Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2408.11399

On the Quantization of the Motion of Liquids

Nadine Suzan Cetin

Published Aug 21, 2024• 0 citations
arXivHTMLPDF

Abstract

For building up a theory of superfluid Helium-4, Lev Landau ingeniously unified the principles of quantum mechanics with the principles of hydrodynamics. By introducing a velocity operator he was able to derive a quantum analogue of the mass density flux equation and a quantum Euler equation. The article shows that it is possible to formulate a consistent quantum theory of the ideal liquid based on Landau's Quantum Hydrodynamics. This means that it is added the quantum analogue of the momentum density flux equation, the energy density flux equation, the entropy density flux equation, and a quantum vorticity equation. These are all manifestly in agreement with the classical limit of the flux equations of classical ideal liquids. It is also shown that the various aspects that have been criticised about Landau's Quantum Hydrodynamics - e.g. the application of the inverse of the mass density operator - are not justified. Such a consistent quantum theory of an ideal liquid may therefore provide a starting point of a complementary approach in the theoretical research of superfluid Helium 4.

DOI
10.48550/arxiv.2408.11399

Related papers

Current-Algebra Approach to Landau's Quantized Hydrodynamics

Donald H. Kobe, Grant C. Coomer

Landau's quantized hydrodynamics is derived from the microscopic Hamiltonian, which is written in terms of the density and velocity operators. In Euler's equation for the velocity operator, the force per unit mass has a form different from Landau's and is given explicitly in terms of the two-body potential. A quantum pressure term, which does not occur in Landau's theory, also appears. New realizations of the density and velocity operators are given which satisfy the commutation relations, but must act in a Hilbert space larger than Fock space. It is shown that the formalism is without meaning in Fock space owing to the nonexistence of certain operators.

Physical Review A•Apr 1, 1973•20 citations

Wave mechanics equivalent to classical mechanics

Dennis Shay

A nonlinear interaction term is added to the Schr\"odinger equation which makes it exactly equivalent to classical mechanics. In the hydrodynamic formulation of this equation, the usual classical theory is obtained. This theory is a Lagrangian density formulation of hydrodynamics and is used to calculate the density profile of superfluid helium in a line vortex. The theory can be canonically quantized, in the sense of quantum field theory, in terms of the density and velocity fields. The quantized theory is not equivalent to that which one would obtain if the quantization were done in the usual way in terms of the field operations ${\ensuremath{\psi}}^{*}$ and ($\ensuremath{\psi}$). The Schr\"odinger equation, with this interaction, has localized solutions with a definite energy and momentum and does not give quantized energy levels for a one-dimensional harmonic oscillator.

Physical Review A•Jun 1, 1976•2 citations

Theory of a superfluid Fermi liquid

Alexander I. Akhiezer, Красильников Владимир Владимирович, S. V. Peletminskiǐ +1 more

A Fermi-liquid approach to the theory of superfluidity is developed, based on a generalization of the combinatoric expression for the entropy to the case in which the state of the Fermi liquid is described not only by a normal but also by an anomalous distribution function. The energy of the superfluid Fermi liquid is given in the form of a functional of these distribution functions. In the weak-coupling approximation the Fermi-liquid approach is equivalent to the BCS theory. The interaction between the Fermi liquid and an electromagnetic field is gauge invariant. In addition to the singlet-pairing theory the triplet-pairing theory is also developed. A unified variational principle that gives rise to the theory of superfluidity (superconductivity) is used to obtain the London equations and the Ginzburg–Landau equation. Applications of this theory are made to the derivation of an equation for ideal two-component hydrodynamics.

Physics-Uspekhi•Feb 28, 1993•13 citations

On the microscopic theory of superfluid liquids

N. N. Bogolyubov, M. Yu. Kovalevskiĭ, A. M. Kurbatov +2 more

This review presents a microscopic construction of the thermodynamics and hydrodynamics of superfluid bosons and fermions with singlet pairing, based on the concept of quasiaverages and the hypothesis of reduced description. Here we do not assume that the Hamiltonian possesses any dynamical symmetry. This has permitted obtaining results pertaining to both Galilean-invariant and relativistic systems. Account is taken of dissipative processes. The kinetic coefficients are presented in terms of the correlation functions of the flux operators. The approach is extended to solutions of quantum liquids. The influence of an external ac field on superfluid systems is studied, and the low-frequency asymptotic behavior of the Green's function is found in the hydrodynamic approximation. The symmetry properties of the equilibrium state are formulated, and the thermodynamics is constructed for superfluid Fermi systems with triplet pairing (the superfluid phases 3 He-B and 3 He-A). For the latter the flux densities of the additive integrals of motion are found in a state of equilibrium and the equations of "ideal" hydrodynamics are derived.

