Which of the functions below is a common eigenfunction of $\frac{\mathrm{d}}{\mathrm{dx}}$ and $\frac{\mathrm{d}^{2}}{\mathrm{dx}^{2}}$ operators?
(a)$\cos \mathrm{x}$
(b)kx
(c)$\mathrm{e}^{\mathrm{ix}}$
(d)$\mathrm{e}-\mathrm{x}^{2}$
Answer
Answer: C ✓ checked by 4AB · confidence high
The source book printed no answer; this one was worked out and checked — see the explanation.
Explanation
d/dx e^{ix} = i e^{ix} and d²/dx² e^{ix} = −e^{ix}: e^{ix} is an eigenfunction of both. cos x is not an eigenfunction of d/dx, and kx and e^{−x²} of neither.