Solving Matrix Eigenvalue Equation for ψ_{200} and ψ_{210} States

[M. next]

In order to apply perturbation theory to the ψ$_{200}$ and ψ$_{210}$ states, we have to solve the matrix eigenvalue equation.

Ux=λx where U is the matrix of the matrix elements of H$_{1}$= eEz between these states.

Please see the matrix in attachment 1.

where <2,0,0|z|2,1,0>=<2,1,0|z|2,0,0>=3a$_{o}$

Solving this matrix we get, λ$_{1}$=3ea$_{o}$|E| and λ$_{2}$= -3ea$_{o}$|E|

Then we find eigenvectors to get x$_{1}$ =(1/√2 1/√2)$^{T}$ and x$_{2}$= (1/√2 -1/√2)$^{T}$

**** They finally said that ψ$_{1}$ = (ψ$_{200}$ + ψ$_{210}$)/√2

and ψ$_{2}$ = (ψ$_{200}$ - ψ$_{210}$)/√2

How did they get this? How did they combine ψ$_{1}$ and ψ$_{2}$ as follows? It is just the linear combination that I don't get. Thank you.

 

[M. next]

Please note that this is for Hydrogen atom, n=2 where we have degeneracy!