Mathematical Foundations of Quantum Hydrodynamics

This advanced textbook introduces the concept of quantum hydrodynamics, emphasizing the role of water flow in influencing quantum states. It covers the derivation of governing equations, explores the analogies between hydrodynamic and quantum wavefunctions, and applies group theory to analyze symmetry breaking and coherence in quantum systems induced by fluid motion.

Key Features:

- Clear and concise explanations of quantum hydrodynamics and its core principles.
- Integration of quantum mechanics and fluid dynamics through innovative methods.
- Practical Python coding examples for real-world application and deeper comprehension.
- In-depth explorations of symmetry, coherence, and quantum phenomena influenced by fluid flow.
- Thorough analysis of relevant mathematical techniques including group theory.

What You Will Learn:

- Grasp the foundational postulates of quantum mechanics.
- Understand the classical principles of fluid dynamics.
- Bridge quantum mechanics and fluid dynamics via the Madelung transformation.
- Derive the Madelung equations from the Schrödinger equation.
- Explore Bohmian mechanics and its connection to quantum hydrodynamics.
- Analyze the quantum potential's effect on fluid flow and particle trajectories.
- Apply the continuity equation to probability flow in quantum systems.
- Formulate Euler equations for describing quantum pressure in fluids.
- Interpret wavefunctions as fluid-like entities, understanding amplitude and phase.
- Investigate vorticity and its implications in quantum fluids.
- Examine phase and group velocities within a quantum hydrodynamic context.
- Study the nonlinear Schrödinger equation in quantum fluid scenarios.
- Discover soliton solutions as stable quantum fluid phenomena.
- Dive into Bose-Einstein condensates and their hydrodynamic modelling.
- Analyze superfluidity and quantized vortices in quantum systems.
- Assess stability criteria for superfluidity using hydrodynamic models.
- Examine wavefunction topology and fluid flow patterns.
- Discuss quantum tunneling phenomena through hydrodynamic analogies.
- Apply group theory to analyze symmetries in quantum fluids.
- Correlate symmetry operations with conservation laws in quantum hydrodynamics.
- Use Noether's theorem to derive conservation laws in both fluid and quantum systems.
- Study the roles of Lie groups and algebras in quantum hydrodynamics.
- Utilize representation theory for understanding quantum state transformations.
- Analyze spontaneous symmetry breaking in quantum systems.
- Investigate coherence and the impact of fluid interactions on decoherence.
- Explore gauge symmetries and their role in quantum hydrodynamic equations.
- Understand quantization of circulation in superfluids.
- Examine phase transitions and critical points in quantum fluids.
- Study hydrodynamic instabilities such as Kelvin-Helmholtz and Rayleigh-Taylor.
- Apply density functional theory to describe inhomogeneous quantum fluids.