1. The problem statement, all variables and given/known data Consider the Parity Operator, P', of a single variable function, defined as P'ψ(x)=P'(-x). Let ψ1=(1+x)/(1+x^2) and ψ2=(1+x)/(1+x^2). I have already shown that these are not eigenfunctions of P'. The question asks me to find what linear combinations, Θ=aψ1+bψ2 are eigenfunctions of P', and what are their eigenvalues. 2. Relevant equations I have already done most of the calculations through this method: Taken P'Θ=P*Θ=P*(aψ1+bψ2) [P here is the eigenvalue], expanded and equated coefficients of the resultant polynomial. This has given me: (a+b)(P-1)=0 and (P+1)(b-a)=0 So the solutions are a=-b, P=1 (from the first equation) and P=-1 and a=b (from the second). 3. The attempt at a solution Now, I understand that when a=b, the linear combination will have even parity, and this should have an eigenvalue of P=1. What confuses me is that the equation that gives the a=b solution also gives P=-1, which corresponds to odd parity. I feel like there is something simple I am missing, as I seem to have the right relations between a and b, and eigenvalues, but I've somehow mixed them up. !!Further more!!, the following questions asks "what can we add to a given wave function, which is not an eigenfunction of P', to make parity eigenfunctions?". I feel this might be more obvious when I understand the above question, but my reasoning would be that we should add the complex conjugate of the wave function. This is because this would cancel out the imaginary part. Am I on the right track here? I would greatly appreciate any guidance in making sense of my results. Many thanks.