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I know that a quantum system can never have 0 energy due the Uncertainty Principle, and its lowest energy is called the Zero point Energy. However, Energy is a relative quantity (atleast in classical mechanics). What is this zero point energy relative to?

When I calculate the energy for a harmonic oscillator, I don't seem to define a ground state, unlike mechanical problems, where I take potential energy to be 0 explicitly at ∞ or at some elevation. Without this chosen ground state, what is the physical significance of this energy?

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It is indeed true that only relative energy matters. For a perfectly closed system, zero point energy doesn't matter. However, in other cases, the effective Hamilton can contain parameters and the zero point energy is not a fixed reference but a function of those parameters.

The casimir effect gives an effective Hamilton and a zero point energy that depends on the spacing of the two plates. This result to the casimir force.

Another example is that the binding energy of a molecule not only depends on the electronic ground state (given by say density functional theory), but also depends on the zero point energy of atom/nuclei vibrations. The true binding energy of a molecule also depends on how strong those bonds vibrate. The stronger the bond, the higher the zero point energy. This zero point energy contribution to binding energy is typical way smaller than the electronic and ionic contribution to binding energy, but it's non zero.

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