- #1

- 430

- 2

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?

- Thread starter aim1732
- Start date

- #1

- 430

- 2

ψ - (U

Can somebody prove the above function's orthogonality to U

- #2

cgk

Science Advisor

- 521

- 42

[tex]P = \bigg(1 - \sum\nolimits_{i=1}^M |i\rangle\langle i|\bigg)[/tex]

to the trial wave function psi, where

[tex]\langle i | j \rangle = \delta_{ij}[/tex]

(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

[tex]\langle i|P \psi\rangle=0[/tex]

for any lower eigenstate

- #3

- 430

- 2

Thanks.Got it.

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