Soviet Physics Uspekhi•Dec 31, 1989•24 citations

Landau's theory of superfluidity

Allan Griffin, Tetsuro Nikuni, E. Zaremba

In this chapter, we review the famous Landau theory of superfluidity at finite temperatures. This theory is based on coupled hydrodynamic equations for the superfluid and normal fluid components. Landau's two-fluid description is only valid when collisions among the thermal excitations making up the normal fluid are strong enough to produce local hydrodynamic equilibrium. These two-fluid equations were originally developed for liquid. He but are thought to be generic in form, describing the collision-dominated hydrodynamic region of all Bose superfluids. In this chapter, we will consider the solutions of the two-fluid equations mainly for a uniform superfluid. We discuss the existence of second sound (involving the out-of-phase motion of the superfluid and normal fluid components) as a characteristic feature of a Bose superfluid at finite temperatures. This chapter gives background material needed for Chapters 15–19. In Chapter 15, we will show that, in the appropriate limit, the Landau two-fluid equations can be derived from the ZNG coupled equations given in Chapter 3 for a trapped dilute Bose-condensed gas. In Chapters 17–19, we extend this discussion and derive the Landau–Khalatnikov two-fluid equations, which include hydrodynamic damping associated with various transport coefficients. Useful reviews of the two-fluid equations in the context of dilute spatially uniform Bose-condensed gases are given by Pethick and Smith (2008, Chapter 10) and Pitaevskii and Stringari (2003, Chapter 6). History of two-fluid equations The original discovery of superfluidity in liquid He was dramatically announced with the publication of the famous back-to-back papers of Kapitza (1938) in Moscow and Allen and Misener (1938) in Cambridge. These and subsequent experiments in the next few years showed that superfluid He could exhibit very bizarre hydrodynamic behaviour compared to classical liquids.

Cambridge University Press eBooks•Feb 19, 2009•1 citation

Superfluidity

Allan Griffin

In January 1938, the Russian physicist Pyotr Kapitza reported that, at temperatures below 2.2 degrees above absolute zero, liquid helium seemed to flow with no loss of energy due to friction. The term “superfluid” was suggested to describe this new behavior—which was eventually understood as a dramatic illustration of the quantum-mechanical wave nature of matter. The study of such macroscopic quantum effects now lies at the heart of modern physics. The complete analogue of the Landau two-fluid hydrodynamic equations, including various transport coefficients associated with the normal fluid, has recently been derived for the trapped atomic Bose gases. In the next few years, experiments should be able to probe this two-fluid domain in Bose gases, a region first explored with such spectacular success in superfluid helium by Kapitza and Allen and Misener. The complexity of analyzing any liquid at a microscopic level has resulted in many aspects of the theory of superfluid.

