Madelung equations

In the fall of 1926, Erwin Madelung reformulated{{sfn | Madelung | 1926}}{{sfn | Madelung | 1927}} 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.{{sfn | Kragh | Carazza | 2000 | p=52}} The most influential of these theories was ultimately de Broglie's through the 1952 work of David Bohm{{sfn | Kragh | Carazza | 2000 | p=53}} now called Bohmian mechanics.

In 1994 Timothy C. Wallstrom showed{{Cite journal |last=Wallstrom |first=Timothy C. |date=1994-03-01 |title=Inequivalence between the Schrödinger equation and the Madelung hydrodynamic equations |url=https://journals.aps.org/pra/abstract/10.1103/PhysRevA.49.1613 |journal=Physical Review A |volume=49 |issue=3 |pages=1613–1617 |doi=10.1103/PhysRevA.49.1613|url-access=subscription }} that an additional ad hoc quantization condition must be added to the Madelung equations to reproduce Schrodinger's work. His analysis paralleled earlier work{{Cite journal |last=Takabayasi |first=Takehiko |date=1952-08-01 |title=On the Formulation of Quantum Mechanics associated with Classical Pictures* |url=https://academic.oup.com/ptp/article/8/2/143/1883374 |journal=Progress of Theoretical Physics |volume=8 |issue=2 |pages=143–182 |doi=10.1143/ptp/8.2.143 |issn=0033-068X}} by Takehiko Takabayashi on the hydrodynamic interpretation of Bohmian mechanics. The mathematical foundations of the Madelung equations continue to be a topic of research.{{Cite journal |last=Reddiger |first=Maik |last2=Poirier |first2=Bill |date=2023-04-19 |title=Towards a mathematical theory of the Madelung equations: Takabayasi’s quantization condition, quantum quasi-irrotationality, weak formulations, and the Wallstrom phenomenon |journal=Journal of Physics A: Mathematical and Theoretical |volume=56 |issue=19 |pages=193001 |doi=10.1088/1751-8121/acc7db |issn=1751-8113|arxiv=2207.11367 }}

The Madelung equations are quantum Euler equations:{{cn|date=September 2024}}

$$\backslash begin\{align\}$$

& \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{align}

where

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

The Madelung equations answer the question whether $\backslash mathbf\{v\}(\backslash mathbf\{x\},t)$ obeys the continuity equations of hydrodynamics and, subsequently, what plays the role of the stress tensor.{{sfn | Białynicki-Birula | Cieplak | Kaminski | 1992 | p=88}}

The circulation of the flow velocity field along any closed path obeys the auxiliary quantization condition $\backslash Gamma\; \backslash doteq\; \backslash oint\{m\backslash mathbf\{v\}\; \backslash cdot\; d\backslash mathbf\{l\}\}\; =\; 2\backslash pi\; n\backslash hbar$ for all integers {{mvar|n}}.{{sfn | Białynicki-Birula | Cieplak | Kaminski | 1992 | pp=90-91}}{{sfn | Wyatt | 2005 | loc=§13.5}}

The Madelung equations are derived by first writing the wavefunction in polar form{{sfn | Białynicki-Birula | Cieplak | Kaminski | 1992 | pp=88-89}}{{sfn | Wyatt | 2005 | loc=§2.2}}

$$\backslash psi(\backslash mathbf\{x\},\; t)\; =\; R(\backslash mathbf\{x\},\; t)\; e^\{\; i\; S(\backslash mathbf\{x\},\; t)/\backslash hbar\},$$

with $R\; \backslash geq\; 0$ and $S$ both real and

$$\backslash rho(\backslash mathbf\{x\},t)=\backslash psi(\backslash mathbf\{x\},t)^*\backslash psi(\backslash mathbf\{x\},t)=R^2(\backslash mathbf\{x\},t),$$

the associated probability density. Substituting this form into the probability current gives:

$$\backslash mathbf\{J\}\; =\; \backslash frac\{\backslash hbar\}\{2mi\}(\backslash psi^*\; \backslash nabla\; \backslash psi\; -\; \backslash psi\; \backslash nabla\; \backslash psi^*)\; =\; \backslash frac\{1\}\{m\}\backslash rho(\backslash mathbf\{x\},t)\backslash nabla\; S(\backslash mathbf\{x\},t)\; =\; \backslash rho(\backslash mathbf\{x\},t)\backslash mathbf\{v\}(\backslash mathbf\{x\},t),$$

where the flow velocity is expressed as

$$\backslash mathbf\{v\}(\backslash mathbf\{x\},t)=\backslash frac\{1\}\{m\}\backslash nabla\; S(\backslash mathbf\{x\},t).$$

