Madelung equations

In theoretical physics, the Madelung equations, or the equations of quantum hydrodynamics, are Erwin Madelung's equivalent alternative formulation of the Schrödinger equation for a spinless non relativistic particle, written in terms of hydrodynamical variables, similar to the Navier–Stokes equations of fluid dynamics. The derivation of the Madelung equations is similar to the de Broglie–Bohm formulation, which represents the Schrödinger equation as a quantum Hamilton–Jacobi equation.

History

In the fall of 1926, Erwin Madelung reformulated Schrödinger's quantum equation in a more classical and visualizable form resembling hydrodynamics. His paper was one of numerous early attempts at different approaches to quantum mechanics, including those of Louis de Broglie and Earle Hesse Kennard. The most influential of these theories was ultimately de Broglie's through the 1952 work of David Bohm now called Bohmian mechanics

Equations

The Madelung equations are quantum Euler equations: $${\begin{aligned}&\partial _{t}\rho _{m}+\nabla \cdot (\rho _{m}\mathbf {v} )=0,\\[4pt]&{\frac {d\mathbf {v} }{dt}}=\partial _{t}\mathbf {v} +\mathbf {v} \cdot \nabla \mathbf {v} =-{\frac {1}{m}}\mathbf {\nabla } (Q+V),\end{aligned}}$$ where

• $\mathbf {v}$ is the flow velocity,
• $\rho _{m}=m\rho =m|\psi |^{2}$ is the mass density,
• $Q=-{\frac {\hbar ^{2}}{2m}}{\frac {\nabla ^{2}{\sqrt {\rho }}}{\sqrt {\rho }}}=-{\frac {\hbar ^{2}}{2m}}{\frac {\nabla ^{2}{\sqrt {\rho _{m}}}}{\sqrt {\rho _{m}}}}$ is the Bohm quantum potential,
• V is the potential from the Schrödinger equation.

The Madelung equations answer the question whether $\mathbf {v} (\mathbf {x} ,t)$ obeys the continuity equations of hydrodynamics and, subsequently, what plays the role of the stress tensor.

The circulation of the flow velocity field along any closed path obeys the auxiliary quantization condition ${\textstyle \Gamma \doteq \oint {m\mathbf {v} \cdot d\mathbf {l} }=2\pi n\hbar }$ for all integers n.

Derivation

The Madelung equations are derived by first writing the wavefunction in polar form $$\psi (\mathbf {x} ,t)=R(\mathbf {x} ,t)e^{iS(\mathbf {x} ,t)/\hbar },$$ with $R\geq 0$ and $S$ both real and $$\rho (\mathbf {x} ,t)=\psi (\mathbf {x} ,t)^{*}\psi (\mathbf {x} ,t)=R^{2}(\mathbf {x} ,t),$$ the associated probability density. Substituting this form into the probability current gives: $$\mathbf {J} ={\frac {\hbar }{2mi}}(\psi ^{*}\nabla \psi -\psi \nabla \psi ^{*})={\frac {1}{m}}\rho (\mathbf {x} ,t)\nabla S(\mathbf {x} ,t)=\rho (\mathbf {x} ,t)\mathbf {v} (\mathbf {x} ,t),$$ where the flow velocity is expressed as $$\mathbf {v} (\mathbf {x} ,t)={\frac {1}{m}}\nabla S(\mathbf {x} ,t).$$ However, the interpretation of $\mathbf {v}$ as a "velocity" should not be taken too literal, because a simultaneous exact measurement of position and velocity would necessarily violate the uncertainty principle.

Next, substituting the polar form into the Schrödinger equation $$i\hbar {\frac {\partial }{\partial t}}\psi (\mathbf {x} ,t)=\left[{\frac {-\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {x} )\right]\psi (\mathbf {x} ,t),$$ and performing the appropriate differentiations, dividing the equation by $e^{iS(\mathbf {x} ,t)/\hbar }$ and separating the real and imaginary parts, one obtains a system of two coupled partial differential equations: $${\begin{aligned}&\partial _{t}R(\mathbf {x} ,t)+{\frac {1}{m}}\nabla R(\mathbf {x} ,t)\cdot \nabla S(\mathbf {x} ,t)+{\frac {1}{2m}}R(\mathbf {x} ,t)\Delta S(\mathbf {x} ,t)=0,\\&\partial _{t}S(\mathbf {x} ,t)+{\frac {1}{2m}}\left[\nabla S(\mathbf {x} ,t)\right]^{2}+V(\mathbf {x} )={\frac {\hbar ^{2}}{2m}}{\frac {\Delta R(\mathbf {x} ,t)}{R(\mathbf {x} ,t)}}.\end{aligned}}$$ The first equation corresponds to the imaginary part of Schrödinger equation and can be interpreted as the continuity equation. The second equation corresponds to the real part and is also referred to as the quantum Hamilton-Jacobi equation. Multiplying the first equation by $2R$ and calculating the gradient of the second equation results in the Madelung equations: $${\begin{aligned}&\partial _{t}\rho (\mathbf {x} ,t)+\nabla \cdot \left[\rho (\mathbf {x} ,t)v(\mathbf {x} ,t)\right]=0,\\&{\frac {d}{dt}}\mathbf {v} (\mathbf {x} ,t)=\partial _{t}v(\mathbf {x} ,t)+\left[v(\mathbf {x} ,t)\cdot \nabla \right]v(\mathbf {x} ,t)=-{\frac {1}{m}}\nabla \left[V(\mathbf {x} )-{\frac {\hbar ^{2}}{2m}}{\frac {\Delta {\sqrt {\rho (\mathbf {x} ,t)}}}{\sqrt {\rho (\mathbf {x} ,t)}}}\right]=-{\frac {1}{m}}\nabla \left[V(\mathbf {x} )+Q(\mathbf {x} ,t)\right].\end{aligned}}$$ with quantum potential $$Q(\mathbf {x} ,t)=-{\frac {\hbar ^{2}}{2m}}{\frac {\Delta {\sqrt {\rho (\mathbf {x} ,t)}}}{\sqrt {\rho (\mathbf {x} ,t)}}}.$$

Alternatively, the quantum Hamilton-Jacobi equation can be written in a form similar to the Cauchy momentum equation: $${\frac {d}{dt}}\mathbf {v} =\mathbf {f} -{\frac {1}{\rho _{m}}}\nabla \cdot \mathbf {p} _{Q},$$ with an external force defined as $$\mathbf {f} (\mathbf {x} )=-{\frac {1}{m}}\nabla V(\mathbf {x} ),$$ and a quantum pressure tensor $$\mathbf {p} _{Q}=-(\hbar /2m)^{2}\rho _{m}\nabla \otimes \nabla \ln \rho _{m}.$$

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.

Quantum energies

The thermodynamic definition of the quantum chemical potential $$\mu =Q+V={\frac {1}{\sqrt {\rho _{m}}}}{\widehat {H}}{\sqrt {\rho _{m}}}$$ follows from the hydrostatic force balance above: $$\nabla \mu ={\frac {m}{\rho _{m}}}\nabla \cdot \mathbf {p} _{Q}+\nabla V.$$ 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 $$\varepsilon =\mu -\operatorname {tr} (\mathbf {p} _{Q}){\frac {m}{\rho _{m}}}=-{\frac {\hbar ^{2}}{8m}}(\nabla \ln \rho _{m})^{2}+U$$ and is related to the local Carl Friedrich von Weizsäcker correction.

See also

• Quantum potential
• Quantum hydrodynamics
• Bohmian quantum mechanics
• Pilot wave theory

References

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