Mathematical Foundations of Quantum Hydrodynamics

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.



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