Question

Prove that average kinetic energy/ potential energy of a harmonic oscillator is one-half of its total energy.

Answer 1

In Newtonian mechanics, one may derive a virial theorem, which says that the (time-averaged) kinetic and potential energies are related as

2⟨T⟩ = p⟨V⟩,(1)

(1)2⟨T⟩ = p⟨V⟩,

if the potential V(r)∝rpV(r)∝rp is a power law. Thus for the classical harmonic oscillator (HO)

⟨T⟩ = ⟨V⟩.(2)

(2)⟨T⟩ = ⟨V⟩.

In QM the averaging procedure is of a different nature, but even so, it turns out that there are no quantum mechanical corrections to (2) in the HO case.

Answer 2

In Newtonian mechanics, one may derive a virial theorem, which says that the (time-averaged) kinetic and potential energies are related as

2⟨T⟩ = p⟨V⟩,(1)

(1)2⟨T⟩ = p⟨V⟩,

if the potential V(r)∝rpV(r)∝rp is a power law. Thus for the classical harmonic oscillator (HO)

⟨T⟩ = ⟨V⟩.(2)

(2)⟨T⟩ = ⟨V⟩.

In QM the averaging procedure is of a different nature, but even so, it turns out that there are no quantum mechanical corrections to (2) in the HO case.