Planck’s constant: Why are atom vibration frequencies integers only?

[Martyn Arthur]

TL;DR

Why do atom vibration frequencies integers only

Given the definition of whole numbers as integers, https://www.google.com/search?q=wha...7j0i512l9.11619j0j15&sourceid=chrome&ie=UTF-8
Is it known why atom vibrations are only at whole numbers ( ref plank’s constant)
https://edulab.com/plancks-constant-what-is-it-and-why-does-it-matter/
thanks
martyn

 

[Nugatory]

Is it known why atom vibrations are only at whole numbers ( ref plank’s constant)
https://edulab.com/plancks-constant-what-is-it-and-why-does-it-matter/

Yes, in that we know how to apply the laws of physics, especially Schrödinger’s equation, to calculate the behavior of these atoms.

The details of the calculation are not trivial - you would have to go through an entire college-level class in QM before you could take this problem on. However, most vibrating systems can be modeled more or less accurately as a harmonic oscillator and “vibrating atoms” are no exception. Thus, looking at and understanding the solution to Schrödinger’s equation for the energy levels of a harmonic oscillator will go a long ways towards explaining why integer multiples of things appear all through quantum mechanics.

 

[Martyn Arthur]

Thank you; I am at an early interest/curious stage.
I am working towards Schrödinger....
But for now
is there a reason known to us...in nature..why the vibrations are at integers only..excluding fractions?

 

[Nugatory]

is there a reason known to us...in nature..why the vibrations are at integers only..excluding fractions?

Why integer multiples? The only answer science will give us is that's what we find when we apply the laws of quantum mechanics to a given system. Science can't tell us why the universe is governed by these laws and not some other laws.

It's not just quantum mechanics either. For example, we apply Newton's laws (including gravity) to the solar system and we see that the planets will follow elliptical orbits. It's tempting to say that we now know the reason in nature that planets follow elliptical orbits.... but all we've really done is push the "why?" question down one level. Why should the universe behave according to Newton's laws that predict elliptical orbits instead of some other laws that predict something else?

 

[Hyperfine]

But for now
is there a reason known to us...in nature..why the vibrations are at integers only..excluding fractions?

Consideration of the particle in a box might prove helpful to you. See for example Particle in a box (Wikipedia).

 

[PeroK]

Thank you; I am at an early interest/curious stage.
I am working towards Schrödinger....
But for now
is there a reason known to us...in nature..why the vibrations are at integers only..excluding fractions?

The theory of quantum angular momentum proves, given certain assumptions, that angular momentum occurs in half-integer multiples of $\hbar$. The integer values are found in orbital angular and the half-integer values occur in spin angular momentum.

 

[Martyn Arthur]

Thank you' from my perspectve the answer is as good as it gets.
martyn

 

[PeroK]

Thank you' from my perspectve the answer is as good as it gets.
martyn

Here's a relatively painless derivation of the integer quantization for "orbital" angular momentum:

https://www2.ph.ed.ac.uk/~ldeldebb/docs/QM/lect8.pdf

 

[malawi_glenn]

It has to do with how differential equations works.

 

[PeterDonis]

atom vibrations

I'm not sure what you mean by "atom vibrations".

In general in quantum mechanics, the energy eigenstates of bound systems are discrete (hence the discrete energy levels of atoms), while the energy eigenstates of unbound systems are continuous (hence a free electron, for instance, can have a continuous range of energies).

If by "atom vibrations" you are referring to the energy levels of atoms, then, per the above, those are discrete because atoms are bound states and that's how quantum mechanics works for bound states.