The integral energy stored in the quantum pressure tensor is proportional to the Fisher information, which accounts for the quality of measurements. Thus, according to the Cramér–Rao bound, the Heisenberg uncertainty principle is equivalent to a standard inequality for the efficiency of measurements. The thermodynamic definition of the quantum chemical potential μ μ = Q + V = 1 ρ ρ m H ^ ^ ρ ρ m follows from the hydrostatic force balance above: ∇ ∇ μ μ = m ρ ρ m ∇ ∇ ⋅ ⋅ p Q + ∇ ∇ V . According to thermodynamics, at equilibrium the chemical potential is constant everywhere, which corresponds straightforwardly to the stationary Schrödinger equation. Therefore, the eigenvalues of the Schrödinger equation are free energies, which differ from the internal energies of the system. The particle internal energy is calculated as ε ε = μ μ − − tr ⁡ ⁡ ( p Q ) m ρ ρ m = − − ℏ ℏ 2 8 m ( ∇ ∇ ln ⁡ ⁡ ρ ρ m ) 2 + U and is related to the local Carl Friedrich von Weizsäcker correction. In the case of a quantum harmonic oscillator, for instance, one can easily show that the zero-point energy is the value of the oscillator chemical potential, while the oscillator internal energy is zero in the ground state, ε ε = 0 . Hence, the zero point energy represents the energy to place a static oscillator in vacuum, which shows again that the vacuum fluctuations are the reason for quantum mechanics.