Variation Method for Higher Energy States

[aim1732]

The variation method for approximating the the ground state eigenvalue, when applied to higher energy states requires that the trial function be orthogonal to the lower energy eigenfunctions.In that respect this book I am referring(by Leonard Schiff) mentions the following function as the general function orthogonal to the previous level eigenfunction:

ψ - (U_Eo × ∫((U_Eo[conjug])ψ dτ)) for any general fn ψ

Can somebody prove the above function's orthogonality to U_Eo ? I have tried to do it but could not come up with it?

 

[cgk]

This is simply the application of a projector
$$P = \bigg(1 - \sum\nolimits_{i=1}^M |i\rangle\langle i|\bigg)$$
to the trial wave function psi, where i sums over the lower energy eigenstates. For this to be valid, the lower states need to be orthonormal:
$$\langle i | j \rangle = \delta_{ij}$$
(it is always possible to choose a set of eigenvectors of a Hermitian operator in such a way that they are orthonormal to each other; for non-degenerate eigenstates the orthogonality comes automatically).

Now take a closer look at this projector and take into account the lower state orthogonality. You will easily see that if you add any linear combination of a lower eigenstate to psi, this linear combination will be zeroed out by P. Thus
$$\langle i|P \psi\rangle=0$$
for any lower eigenstate i and any wave function psi.

 

[aim1732]

Thanks.Got it.