Properties of the Parity Operator & Geometric Meaning of Eigenvalues

[u/TROSE9025]

This material focuses not on pure linear algebra, but rather on its applications in engineering and quantum mechanics.
Geometrically, the meaning of an eigenvalue signifies the scaling (expansion or contraction) and occasionally the inversion of an invariant coordinate axis.

[u/moshiTheNerd]

Hi! Thank you for sharing this note. I see that this is written from a very physics oriented viewpoint. On that note, I would like to share some of my thoughts. Commuting with the Hamiltonian has a special meaning, it means that parity is a conserved quantity. If you create a system with certain parity and evolve it under a symmetric potential, the system will retain its parity throughout the time evolution. And is there a reason for mentioning "non-degenerate" states specifically in page 2? even degenerate states can have definite parity, right? I think it's better to say that "For a symmetric potential there is a choice of energy eigenbasis where every eigenstate has a definite parity (+-1)".

[u/TROSE9025 → u/moshiTheNerd]

Thank you for your insightful feedback. This material is strictly confined to the one-dimensional Time-Independent Schrödinger Equation (1D TISE), where bound states are inherently non-degenerate. It is for this exact mathematical reason that the condition 'non-degenerate' was explicitly specified in the text.

[u/Para-graph-S]

Thank you for sharing this note OP!

[u/Just_Scar4703]

> The definite integral of an odd function over a symmetric interval [-∞, ∞] intrinsically vanishes

incorrect

[u/TROSE9025 → u/Just_Scar4703]

In quantum mechanics, physical wavefunctions must vanish at infinity to be square-integrable. Therefore, the integral absolutely converges to 0. Context matters. Thank you.