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!