Answer to Question #246373 in Quantum Mechanics for hanata yuji

Question

show that linear combination of e^ikx and e^-ikx is a eigen function of d²/dx²

Answer 1

Let us consider the linear combination $\psi(x) = C_1 e^{i k x} + C_2 e^{- i k x}$, where $C_1, C_2$ - constants. Then,

$\frac{d^2}{d x^2} \psi(x) = C_1 (i k)^2 e^{i k x} + C_2 (-i k)^2 e^{-i k x} = -k^2(C_1 e^{i k x} + C_2 e^{-i k x}) = - k^2 \psi(x)$.

Hence, $\psi(x)$ is an eigenfunction of differential operator $\frac{d^2}{d x^2}$ with corresponding eigenvalue $-k^2$.