# When is the vibration energy quantized?

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, $$\hbar\omega\left(n + 1/2 \right)$$.

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?

• Have you tried to estimate the difference in energy levels of a macroscopic HO? Jul 18, 2020 at 8:10
• @Qmechanic The neighboring energy level difference would be extremely small comparing to the vibrational energy if it's quantized but does it prove that it's quantized? – Jul 18, 2020 at 8:19

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.