# What happens when the DeBroglie wavelength becomes smaller than the Planck length?

A particle around the size of a grain of salt would have a DeBroglie wavelength close the plank length. Does this mean quantum phenomena become hidden for larger particles since a DeBroglie wavelength smaller than the plank length makes no sense? Does the quantum information become hidden under a sort of Schwartzchild radius similar to when a black hole is formed and thus classical mechanics is all that's left?

A grain of salt can have a de Broglie wavelength much smaller than its size, and this is part of the explanation for why we don't see macroscopic quantum behavior of the grain of salt. This doesn't really have anything to do with the Planck length, though. Any time the de Broglie wavelength is smaller than our instruments are able to resolve, we'd have a hard time doing anything like a double-slit experiment with that object (for example).

Regarding the Schwarzschild radius and the Planck length:

• An object will (presumably) form a black hole if its mass $$M$$ is all concentrated into a region smaller than its Schwarzschild radius $$2GM/c^2$$, which doesn't involve Planck's constant.

• The Planck length $$\sqrt{G\hbar/c^3}$$ does involve Planck's constant, of course.

If we choose the mass so that these two scales are roughly equal to each other, then we have $$M\sim\sqrt{\hbar c/G}$$, which is the Planck mass. According to https://www.physlink.com/education/askexperts/ae342.cfm, a typical grain of salt has a mass of $$\sim 6\times 10^{-5}$$ grams (subject to lots of variation, of course), which is comparable to the Planck mass $$\approx 2\times 10^{-5}$$ grams.

An object may have a de Broglie wavelength much less than its Schwarzschild radius, but that doesn't mean it's a black hole or that gravity matters at all. The de Broglie wavelength only indicates the kind of scale that we'd need to resolve in order to observe quantum interference with that object, and has little to do with the object's physical size.

But if we squeezed all of the mass of a grain of salt into a space comparable to the Planck length, then it would become a black hole. At least, that's what classical general relativity would predict, but this is surely pushing classical general relativity beyond its domain of validity, because quantum-gravity effects are presumably essential at the Planck scale, so the true result would probably be even less like "classical physics" than either ordinary black holes or ordinary quantum objects.

The DeBroglie wavelength $$\lambda$$ depends on the speed of the particle: $$\lambda = \dfrac{h}{mv}$$

So a particle that has a mass like a grain of salt can have an arbitrarily large DeBroglie wavelength if it moves slowly enough. And likewise, a tiny particle like an electron can have an arbitrarily small wavelength if it moves fast enough.

• Your answer makes a good point, and I would add this: For the center-of-mass of the grain of salt to have a location that is defined to within $<\Delta x$, it's momentum cannot be defined much more sharply than $\Delta p\sim \hbar/\Delta x$. So it doesn't really have a single de Broglie wavelength, which would be $\lambda=h/p$ if the momentum were exactly $p$ (in which case it would be completely delocalized). Instead, it has a spread of wavelengths, with characteristic wavelength $\lambda\sim h/\Delta p\sim\Delta x$, where $\Delta x$ is normally microscopic for something like a grain of salt. Commented Jan 13, 2019 at 22:57