Madelung equations explained

In theoretical physics, the Madelung equations, or the equations of quantum hydrodynamics, are Erwin Madelung's equivalent alternative formulation of the Schrödinger equation, 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.

Equations

The Madelung equations^[1] are quantum Euler equations:^[2] where

• is the flow velocity,

\rho _{m} = m \rho = m |\psi| ^{2} is the mass density,Q =- (\hbar ^{2})/(2 m) (\nabla ^{2} \sqrt{\rho)/() } = -\frac \frac is the Bohm quantum potential,

• is the potential from the Schrödinger equation.

The circulation of the flow velocity field along any closed path obeys the auxiliary condition $\backslash Gamma\; \backslash doteq\; \backslash oint\; =\; 2\backslash pi\; n\backslash hbar$ for all integers .^[3]

Derivation

The Madelung equations are derived by writing the wavefunction in polar form:$$\backslash psi(\backslash mathbf,\; t)\; =\; \backslash sqrt\; e^,$$and substituting this form into the Schrödinger equation$$i\backslash hbar\backslash frac\; \backslash psi(\backslash mathbf,t)\; =\; \backslash left[\backslash frac\{-\backslash hbar^2\}\{2m\}\; \backslash nabla^2\; +\; V(\backslash mathbf\{x\},t)\; \backslash right]\; \backslash psi(\backslash mathbf,\; t).$$

The flow velocity is defined by$$\backslash mathbf(\backslash mathbf,\; t)\; =\; \backslash frac\; \backslash mathbf\; S,$$from which we also find that$$\backslash frac\; \backslash rho\_m\; \backslash mathbf\; =\; \backslash mathbf\; =\; \backslash frac\; [\backslash psi^*(\backslash nabla\backslash psi)\; -\; \backslash psi(\backslash nabla\backslash psi^*)],$$where

j is the probability current of standard quantum mechanics.

The quantum force, which is the negative of the gradient of the quantum potential, can also be written in terms of the quantum pressure tensor:$$\backslash mathbf\; =\; -\backslash mathbf\; Q\; =\; -\; \backslash frac\; \backslash nabla\; \backslash cdot\; \backslash mathbf\; p\_Q,$$where $$\backslash mathbf\; p\_Q\; =\; -(\backslash hbar/2m)^2\; \backslash rho\_m\; \backslash nabla\; \backslash otimes\; \backslash nabla\; \backslash ln\; \backslash rho\_m.$$

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. The thermodynamic definition of the quantum chemical potential$$\backslash mu\; =\; Q\; +\; V\; =\; \backslash frac\; \backslash widehat\; H\; \backslash sqrt$$follows from the hydrostatic force balance above:$$\backslash nabla\; \backslash mu\; =\; \backslash frac\; \backslash nabla\; \backslash cdot\; \backslash mathbf\; p\_Q\; +\; \backslash nabla\; V.$$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$$\backslash varepsilon\; =\; \backslash mu\; -\; \backslash operatorname(\backslash mathbf\; p\_Q)\; \backslash frac\; =\; -\backslash frac\; (\backslash nabla\; \backslash ln\; \backslash rho\_m)^2\; +\; U$$and is related to the local Carl Friedrich von Weizsäcker correction.^[4] 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,

\varepsilon = 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
• Pilot wave theory

Further reading

• 10.1007/BF02820368 . M. . Schönberg . Mário Schenberg . On the hydrodynamical model of the quantum mechanics . Il Nuovo Cimento . 1954 . 12 . 103–133 . 1954NCim...12..103S . 1 . 123655548 .
• Book: Wyatt, Robert E. . Corey J. . Trahan . Quantum Dynamics with Trajectories : Introduction to Quantum Hydrodynamics . New York . Springer . 2005 . 0-387-22964-7 . The Bohmian Route to the Hydrodynamic Equations . 40–61 .

Notes and References

1. 10.1007/BF01504657 . E. . Madelung . Eine anschauliche Deutung der Gleichung von Schrödinger . de . Naturwissenschaften . 1926 . 14 . 1004 . 1926NW.....14.1004M . 45 . 39430240 .
2. 10.1007/BF01400372 . E. . Madelung . Quantentheorie in hydrodynamischer Form . de . Z. Phys. . 1927 . 40 . 322–326 . 1927ZPhy...40..322M . 3–4 . 121537534 .
3. .
4. Tsekov . R. . 0903.3644 . Dissipative Time Dependent Density Functional Theory. 2009 . 10.1007/s10773-009-0054-6 . 48 . International Journal of Theoretical Physics . 9 . 2660–2664 . 2009IJTP...48.2660T . 119252668 .