Question

If Hyı = Evi, Hv2 = EV2, and HVs = 3by3, Show whether each function below is an eigenfunction of H and if so state the eigenvalue. Note cı, c2, c3, are constants.

Refer to the integrals and equations below when/if needed.

IMPORTANT: *Typed* step by step answers preferred. If hand written, please write clearly and legibly. Please be sure answer is CORRECT. Thanks!

Answer 1

H psi_1 = E psi_1: H psi_2 = E psi_2: H psi_3 = 3E psi_3 a) given psi = pi C_1 psi_1 + pi C_2 psi_2 So, H psi = H(pi C_1 psi_1 + pi C_2 psi_2) = pi C_1 E psi_1 + pi C_2 = E psi_2 = E(pi C_1 psi_1 + pi C_2 psi_2) = E psi therefore psi = pi C_1 psi_1 + pi C_2 psi_2 is an eigenfunction of H with eigenvalue E. b) psi = C_3 psi_3/6 - C_1 psi_1 So, H psi = H(C_3 psi_3/6 - C_1 psi_1) = C_3/6 3E psi_3 - C_1 E psi_1 = E(C_3/2 psi_3 - C_1 psi_1) therefore = C_3 psi_3/6 - C_1 psi_1 is not the eigenfunction of H. c) psi = 3C_2 psi_2 + C_3 psi_3 So, H psi = H(3C_2 psi_2 + C_3 psi_3) = 3C_2 E psi_2 + 3E C_3 psi_3 = 3E(C_2 psi_2 + C_3 psi_3) therefore psi = 3C_2 psi_2 + C_3 psi_3 is not an eigenfunction of H.