Madelung Equations
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theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...
, the Madelung equations, or the equations of quantum hydrodynamics, are Erwin Madelung's alternative formulation of the
Schrödinger equation The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...
for a spinless non relativistic particle, written in terms of hydrodynamical variables, similar to the
Navier–Stokes equations The Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician Georg ...
of
fluid dynamics In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in motion ...
. 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. In both cases the hydrodynamic interpretations are not equivalent to Schrodinger's equation without the addition of a quantization condition. Recently, the extension to the relativistic case with spin was done by having the
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1/2 massive particles, called "Dirac ...
written with hydrodynamic variables. In the relativistic case, the Hamilton–Jacobi equation is also the guidance equation, which therefore does not have to be postulated.


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 Louis Victor Pierre Raymond, 7th Duc de Broglie (15 August 1892 – 19 March 1987) was a French theoretical physicist and aristocrat known for his contributions to quantum theory. In his 1924 PhD thesis, he postulated the wave nature of elec ...
and Earle Hesse Kennard. The most influential of these theories was ultimately de Broglie's through the 1952 work of
David Bohm David Joseph Bohm (; 20 December 1917 – 27 October 1992) was an American scientist who has been described as one of the most significant Theoretical physics, theoretical physicists of the 20th centuryDavid Peat Who's Afraid of Schrödinger' ...
now called Bohmian mechanics. In 1994 Timothy C. Wallstrom showed that an additional ''ad hoc'' quantization condition must be added to the Madelung equations to reproduce Schrodinger's work. His analysis paralleled earlier work by Takehiko Takabayashi on the hydrodynamic interpretation of Bohmian mechanics. The mathematical foundations of the Madelung equations continue to be a topic of research.


Equations

The Madelung equations are quantum
Euler equations In mathematics and physics, many topics are eponym, named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include their own unique function, e ...
: \begin & \partial_t \rho_m + \nabla\cdot(\rho_m \mathbf v) = 0, \\ pt& \frac = \partial_t\mathbf v + \mathbf v \cdot \nabla\mathbf v = -\frac \mathbf(Q + V), \end where * \mathbf v is the
flow velocity In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
, * \rho_m = m \rho = m , \psi, ^2 is the
mass density Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek language, Greek letter rho), although the Latin letter ''D'' (or ''d'') ...
, * Q = -\frac \frac = -\frac \frac is the Bohm quantum potential, * is the potential from the Schrödinger equation. The Madelung equations answer the question whether \mathbf(\mathbf,t) obeys the
continuity equations A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantit ...
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 \Gamma \doteq \oint = 2\pi n\hbar for all
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s .


