Question

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.

Answer 1

Answer:

• 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)