− − ℏ ℏ 2 2 m d 2 ψ ψ 1 d x 2 + V ψ ψ 1 = E ψ ψ 1 − − ℏ ℏ 2 2 m d 2 ψ ψ 2 d x 2 + V ψ ψ 2 = E ψ ψ 2 Multiplying the first equation by ψ ψ 2 and the second by ψ ψ 1 and subtracting one from the other, we get: ψ ψ 1 d 2 d x 2 ψ ψ 2 − − ψ ψ 2 d 2 d x 2 ψ ψ 1 = 0 Integrating both sides ψ ψ 1 d ψ ψ 2 d x − − ψ ψ 2 d ψ ψ 1 d x = constant In case of well-defined and normalizable wave functions, the above constant vanishes, provided both the wave functions vanish at at least one point, and we find: ψ ψ 1 ( x ) = c ψ ψ 2 ( x ) where c is, in general, a complex constant. For bound state eigenfunctions (which tend to zero as x → → ∞ ∞ ), and assuming V and E satisfy the condition given above, it can be shown that also the first derivative of the wave function approaches zero in the limit x → → ∞ ∞ , so that the above constant is zero and we have no degeneracy.