Analytic solutions of the Madelung equation

We present analytic self-similar solutions for the one, two and three dimensional Madelung hydrodynamical equation for a free particle. There is a direct connection between the zeros of the Madelung fluid density and the magnitude of the quantum potential.


I. INTRODUCTION
Finding classical physical basements of quantum mechanics is a great challenge since the advent of the theory. Madelung was one among the firsts who gave one explanation, this was the hydrodynamical foundation of the Schrödinger equation [1,2]. His exponential transformation simply indicates that one can model quantum statistics hydrodynamically.
The transformed equation has an attractive feature that the Planck's constant appears only once, as the coefficient of the quantum potential or pressure. Thus, the fluid dynamicist can gather experience of its effects by translating some of the elementary situations of the quantum theory into their corresponding fluid mechanical statements and vice versa.
The quantum potential also appears in the de Broglie-Bohm pilot wave theory [3,4] (in other context) which is a non-mainstream attempt to interpret quantum mechanics as a deterministic non-local theory. In the case of → 0 the Euler equation transforms to the Hamilton-Jacobi equation.
Later, many authors tried to generalize this Madelung's description for more compound quantum mechanical systems (like particles in external electromagnetic fields or with spin 1/2 ) [5]. Takabayashi [6] tried to interpret the Ansatz of Madelung as an ensemble of trajectories. Schönberg [7] developed a new type of hydrodynamical model for the quantum mechanics for any values of spin. The quantum potential appears as a combination of a pressure term arising from the turbulence.
In spite of the flaws of Madelung hydrodynamics (it cannot give a proper solution of the problem of atomic eigenstates and to the quantum description of emission or absorption processes), this approach turns to be fruitful in a number of applications like the stochastic quantum mechanics [8], quantum cosmology [9], description of quantum-like systems [10], the coherent properties of high-energy charged particle beams [11,12].
Terlecki applied the fluid dynamical interpretation of the quantum mechanical probability density and current for the trajectory method and evaluated the solution of the timedependent Schrödinger equation in atomic physics and calculated ionization and electron transfer cross sections for proton-hydrogen collision [13].
As an interesting peculiarity Wallstrom showed with mathematical means that the initialvalue problem of the Madelung equation is not well-defined and additional conditions are needed [14].
Tsubota summarized the hydrodynamical descriptions of quantum condensed fluids such as superfluid helium and Bose-Einstein condensates as quantum hydrodynamics based on the original Ansatz of Madelung [15].
Nowadays, hydrodynamical description of quantum mechanical systems is a popular technical tool in numerical simulations. Review articles on quantum trajectories can be found in a booklet edited by Huges in 2011 [16].
From general concepts as the second law of thermodynamics a weakly non-local extension of ideal fluid dynamics can be derived which leads to the Schrödinger-Madelung equation as well [17].
In the following study we investigate the Madelung equation with the self-similar Ansatz and present analytic solutions with discussion.
This way of investigation is a powerful method to study the global properties of the solutions of various non-linear partial differential equations(PDEs) [18]. Self-similar Ansatz describes the intermediate asymptotic of a problem: it is hold when the precise initial conditions are no longer important, but before the system has reached its final steady state. This is much more simpler than the full solutions and so easier to understand and study the different regions of the parameter space. A final reason for studying them is that they are solutions of a system of ordinary differential equations(ODEs) and hence do not suffer the extra inherent numerical problems of the full PDEs. In some cases self-similar solutions help to understand diffusion-like properties or the existence of compact supports of the solution.
In the last years we successfully applied the multi-dimensional generalization of the selfsimilar Ansatz to numerous viscous fluid equations [19,20] ending up with a book chapter of [21].  [23] however that is not the field of our present interest.

