the kinetic energy operator is proportional tod 2 /dx 2 . which of the following

Question

the kinetic energy operator is proportional tod2/dx2. which of the following functions areeigenfunctions of d2/dx2 :
a) eax   b) e -a(x^2) c) k d) kx2 e) ax + b f) cos(kx + 5) ? Give the corresponding eigenvalue where appropriate. the kinetic energy operator is proportional tod2/dx2. which of the following functions areeigenfunctions of d2/dx2 :
a) eax   b) e -a(x^2) c) k d) kx2 e) ax + b f) cos(kx + 5) ? Give the corresponding eigenvalue where appropriate.

Explanation / Answer

Let T = ke operator x is an eigenfunction of T if    T*x =t*x,    where t is a scalar (the eigenvalue) a)   yes,     T*exp(ax) = a^2*exp(ax)    eigenvalue is a^2 b)   T*exp(-ax^2) = d(-2ax*exp(-ax^2) )/dx =-2a*exp(-ax^2) + 4a^2x^2*exp(-ax^2)     =>not eigenfunction c)   T*(k) = d2(k)/dx2 = 0 => noteigenfucntion d)   T*(kx^2) = 2k   => noteigenfunction e)   T*(ax+b) = 0   => not eigenfunction f)    T*cos(kx+5) =-k2*cos(kx+5)       YES =>   eigenvalue is -k^2

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