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It seems that we have quantum description of solid and gas, but there seem to be few quantum models of liquid, with the exception of Liquid Helium or perhaps Fermi Liquid.

For solids, we often study crystal, which is easier due to the lattice symmetry. For gas, particles often interact weakly so it's easy to formulate a non-interacting model. In this sense, I can see why people choose to study these systems first.

But are there any models that make use of quantum mechanics to explore the liquid phase? Perhaps a quantum mechanical description of water. Maybe we can combine Shrodinger Equation and Navier Stokes equation in some way. Of course Navier Stokes equation itself is already hard enough, so perhaps we need to go to certain limits.

Qmechanic
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Liuke LYU
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2 Answers2

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Fermi liquid, Luttinger liquid, Bose-Einstein condensate, Superconductivity and similar states of matter are known as quantum fluids or quantum liquids, and there are books written on the subject, see, e.g., Interactions in quantum fluids. The word liquid/fluid is however somewhat ambiguous, as it may mean different (although not mutually exclusive) things:

  • Fluid/liquid as a phase of matter is studied in the theory of phase-transitions / critical-phenomena. Water is a well-known and well-studied example of a sysmtem exhibiting a phas etransition,a nd it shows many interesting properties in its liquid phase (see, e.g., Superheated water and Supercooling). However, beyond the derivation of the intermolecular potentials (the subject of quantum chemistry), quantum effects are of real relevance to critical phenomena (see, e.g., Lectures On Phase Transitions And The Renormalization Group).
  • Mechanical properties of liquids, such as studied using Navier-Dtokes equation. Superfluid Helium and atomic Bose-Einstein condensates do fall into this category.
  • Quantum liquids are often not really liquids in the above sense, as matter phases or mechanical entities (although one could stretch the analogy rather far - e.g., if we consider Wigner cristals as a solid phase of electron liquid). Fermi liquid was called this way as an extension of Fermi gas description, because it accounts for the interactions between particles - the same distinction as exist between the ideal gas of atoms and a liquid in statistical physics. The analogy was thus grounded in the analogy of the mathematical/physical description, rather than physical similarities. Luttinger liquid then received a separate name, because it is different from the Fermi liquid, while also being a liquid of electrons. It is also common to speak more generally about non-Fermi-liquid behavior, meaning the electronic properties. Similarly, Atomic Bose-Einstein condensates may exhibit physical properties similar to liquids in conventional everyday sense, but their close brothers - Bose-Einstein condensates of photons (in lasers and masers), excitons or the superconducting state have physical properties very different from water or orange juice.
Roger V.
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In a gas the chemical potential is 0 because interaction between the molecules are considered negligible .However in a liquid this is not the case so the chemical potential will be different from 0.

For a big number of molecules the distribution of molecules over energy will be Dirac-Fermi or Bose-Einstein distribution depending on the molecule and you just apply the formulas for these distributions.

Jun Seo-He
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