However, the interpretation of $\backslash mathbf\{v\}$ as a "velocity" should not be taken too literal, because a simultaneous exact measurement of position and velocity would necessarily violate

the uncertainty principle.{{sfn | Sakurai | Napolitano | 2020 | p=95}}

Next, substituting the polar form into the Schrödinger equation

$$i\backslash hbar\backslash frac\{\backslash partial\}\{\backslash partial\; t\}\; \backslash psi(\backslash mathbf\{x\},t)\; =\; \backslash left[\; \backslash frac\{-\backslash hbar^2\}\{2m\}\; \backslash nabla^2\; +\; V(\backslash mathbf\{x\})\; \backslash right]\; \backslash psi(\backslash mathbf\{x\},\; t),$$

and performing the appropriate differentiations, dividing the equation by $e^\{\; i\; S(\backslash mathbf\{x\},\; t)/\backslash hbar\}$ and separating the real and imaginary parts, one obtains a system of two coupled partial differential equations:

$$\backslash begin\{align\}$$

&\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{align}

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.{{sfn | Wyatt | 2005 | p=48}}

Multiplying the first equation by $2R$ and calculating the gradient of the second equation results in the Madelung equations:

$$\backslash begin\{align\}$$

&\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{align}

with quantum potential

$$Q(\backslash mathbf\{x\},t)\; =\; -\; \backslash frac\{\backslash hbar^2\}\{2m\}\backslash frac\{\backslash Delta\; \backslash sqrt\{\backslash rho(\backslash mathbf\{x\},t)\}\}\{\backslash sqrt\{\backslash rho(\backslash mathbf\{x\},t)\}\}.$$

Alternatively, the quantum Hamilton-Jacobi equation can be written in a form similar to the Cauchy momentum equation:

$$\backslash frac\{d\}\{dt\}\backslash mathbf\{v\}=\; \backslash mathbf\{f\}\; -\; \backslash frac\{1\}\{\backslash rho\_m\}\; \backslash nabla\; \backslash cdot\; \backslash mathbf\{p\}\_Q,$$

with an external force defined as

$$\backslash mathbf\{f\}(\backslash mathbf\{x\})=-\backslash frac\{1\}\{m\}\backslash nabla\; V(\backslash mathbf\{x\}),$$

and a quantum pressure tensor{{sfn | Tsekov | 2011}}

$$\backslash mathbf\{p\}\_Q\; =\; -\; (\backslash hbar/2m)^2\; \backslash rho\_m\; \backslash nabla\; \backslash otimes\; \backslash nabla\; \backslash ln\; \backslash rho\_m.$$

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.{{sfn | Reginatto | 1998}}{{sfn | Heifetz | Cohen | 2015}}

The thermodynamic definition of the quantum chemical potential

$$\backslash mu\; =\; Q\; +\; V\; =\; \backslash frac\{1\}\{\backslash sqrt\{\backslash rho\_m\}\}\; \backslash widehat\; H\; \backslash sqrt\{\backslash rho\_m\}$$

follows from the hydrostatic force balance above:

$$\backslash nabla\; \backslash mu\; =\; \backslash frac\{m\}\{\backslash rho\_m\}\; \backslash nabla\; \backslash cdot\; \backslash mathbf\; p\_Q\; +\; \backslash nabla\; V.$$

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

$$\backslash varepsilon\; =\; \backslash mu\; -\; \backslash operatorname\{tr\}(\backslash mathbf\; p\_Q)\; \backslash frac\{m\}\{\backslash rho\_m\}\; =\; -\backslash frac\{\backslash hbar^2\}\{8m\}\; (\backslash nabla\; \backslash ln\; \backslash rho\_m)^2\; +\; U$$

and is related to the von Weizsäcker correction of density functional theory.{{sfn | von Weizsäcker | 1935}}{{sfn | Tsekov | 2009}}

