Discussion Overview
The discussion revolves around the implications of left-right symmetry on the eigenstates of a two-electron system with identical orbitals. Participants explore whether eigenstates can exist in a form that does not respect this symmetry, particularly in the context of fermionic operators and Hamiltonians.
Discussion Character
- Debate/contested
- Technical explanation
- Conceptual clarification
Main Points Raised
- One participant questions whether an eigenstate of the form ##|1,1,0,0>## can exist in a left-right symmetric system or if it must be expressed as a symmetric combination like ##\frac{|1,1,0,0>\pm|0,0,1,1>}{\sqrt(2)}##.
- Another participant states that the only symmetry requirement for states is related to the interchange of identical particles, suggesting that general states do not need to adhere to specific symmetry forms unless they are eigenstates of certain operators.
- A participant seeks clarification on whether left-right symmetry necessitates eigenstates of the Hamiltonian to be in a symmetric form or if non-symmetric forms are permissible.
- There is a reiteration that for left-right symmetric systems, eigenstates can exist that do not exhibit this symmetry, particularly if there is degeneracy involved.
- One participant draws an analogy to parity in one-particle systems, indicating that without degeneracy, eigenstates must have even or odd symmetry, while otherwise, symmetry is not required.
Areas of Agreement / Disagreement
Participants express differing views on the necessity of symmetry in eigenstates, with some arguing that symmetry is required under certain conditions, while others contend that it is not a strict requirement. The discussion remains unresolved regarding the implications of left-right symmetry on eigenstate forms.
Contextual Notes
Participants reference the role of degeneracy and the nature of Hamiltonians in determining the symmetry of eigenstates, but the specific conditions under which these factors apply remain unclear.
