1. The problem statement, all variables and given/known data Show that for a 1d potential V (-x)=-V (x), the eigen functions of the Schrödinger equation are either symmetric/ anti-symmetric functions of x. 2. Relevant equations 3. The attempt at a solution I really don't know how to do it for odd potential. Let me show you how I am doing it for even potential. V (x)=V (-x) - (h2 / 2μ) [d2ψ(x)/dx2] + V (x) ψ(x)=E ψ (x).... (1) Making x to -x transformation we get - (h2 / 2μ) [d2ψ(-x)/dx2] + V (x) ψ(-x)=E ψ (-x)..(2) where i use V (-x)=V (x) Comparing (1) and (2) we see that ψ (x) and ψ(-x) eigenfunctions belongs to the same energy E. For a non degenerate state, ψ (-x ) must be a multiple of ψ (x): Ψ (-x) = λψ (x). Clearly ψ (x)= λψ (-x)=λ2ψ (x). λ2=1 or λ=+/- 1 So ψ(-x)=+/- ψ(x) Now if i use odd potential in (2) eigenfunctions no longer belong to same energy E. The hamiltonian becomes weird for negative potential. Please help.