II. THEORY AND RESULTS
Following the original paper of Madelung [2] the time-dependent Schrödinger equation where Ψ, U, m, h are the wave function, potential, mass and Planck's constant, respectively.
Taking the following Ansatz Ψ = √ ρe iS where ρ(x, t) and S(x, t) are time and space dependent functions. Substituting this trial function into Eq. (1) going through the derivations the real and the imaginary part give us the following continuity and Euler equations with the form of with v = m ∇S. The ρ is the density of the investigated fluid and v is the velocity field.
Madelung also showed that this is a rotation-free flow. The transformed equations has an attractive feature that the Planck's constant appears only once, at the coefficient of the quantum potential or pressure, which is the first term of the right hand side of the second equation. Note, that these are most general vectorial equation for the velocity field v which means that one, two or three dimensional motions can be investigated as well.
In the following we will consider the two dimensional flow motion v = (u, v) in Cartesian coordinates without any external field U = 0. The functional form of the three and one dimensional solutions will be mentioned briefly as well.
We are looking for the solution of Eqs. (2) with self-similar Ansatz which is well-known from [18] ρ(x, y, t where f, g and h are the shape functions of the density and the velocity field, respectively.
The similarity exponents α, β, δ, are of primary physical importance since α, δ, represents the rate of decay of the magnitude of the shape function while β represents the spreading.
More about the general properties of the Ansatz can be found in our former papers [19,20].
Except some pathological cases all positive similarity exponents mean physically relevant dispersive solutions with decaying features at x, y, t → ∞. Substituting the Ansatz (3) into (2) and going through some algebraic manipulation the following ODE system can be obtained for the shape functions Note that the particle mass appears in the denominator of the quantum potential term which is consistent with the experience of regular quantum mechanics that quantum features are relevant at small particle masses.
All the similarity exponents have the fixed value of + 1 2 which is usual for regular heat conduction, diffusion or for Navier-Stokes equations [21]. Note, that the two remaining free parameters are the mass of the particle m and which is the Planck's constant divided by 2π. For a better transparency we fix = 1.
At this point it is worth to mention, that the obtained ODE for the density shape function is very similar to Eq. (5) for different space dimensions, the only difference is a constant in the last term. For one, two or three dimensions the denominator has a factor of 1,2 or 3, The solution of (5) can be expressed with the help of the Bessel functions of the first and second kind [24] and has the following form of where c 1 and c 2 are the usual integration constants. The correctness of this solutions can be easily verified via back substitution into the original ODE.
To imagine the complexity of these solutions Figure 1 presents f (η) for various m values.
It has a strong decay with a stronger and stronger oscillation at large arguments. The function is positive for all values of the argument, (which is physical for a fluid density), but such oscillatory profiles are completely unknown in regular fluid mechanics [21]. The most interesting feature is the infinite number of zero values which cannot be interpreted physically for a classical real fluid.  Both J ν and Y ν Bessel functions with linear argument form an orthonormal set, therefore integrable over the L 2 space. In our case, the integral ∞ 0 f (η)dη is finite as well, unfortunately can be evaluated only by numerical means. This is good news because f (η) is the density function of the original Schrdinger equation. In this sense √ f is the fluid mechanical analogue of the real part of the wave function of the free quantum mechanical particle which can be described with a Gaussian wave packet. To obtain the complete original wave function, the imaginary part has to be evaluated as well. It is trivial from η = x+y t 1/2 = 2(g + h) that Ψ(x, y, t) = √ 2t · e mi (x+y) 2 4t (8) Figure 2 shows the projection of the real part wave function to the x, t sub-space. At small times the oscillations are clear to see, however at larger times the strong damping is evident.
For arbitrary quantum systems, the wave function can be evaluated according to the Schrödinger equation, however we never know directly how large is the quantum contribution to the classical one. Now, it is possible for a free particle to get this contribution. .
(9) Figure 3 shows the shape function of quantum potential Q(η) comparing to the shape function of the density f (η). Note, that where the density has zeros the quantum potential is singular. Such singular potentials might appear in quantum mechanics, however the corresponding wave function should compensate this effect. This is the main message of our study. Unfortunately, we cannot squarely state, that this kind of property is true in general for all the Madelung quantum potentials (e.g. for other Ansätze). Therefore additional investigations should be made to clarify this conjecture.

III. SUMMARY AND OUTLOOK
After reviewing the historical development and interpretation of the Madelung equation we introduced the self-similar Ansatz which is a not-so-well-known but powerful tool to investigate non-linear PDEs. The free particle Madelung equation was investigated in two dimensions with this method, (the one and three dimensional solutions were mentioned as well.) We found analytic solution for the fluid density, velocity field and the original wave function. All can be expressed in terms of the Bessel functions. The classical fluid density has interesting properties, oscillates and has infinite number of zeros which is quite unusual and not yet been seen in such analysis.