Quantum momentum decomposition

Question

Consider the time-dependent Schroedinger equation for a single non-relativistic particle with mass $m = 1$. The kinetic energy of the system is [ref] $$ T := \tfrac{\hbar^2}{2} \int |\nabla \psi|^2 dx . $$

Setting $\psi =: \sqrt{\rho} e^{i S / \hbar}$ (see pilot-wave formalism) one finds the decomposition $$ T = A + B := \tfrac12 \int |\nabla S|^2 \rho dx + \tfrac{\hbar^2}{8} I(\rho) $$ where $$ I(\rho) = \int |\nabla \rho|^2 \rho^{-1} dx $$ is the Fisher information measure (whose derivative is proportional the Bohm potential).

The first term $A$ has the obvious classical interpretation of kinetic energy of the density $\rho$ propelled by the velocity field $\nabla S$.

My main question is:

• Could you please point to some references where this decomposition appears?

Additionally, I would like to know

• What is a possible meaning of the second term $B$?
• Are there similar decompositions for other quantities?

Answer 1

Your guide in this might as well be P Holland's classic text The Quantum Theory of Motion: An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics, Chapter 3, by Peter R. Holland, ISBN-13: 978-0521485432.

In your conventions, T is the expectation of the kinetic energy, and its $\hbar^2 I(\rho)/8$ piece is the expectation of the quantum potential term, up to an normally ignorable surface term, $$ \int dx~ \rho ~Q = \frac{\hbar^2}{8} \int dx ~\left ( -2 \nabla^2 \rho + \frac{(\nabla \rho)^2}{\rho} \right ). $$ As you appear to be aware, Bohm and Hiley characterize it as an "information potential energy". Like the classical potential V, to which it is added to guide the quantum particle's motion, it has universal characteristics and does not care about the behavior of the particle (even though it is aware of its mass m you/we set to 1.

Leaving aside the recondite Bohemian mechanics aspects of it, it is actually a broadly intuitive quantity, and was, in fact, a celebrated correction introduced by von Weizsäcker in correction to the kinetic energy of the Thomas–Fermi theory of atoms (Zur Theorie der Kernmassen, Zeitschrift für Physik, Volume 96, pp. 431-458, 1935). It is responsible for the molecular bonding! See doi:10.1007/s10773-009-0054-6 which has a nice discussion of its prosaic, nitty-gritty, physical significance---no fundamentalism and interpretational stuff.

For further references see Benguria et al., and the WP article on Thomas-Fermi.