Low quantum numbers, high energy, and distance scales.

[TomServo]

I understand how we associate high energies with small wavelengths and thus small distance scales, but we also tend to associate small distance scales with ordinary quantum mechanics, and hence low quantum numbers (low energy). Also, many high-energy processes are active across large distance scales, such as binary black hole mergers, neutron star mergers, the LHC, etc.

So what, really, are the "rules" (beyond the de Broglie wavelength equation) for associating large/small distance scales with large/small energy scales?

 

[mfb]

Binary black holes have a large overall energy but not large energies per particle, in general there are also not many particles around.
LHC collisions are tiny, the size of the accelerator does not matter here.
For processes in neutron star mergers you can have high energies per particle.

To see if quantum mechanics is relevant, find pairs of relevant coordinates that multiply to an action (same units as the Planck constant). If it is small compared to the Planck constant quantum mechanics will be relevant, otherwise probably not.

 

[TomServo]

To see if quantum mechanics is relevant, find pairs of relevant coordinates that multiply to an action (same units as the Planck constant). If it is small compared to the Planck constant quantum mechanics will be relevant, otherwise probably not.

Okay, that's interesting, but how would I show that?

 

[mfb]

That depends on your system.