Question

Consider two normalized eigen functions ψ1 and ψ2 , corresponding to the same eigen value. If

∫ψ∗1ψ2dτ=d ,

where d is real

Find a normalized linear combinations of ψ1 and ψ2 that are orthogonal to

(a) ψ1

(b) ψ1+ψ2

Note: The coefficients of the linear combinations need not be real

Answer 1

ψ(x) = ψ1(x) + ψ2(x)

ψ⁺ = 1/1.41 * (ψ1 + xψ2);

ψ^- = 1/1.41 * (ψ1 - xψ2);

d/dT = $\int$ (ψ₁ + ψ₂) * -(ψ₂)