Madelung equations

The Madelung equations are Erwin Madelung's equivalent alternative formulation of the Schrödinger equation.

Equations

The Madelung equations are quantum Euler equations:

$\partial _{t}\rho +\nabla \cdot (\rho {\mathbf {u}})=0,$

$\partial _{t}{\mathbf {u}}+{\mathbf {u}}\cdot \nabla {\mathbf {u}}=-{\frac {1}{m}}\nabla \left({\frac {1}{\sqrt {\rho }}}{\hat {H}}{\sqrt {\rho }}\right)=-{\frac {1}{\rho }}\nabla \cdot {\mathbf {p}}_{Q}-{\frac {1}{m}}\nabla U,$

where ${\mathbf {u}}$ is the flow velocity in the quantum probability space with mass density ${\mathbf {\rho }}=m|\psi |^{2}$. The circulation of the flow velocity field along any closed path obeys the auxiliary condition ${\begin{matrix}\Gamma \doteq \oint {m\mathbf {u} \cdot d\mathbf {l} }=2\pi n\hbar ,&n\in \mathbb {Z} \\\end{matrix}}$. The term in the brackets represents a quantum chemical potential $\mu$ in vacuum. The kinetic energy operator from the Hamiltonian ${\hat {H}}$ results in a non-local quantum pressure tensor

${\mathbf {p}}_{Q}=-(\hbar /2m)^{2}\rho \nabla \otimes \nabla \ln \rho$

which is related to the Bohm quantum potential $Q$. While the latter is the icon of the de Broglie–Bohm theory, ${\mathbf {p} }_{Q}$ is the quantum symbol of the Madelung hydrodynamics. 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 (statistics) of measurements. The thermodynamic definition of the quantum chemical potential $\mu =Q+U$ follows from the hydrostatic force balance above $\ \nabla \mu =(m/\rho )\nabla \cdot {\mathbf {p}}_{Q}+\nabla U$. According to thermodynamics, at equilibrium the chemical potential is constant everywhere, which corresponds straightforward 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 via $\ \epsilon =\mu -tr({\mathbf {p}}_{Q})(m/\rho )=-\hbar ^{2}(\nabla \ln \rho )^{2}/8m+U$ and it 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, $\ \epsilon =0$. 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.

See also

• Quantum potential
• Quantum hydrodynamics
• Bohmian quantum mechanics

Further reading

• Schönberg, M. (1954). "On the hydrodynamical model of the quantum mechanics". Il Nuovo Cimento. 12 (1): 103–133. doi:10.1007/BF02820368.