SUMMARY
The discussion centers on the derivation of the energy current in quantum mechanics, specifically in relation to the Hamiltonian density. The continuity equation for energy is established as ∂H/∂t + ∇·j_E(𝑟,t) = 0, where H represents the Hamiltonian density. Participants reference Weinberg's QFT treatise, particularly Chapter 7.3, to derive general results but express a need for a more straightforward expression for energy current analogous to the particle current. The conversation emphasizes the importance of the energy-momentum tensor T^{μν} and the necessity of Hermitian operators in defining the Hamiltonian density.
PREREQUISITES
- Understanding of quantum mechanics continuity equations
- Familiarity with Hamiltonian density and energy-momentum tensor
- Knowledge of Noether's theorem and its application in field theory
- Proficiency in tensor calculus and four-vector notation
NEXT STEPS
- Study the derivation of energy current using the energy-momentum tensor
T_{0μ}(x)
- Explore Chapter 7.3 of Weinberg's QFT treatise for insights on energy current
- Learn about the application of Noether's theorem in deriving conserved currents
- Investigate the relationship between Hamiltonian density and Hermitian operators in quantum mechanics
USEFUL FOR
Physicists, quantum mechanics students, and researchers focusing on energy transport phenomena and thermal conductivity in quantum systems.