Derivation

The Madelung equations are derived by first writing the
wavefunction In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
in polar form \psi(\mathbf, t) = R(\mathbf, t) e^, with R \geq 0 and S both real and \rho(\mathbf,t)=\psi(\mathbf,t)^*\psi(\mathbf,t)=R^2(\mathbf,t), the associated probability density. Substituting this form into the probability current gives: \mathbf = \frac(\psi^* \nabla \psi - \psi \nabla \psi^*) = \frac\rho(\mathbf,t)\nabla S(\mathbf,t) = \rho(\mathbf,t)\mathbf(\mathbf,t), where the
flow velocity In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
is expressed as \mathbf(\mathbf,t)=\frac\nabla S(\mathbf,t). However, the interpretation of \mathbf as a "velocity" should not be taken too literal, because a simultaneous exact measurement of position and velocity would necessarily violate the
uncertainty principle The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position a ...
. Next, substituting the polar form into the
Schrödinger equation The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...
i\hbar\frac \psi(\mathbf,t) = \left \frac \nabla^2 + V(\mathbf) \right\psi(\mathbf, t), and performing the appropriate differentiations, dividing the equation by e^ and separating the real and imaginary parts, one obtains a system of two coupled partial differential equations: \begin &\partial_R(\mathbf,t) + \frac\nabla R(\mathbf,t)\cdot\nabla S(\mathbf,t) + \frac R(\mathbf,t)\Delta S(\mathbf,t) = 0,\\ &\partial_S(\mathbf,t) + \frac\left nabla S(\mathbf,t)\right2 + V(\mathbf) = \frac\frac. \end The first equation corresponds to the imaginary part of Schrödinger equation and can be interpreted as the
continuity equation A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity ...
. 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 &\partial_\rho(\mathbf,t) + \nabla\cdot\left \rho(\mathbf,t)v(\mathbf,t) \right 0,\\ &\frac\mathbf(\mathbf,t)=\partial_v(\mathbf,t) + \left (\mathbf,t)\cdot \nabla\right(\mathbf,t) = -\frac\nabla \left (\mathbf) - \frac\frac\right=-\frac\nabla \left (\mathbf) + Q(\mathbf,t)\right \end with quantum potential Q(\mathbf,t) = - \frac\frac. Alternatively, the quantum Hamilton-Jacobi equation can be written in a form similar to the
Cauchy momentum equation The Cauchy momentum equation is a vector partial differential equation put forth by Augustin-Louis Cauchy that describes the non-relativistic momentum transport in any continuum. Main equation In convective (or Lagrangian) form the Cauchy moment ...
: \frac\mathbf= \mathbf - \frac \nabla \cdot \mathbf_Q, with an external force defined as \mathbf(\mathbf)=-\frac\nabla V(\mathbf), and a quantum pressure tensor \mathbf_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 In mathematical statistics, the Fisher information is a way of measuring the amount of information that an observable random variable ''X'' carries about an unknown parameter ''θ'' of a distribution that models ''X''. Formally, it is the variance ...
, which accounts for the quality of measurements. Thus, according to the
Cramér–Rao bound In estimation theory and statistics, the Cramér–Rao bound (CRB) relates to estimation of a deterministic (fixed, though unknown) parameter. The result is named in honor of Harald Cramér and Calyampudi Radhakrishna Rao, but has also been d ...
, the Heisenberg
uncertainty principle The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position a ...
is equivalent to a standard inequality for the
efficiency Efficiency is the often measurable ability to avoid making mistakes or wasting materials, energy, efforts, money, and time while performing a task. In a more general sense, it is the ability to do things well, successfully, and without waste. ...
of measurements.


Quantum energies

The thermodynamic definition of the quantum chemical potential \mu = Q + V = \frac \widehat H \sqrt follows from the hydrostatic force balance above: \nabla \mu = \frac \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(\mathbf p_Q) \frac = -\frac (\nabla \ln \rho_m)^2 + U and is related to the von Weizsäcker correction of density functional theory.


See also

*
Classical limit The classical limit or correspondence limit is the ability of a physical theory to approximate or "recover" classical mechanics when considered over special values of its parameters. The classical limit is used with physical theories that predict n ...
*
De Broglie–Bohm theory The de Broglie–Bohm theory is an interpretation of quantum mechanics which postulates that, in addition to the wavefunction, an actual configuration of particles exists, even when unobserved. The evolution over time of the configuration of all ...
*
Magnetohydrodynamics In physics and engineering, magnetohydrodynamics (MHD; also called magneto-fluid dynamics or hydro­magnetics) is a model of electrically conducting fluids that treats all interpenetrating particle species together as a single Continuum ...
*
Pilot wave theory In theoretical physics, the pilot wave theory, also known as Bohmian mechanics, was the first known example of a hidden-variable theory, presented by Louis de Broglie in 1927. Its more modern version, the de Broglie–Bohm theory, interprets qua ...
* Quantum potential *
Quantum hydrodynamics In condensed matter physics, quantum hydrodynamics (QHD) is most generally the study of hydrodynamic-like systems which demonstrate quantum mechanical behavior. They arise in semiclassical mechanics in the study of metal and semiconductor device ...
*
WKB approximation In mathematical physics, the WKB approximation or WKB method is a technique for finding approximate solutions to Linear differential equation, linear differential equations with spatially varying coefficients. It is typically used for a Semiclass ...


Notes


References

* * * * * * * * * * * * {{DEFAULTSORT:Madelung Equations Partial differential equations Quantum mechanics