ANY HELP IS MUCH APPRECIATED, I ALWAYS RATE, THANKS!!! The kinetic energy operator is proportional tod2/dx2. Which of the following functions areeigenfunctions of d2/dx2? a) eax b) e-ax^2 c) k d) kx2 e) ax + b f) cos (kx + 5) Give the corresponding eignvalues where appropriate. ANY HELP IS MUCH APPRECIATED, I ALWAYS RATE, THANKS!!! The kinetic energy operator is proportional tod2/dx2. Which of the following functions areeigenfunctions of d2/dx2? a) eax b) e-ax^2 c) k d) kx2 e) ax + b f) cos (kx + 5) Give the corresponding eignvalues where appropriate. e) ax + b f) cos (kx + 5) Give the corresponding eignvalues where appropriate.
Explanation / Answer
We Know that : The operator forthe Kinetic energy is given as d2 /dx2 a. d2/ dx2 ( e ax ) = a2 eax b. d2 /dx2 ( e -ax^2 ) = d /dx ( -2ax * e-ax^2 ) = -2a [ x ( -2ax e-ax^2 ) +e-ax^2 ] = 4a2x2 e-ax^2 -2ae-ax^2 c. d2/ dx2 ( K ) = 0 d. d2/ dx2 ( Kx2 ) = 2K e. d2/ dx2 ( ax+b ) = 0 f. d2/ dx2 Cos ( kx+5) = K2 Cos ( Kx+5) b. d2 /dx2 ( e -ax^2 ) = d /dx ( -2ax * e-ax^2 ) = -2a [ x ( -2ax e-ax^2 ) +e-ax^2 ] = 4a2x2 e-ax^2 -2ae-ax^2 c. d2/ dx2 ( K ) = 0 d. d2/ dx2 ( Kx2 ) = 2K e. d2/ dx2 ( ax+b ) = 0 f. d2/ dx2 Cos ( kx+5) = K2 Cos ( Kx+5)