The quantum force, which is the negative of the gradient of the quantum potential, can also be written in terms of the quantum pressure tensor:
where
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
follows from the hydrostatic force balance above:
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
and is related to the local Carl Friedrich von Weizsäcker correction.[4] 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, . Hence, the zero point energy represents the energy to place a static oscillator in vacuum, which shows again that the
Wyatt, Robert E.; Trahan, Corey J. (2005). "The Bohmian Route to the Hydrodynamic Equations". Quantum Dynamics with Trajectories : Introduction to Quantum Hydrodynamics. New York: Springer. pp. 40–61.