When is the vibration energy quantized?

Question

There is a quantum harmonic oscillator such as the vibration of diatomic molecules. [1, 2]
The molecular vibrations are quantized and the energy spectrum of the system is discrete, $hbaromegaleft(n + 1/2 right)$.
On the other hand, we have macroscopic classical harmonic oscillator such as the mass-spring system. I would be surprised if the vibrational energy of the mass-spring system is quantized.  The system is in contact with the environment all the times but the harmonic motion keeps long enough time to prove that it is not affected by decoherence.
Why are the molecular vibrations quantized but not the mass-spring system? Does the size of the system matter? or is the size of the vibrational energy decisive?  or is it the temperature breaking the quantization?
Or is the vibration of mass-spring system quantized actually?

anna v · Answer

Generally solutions of quantum mechanics equations are  necessary when the phase space  of the problem is of dimensions of the order of magnitude of the Planck constant.

The mathematical transition from the microworld of particles, molecules and lattices described by solutions of quantum mechanical equations to the classical dimensions is the density matrix formalism. In this formalism for a macroscopic system to exhibit quantum effects there should be a coherence ( a calculable phase) of at least some  part of the  wavefunctions of the particles  describing the macroscopic system.Macroscopic systems  are composed of a huge number of molecules , of the order of $10^{23}$ molecules per mole.
This coherence can happen and does happen , but not with your classical spring system. There exist macroscopic quantum mechanical systems, as in superfluidity and superconductivity, but they are special cases of coherence up to macroscopic dimensions.

When is the vibration energy quantized?

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