Phase space view of quantum mechanical systems and Fisher information

• Phase-space Fisher information coming from the canonical distribution is derived for the ground state of quantum mechanical systems.
• Quantum mechanical phase-space Fisher information contains an extra term due to the position dependence of the temperature.
• A complete analogy to the classical case is demonstrated for the linear harmonic oscillator.

Pennini and Plastino showed that the form of the Fisher information generated by the canonical distribution function reflects the intrinsic structure of classical mechanics. Now, a quantum mechanical generalization of the Pennini–Plastino theory is presented based on the thermodynamical transcription of the density functional theory. Comparing to the classical case, the phase-space Fisher information contains an extra term due to the position dependence of the temperature. However, for the special case of constant temperature, the expression derived bears resemblance to the classical one. A complete analogy to the classical case is demonstrated for the linear harmonic oscillator.