourier transformation to decompose the wave to monochromatic 3D waves https://en.wikipedia.org/wiki/Sinusoidal_plane_wave, monochromatic waves have infinite "side sizes", as you said, like waves on the beach.
But you have to consider the quantum effects that limit monochromaticity of real wave functions. Our space is continuous and the price of that is the impossibility of creating ideal monochromatic waves in nature (the diffraction phenomena) Applying measurement postulate to a continuous sum of eigenvectors (by analogy)
So, our wave packet must have some different wavelengths and it localizes the particle in some place in average, it's it's no longer the infinite everywhere the same sinusoidal wave: http://31.media.tumblr.com/7cb79638bd5cc6f70ddc1851b04e2f17/tumblr_mi5zsmKUBh1rg05iho1_1280.gif
The minimum sizes of this wave packet are also limited by uncertainty_principle, where $$\Delta p\cdot\Delta x \geq \hbar/2$$ And this works in every dimension, not only in direction of particle velocity. So max side sizes of wave are not limited, but can't be infinite.
-
$\begingroup$ Thank you very much! This was very helpful. In essence, if I understood correctly, the point is simply that the wave of possibility extends in every dimension, not only in the direction of particle velocity. The uncertainty principle applies "naturally" to all its extension. This explains my question completely. $\endgroup$ Commented Sep 23, 2022 at 12:40