Answer to Question #246373 in Quantum Mechanics for hanata yuji

Question #246373

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

1
Expert's answer
2021-10-05T10:07:41-0400

Let us consider the linear combination ψ(x)=C1eikx+C2eikx\psi(x) = C_1 e^{i k x} + C_2 e^{- i k x}, where C1,C2C_1, C_2 - constants. Then,

d2dx2ψ(x)=C1(ik)2eikx+C2(ik)2eikx=k2(C1eikx+C2eikx)=k2ψ(x)\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, ψ(x)\psi(x) is an eigenfunction of differential operator d2dx2\frac{d^2}{d x^2} with corresponding eigenvalue k2-k^2.


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