Is the function Psi(y) = e^([(-6y^2)/2]) an eigenfunction of the operator [^(A)]

Question

Is the function Psi(y) = e^([(-6y^2)/2]) an eigenfunction of the operator [^(A)] = [(d)/(dy)]? ("yes", "no") Give the result of operating on the function Psi (y) = e^([(-6y^2)/2]) with the operator [^(A)] = -[(d^2)/(dy^2)] Is the result of operating on the function Psi (y) = e^([(-6y^2)/2]) with the operator [^(A)] = 36y^2 Is the function Psi (y) = e^([(-6y^2)/2]) an eigenfunction of the operator 36y^2? ("yes", "no") Give the result of operating on the function Psi (y) = e^([(-6y^2)/2]) with the operator [^(A)] = - [(d^2)/(dy^2)]+36y^2 Is the function Psi (y) = e^([(-6y^2)/2]) an eigenfunction of the operator on [^(A)] = -[(d^2)/dy^2]+ 36 y^2?("yes", "no") What is the eigenvalue of [^(A)] = -[(d^2)/dy^2]+ 36 y^2? operating on Psi (y) = e^([(-6y^2)/2])

Explanation / Answer

If A = a, then is called eigen function of the operator A and a is the eigen value.

1) Ans = yes

GIven function (y) = e[(-6y^2)/2]

   Given operator = d/dy

Hence,

         d/dy [ ] = d/dy{e[(-6y^2)/2]}

                     = e[(-6y^2)/2]. d/dy [(-6y^2)/2]     

                      = e[(-6y^2)/2]. [-12y/2]                                   { d/dy[ey] = ey and d/dy[yn] = nyn-1]

                      = -6y e[(-6y^2)/2]

                      = -6y

      d/dy [ ] = -6y

Therefore,

e[(-6y^2)/2] is the eigen function of the operator d/dy.

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