University of Chicago Press eBooks•Jan 1, 2003•0 citations
For building up a theory of superfluid Helium-4, Lev Landau ingeniously unified the principles of quantum mechanics with the principles of hydrodynamics. By introducing a velocity operator he was able to derive a quantum analogue of the mass density flux equation and a quantum Euler equation. The article shows that it is possible to formulate a consistent quantum theory of the ideal liquid based on Landau's Quantum Hydrodynamics. This means that it is added the quantum analogue of the momentum density flux equation, the energy density flux equation, the entropy density flux equation, and a quantum vorticity equation. These are all manifestly in agreement with the classical limit of the flux equations of classical ideal liquids. It is also shown that the various aspects that have been criticised about Landau's Quantum Hydrodynamics - e.g. the application of the inverse of the mass density operator - are not justified. Such a consistent quantum theory of an ideal liquid may therefore provide a starting point of a complementary approach in the theoretical research of superfluid Helium 4.
Landau's quantized hydrodynamics is derived from the microscopic Hamiltonian, which is written in terms of the density and velocity operators. In Euler's equation for the velocity operator, the force per unit mass has a form different from Landau's and is given explicitly in terms of the two-body potential. A quantum pressure term, which does not occur in Landau's theory, also appears. New realizations of the density and velocity operators are given which satisfy the commutation relations, but must act in a Hilbert space larger than Fock space. It is shown that the formalism is without meaning in Fock space owing to the nonexistence of certain operators.
A nonlinear interaction term is added to the Schr\"odinger equation which makes it exactly equivalent to classical mechanics. In the hydrodynamic formulation of this equation, the usual classical theory is obtained. This theory is a Lagrangian density formulation of hydrodynamics and is used to calculate the density profile of superfluid helium in a line vortex. The theory can be canonically quantized, in the sense of quantum field theory, in terms of the density and velocity fields. The quantized theory is not equivalent to that which one would obtain if the quantization were done in the usual way in terms of the field operations ${\ensuremath{\psi}}^{*}$ and ($\ensuremath{\psi}$). The Schr\"odinger equation, with this interaction, has localized solutions with a definite energy and momentum and does not give quantized energy levels for a one-dimensional harmonic oscillator.
A Fermi-liquid approach to the theory of superfluidity is developed, based on a generalization of the combinatoric expression for the entropy to the case in which the state of the Fermi liquid is described not only by a normal but also by an anomalous distribution function. The energy of the superfluid Fermi liquid is given in the form of a functional of these distribution functions. In the weak-coupling approximation the Fermi-liquid approach is equivalent to the BCS theory. The interaction between the Fermi liquid and an electromagnetic field is gauge invariant. In addition to the singlet-pairing theory the triplet-pairing theory is also developed. A unified variational principle that gives rise to the theory of superfluidity (superconductivity) is used to obtain the London equations and the Ginzburg–Landau equation. Applications of this theory are made to the derivation of an equation for ideal two-component hydrodynamics.
This review presents a microscopic construction of the thermodynamics and hydrodynamics of superfluid bosons and fermions with singlet pairing, based on the concept of quasiaverages and the hypothesis of reduced description. Here we do not assume that the Hamiltonian possesses any dynamical symmetry. This has permitted obtaining results pertaining to both Galilean-invariant and relativistic systems. Account is taken of dissipative processes. The kinetic coefficients are presented in terms of the correlation functions of the flux operators. The approach is extended to solutions of quantum liquids. The influence of an external ac field on superfluid systems is studied, and the low-frequency asymptotic behavior of the Green's function is found in the hydrodynamic approximation. The symmetry properties of the equilibrium state are formulated, and the thermodynamics is constructed for superfluid Fermi systems with triplet pairing (the superfluid phases 3 He-B and 3 He-A). For the latter the flux densities of the additive integrals of motion are found in a state of equilibrium and the equations of "ideal" hydrodynamics are derived.
In this chapter, we review the famous Landau theory of superfluidity at finite temperatures. This theory is based on coupled hydrodynamic equations for the superfluid and normal fluid components. Landau's two-fluid description is only valid when collisions among the thermal excitations making up the normal fluid are strong enough to produce local hydrodynamic equilibrium. These two-fluid equations were originally developed for liquid. He but are thought to be generic in form, describing the collision-dominated hydrodynamic region of all Bose superfluids. In this chapter, we will consider the solutions of the two-fluid equations mainly for a uniform superfluid. We discuss the existence of second sound (involving the out-of-phase motion of the superfluid and normal fluid components) as a characteristic feature of a Bose superfluid at finite temperatures. This chapter gives background material needed for Chapters 15–19. In Chapter 15, we will show that, in the appropriate limit, the Landau two-fluid equations can be derived from the ZNG coupled equations given in Chapter 3 for a trapped dilute Bose-condensed gas. In Chapters 17–19, we extend this discussion and derive the Landau–Khalatnikov two-fluid equations, which include hydrodynamic damping associated with various transport coefficients. Useful reviews of the two-fluid equations in the context of dilute spatially uniform Bose-condensed gases are given by Pethick and Smith (2008, Chapter 10) and Pitaevskii and Stringari (2003, Chapter 6). History of two-fluid equations The original discovery of superfluidity in liquid He was dramatically announced with the publication of the famous back-to-back papers of Kapitza (1938) in Moscow and Allen and Misener (1938) in Cambridge. These and subsequent experiments in the next few years showed that superfluid He could exhibit very bizarre hydrodynamic behaviour compared to classical liquids.
In January 1938, the Russian physicist Pyotr Kapitza reported that, at temperatures below 2.2 degrees above absolute zero, liquid helium seemed to flow with no loss of energy due to friction. The term “superfluid” was suggested to describe this new behavior—which was eventually understood as a dramatic illustration of the quantum-mechanical wave nature of matter. The study of such macroscopic quantum effects now lies at the heart of modern physics. The complete analogue of the Landau two-fluid hydrodynamic equations, including various transport coefficients associated with the normal fluid, has recently been derived for the trapped atomic Bose gases. In the next few years, experiments should be able to probe this two-fluid domain in Bose gases, a region first explored with such spectacular success in superfluid helium by Kapitza and Allen and Misener. The complexity of analyzing any liquid at a microscopic level has resulted in many aspects of the theory of superfluid.