• Classical limit
• De Broglie–Bohm theory
• Magnetohydrodynamics
• Pilot wave theory
• Quantum potential
• Quantum hydrodynamics
• WKB approximation

{{reflist}}

• {{cite book | last=Białynicki-Birula | first=Iwo | last2=Cieplak | first2=Marek | last3=Kaminski | first3=Jerzy | title=Theory of Quanta | publisher=Oxford University Press, USA | publication-place=New York | date=1992 | isbn=0-19-507157-3}}
• {{cite journal | last=Heifetz | first=Eyal | last2=Cohen | first2=Eliahu | title=Toward a Thermo-hydrodynamic Like Description of Schrödinger Equation via the Madelung Formulation and Fisher Information | journal=Foundations of Physics | volume=45 | issue=11 | date=2015 | issn=0015-9018 | doi=10.1007/s10701-015-9926-1 | pages=1514–1525| arxiv=1501.00944 }}
• {{cite journal | last=Kragh | first=Helge | last2=Carazza | first2=Bruno | title=Classical Behavior of Macroscopic Bodies from Quantum Principles: Early Discussions | journal=Archive for History of Exact Sciences | volume=55 | issue=1 | date=2000 | issn=0003-9519 | doi=10.1007/s004070000018 | pages=43–56}}
• {{cite journal | last=Madelung | first=E. | title=Eine anschauliche Deutung der Gleichung von Schrödinger | journal=Die Naturwissenschaften | volume=14 | issue=45 | date=1926 | issn=0028-1042 | doi=10.1007/BF01504657 | pages=1004–1004 | language=de}}
• {{cite journal | last=Madelung | first=E. | title=Quantentheorie in hydrodynamischer Form | journal=Zeitschrift für Physik | volume=40 | issue=3-4 | date=1927 | issn=1434-6001 | doi=10.1007/BF01400372 | pages=322–326 | language=de}}
• {{cite journal | last=Reginatto | first=Marcel | title=Derivation of the equations of nonrelativistic quantum mechanics using the principle of minimum Fisher information | journal=Physical Review A | volume=58 | issue=3 | date=1998-09-01 | issn=1050-2947 | doi=10.1103/PhysRevA.58.1775 | pages=1775–1778}}
• {{cite book | last=Sakurai | first=J. J. | last2=Napolitano | first2=Jim | title=Modern Quantum Mechanics | publisher=Cambridge University Press | publication-place=Cambridge | date=2020 | isbn=978-1-108-47322-4}}
• {{cite journal | last=Schönberg | first=M. | title=On the hydrodynamical model of the quantum mechanics | journal=Il Nuovo Cimento | volume=12 | issue=1 | date=1954 | issn=0029-6341 | doi=10.1007/BF02820368 | pages=103–133}}
• {{cite journal | last=Tsekov | first=Roumen | title=Quantum diffusion | journal=Physica Scripta | volume=83 | issue=3 | date=2011 | issn=0031-8949 | doi=10.1088/0031-8949/83/03/035004 | arxiv=1001.1071 }}
• {{cite journal | last=Tsekov | first=Roumen | title=Dissipative Time Dependent Density Functional Theory | journal=International Journal of Theoretical Physics | volume=48 | issue=9 | date=2009 | issn=0020-7748 | doi=10.1007/s10773-009-0054-6 | pages=2660–2664| arxiv=0903.3644 }}
• {{cite journal | last=von Weizsäcker | first=Carl F. | year=1935 | title=Zur Theorie der Kernmassen | language=de | journal=Zeitschrift für Physik | volume=96 | issue=7-8 | pages=431-58 | doi=10.1007/bf01337700 | issn=0044-3328 }}
• {{cite book | last=Wyatt | first=Robert E. | title=Quantum Dynamics with Trajectories | publisher=Springer Science & Business Media | publication-place=New York | date=2005 | isbn=0-387-22964-7}}

{{DEFAULTSORT:Madelung Equations}}

Category:Partial differential equations

Category:Quantum mechanics