Mixed State Eigenfuntion Equations

[kd001]

H(a1u1 + a2u2) = a1E1 u1 + a2 E2u2

H is the Hamiltonian energy operator, a1 and a2 are normalisation constants, u1 and u2 are wave functions, E1 and E2 are the eigenvalues. Is it possible to calculate the values of E1 and E2 from the above equation if everything else is given? It should be possible to calculate the two eigenvalues given the two eigenfunctions shouldn't it? But there are two unknowns and just one equation. I'm not asking for a solution, whether it is possible to do it or not and if yes some hints as how to go about it.

Thanks.

 

[VanOosten]

yes it does look possible perhaps splitting it into 2 equations would help:
H(a1u1)=E1(a1u1)
h(a2u2)=E2(a2u2)
and then solve the eigenvalue problem for E1 and E2

 

[kd001]

yes it does look possible perhaps splitting it into 2 equations would help:
H(a1u1)=E1(a1u1)
h(a2u2)=E2(a2u2)
and then solve the eigenvalue problem for E1 and E2

Thanks. But Would 'h' be the same as 'H'? Also I believe the normalisation constants wouldn't be the same when the equation is split.

 

[VanOosten]

yes i did mean H, that was a typo
and the normalization constants will still be the same because the wave function itself is not being changed you are just looking at the u1 terms first then the u2 terms second