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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?

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    $\begingroup$ Have you tried to estimate the difference in energy levels of a macroscopic HO? $\endgroup$
    – Qmechanic
    Jul 18, 2020 at 8:10
  • $\begingroup$ @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? – $\endgroup$ Jul 18, 2020 at 8:19

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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.

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.

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  • $\begingroup$ With the superfluidity and superconductivity, the molecules attending the phenomenon are quantized. I think it would be more related to the condition if the molecules show the permutation symmetry for the spin-statistics theorem. But with the quantization of harmonic oscillator, it seems the molecules attending to the phenomenon doesn't matter. The quantization of the harmonic oscillator seems more concerned with the vibrational energy of the system as a whole. Am I understanding in the wrong way? $\endgroup$ Jul 18, 2020 at 10:16
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    $\begingroup$ In order to have a macroscopic quantum effect all the zillions of wavefunctions should be coherent, at least in some projection, The vibrations of the classical oscillator have nothing to do with the case.. Take a violin string, you could call the main and harmonics "quantum", because they come in steps, but it has nothing to do with the quantum state of the individual molecules vibrating altogether . $\endgroup$
    – anna v
    Jul 18, 2020 at 10:22
  • $\begingroup$ I think the violin string example is a bit different story. The frequency is quantized, but the amplitude is not and so not the energy either while the harmonic oscillator's energy is quantized with its fixed resonance frequency. $\endgroup$ Jul 18, 2020 at 11:05
  • $\begingroup$ generally classical equations' and quantum mechanical equations' are a different story. $\endgroup$
    – anna v
    Jul 18, 2020 at 11:51
  • $\begingroup$ I feel a huge gap between the quantization of harmonic oscillator and field quantization in QFT. QFT is to quantize the field of fundamental particles but the harmonic oscillator is to quantize complicated molecules. I don't know how the quantization can work for both cases. I felt that the quantization was not the property of constituent of the system. But the vibration energy is quantized somehow. They even talk about the quantization of the vibration energy of the vacuum. (Though I don't know what they are talking of with the vacuum.) $\endgroup$ Jul 18, 2020 at 12:26

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