SUMMARY
The discussion centers on the implications of modifying the Hamiltonian of a simple harmonic oscillator and its effects on the eigenstates represented by ##\ket{n}##. Participants assert that perturbation theory is unnecessary since the matrix elements ##\braket{m|\hat{x}|n}## and ##\braket{m|\hat{x}^2|n}## remain unchanged regardless of Hamiltonian modifications. The completeness of the basis formed by ##\ket{n}## is emphasized, although questions arise regarding potential changes to the Hilbert space size due to modifications.
PREREQUISITES
- Understanding of quantum mechanics, specifically harmonic oscillators
- Familiarity with Hilbert spaces and eigenstates
- Knowledge of perturbation theory in quantum mechanics
- Basic grasp of matrix elements and their significance in quantum systems
NEXT STEPS
- Study the implications of Hamiltonian modifications on quantum systems
- Explore the completeness of bases in Hilbert spaces
- Learn about perturbation theory applications in quantum mechanics
- Investigate the relationship between eigenstates and matrix elements in quantum mechanics
USEFUL FOR
Quantum physicists, students of quantum mechanics, and researchers interested in the mathematical foundations of quantum systems and their modifications.