Determine which of the following functions are eigenfunctions of the inversion operator î (which has the effect of making the replacement x → - x) : (a) x3 - kx, (b) cos kx, (c) x2 + 3x - 1. State the eigenvalue of î when relevant.

x³ - kx is eigenfunction of eigenvalue - 1 cos (kx) is eigenfunction of eigenvalue 1 x³+3x-1 is not an eigenfunction

Step-by-step explanation:

Lets see how each expression is modified by the operator:

1 - > 1 x - > - x x² - > (-x) ² = x² x³ - > (-x) ³ = - x³

Thus,

x³-kx - > - x³+kx = - (x³-kx) - 1 is eigenvalue cos (kx) - > cos (k (-x)) = cos (-kx) = cos (kx) (because the cosine is an even function) 1 is eigenvalue x²+3x-1 - > x²-3x-1 (not an eigenfunction)