I'm reading the Cohen-Tannoudji book and I found somthing I dont understand(adsbygoogle = window.adsbygoogle || []).push({});

in stationary perturbation theory.

the problem the Hamiltonian is split in the known part an the perturbation:

[tex]

H=H_{o}+\lambda \hat{W}

[/tex]

[tex]

H_{o}|\varphi_{p}^{i}\rangle=E_{p}^{o}|\varphi_{p}^{i}\rangle

[/tex] (1)

and we want to solve the problem:

[tex]

H(\lambda)|\Psi(\lambda)\rangle=E(\lambda)\Psi(\lambda)\rangle[/tex] (2)

Expanding in [tex]\lambda[/tex] series equation (2) I get after equating each term:

[tex]

zeroth: (H_{o}-E_{o})|0\rangle=0

[/tex] (3)

[tex]

first: (H_{o}-E_{o})|1\rangle+(\hat{W}-E_{1})|0\rangle=0

[/tex] (4)

[tex]

second: (H_{o}-E_{o})|3\rangle+(\hat{W}-E_{1})|2\rangle

E_{2}|1\rangle-E_{3}|0\rangle=0

[/tex] (5)

from normalizing the wave fuction order by order I get:

[tex]

zeroth: \langle0|0\rangle=1

[/tex] (6)

[tex]

first: \langle0|1\rangle= \langle1|0\rangle=0

[/tex] (7)

[tex]

second: \langle0|2\rangle=

\langle2|0\rangle=-\frac{1}{2}\langle1|1\rangle

[/tex] (8)

Solution for the non-degenerated level[tex]

H_{o}|\varphi_{n}^{o}\rangle=E_{n}^{o}|\varphi_{n}^{o}\rangle

[/tex]

zeroth order:

[tex]

E_{o}=E_{n}^{o}

[/tex]

[tex]

|0\rangle=|\varphi_{n}\rangle

[/tex]

first order projecting (4) onto the vector [tex]|\varphi_{n}\rangle[/tex]

[tex]

E_{n}(\lambda)=E_{n}^{o}+\langle\varphi_{n}|W|\varphi_{n}\rangle

[/tex]

now this is the part that I dont understand:

when finding the eigenvector |1> the project equation (4) onto [tex]|\varphi_{p}^{i}\rangle[/tex] why the putting the supscript i if it is non-degenerated??????

[tex]|\Psi_{n}(\lambda)\rangle=|\varphi_{n}\rangle+\sum_{p\neq

n}\sum_{i}\frac{\langle\varphi_{p}^{i}|W|\varphi_{n}\rangle}{E_{n}^{o}-E_{p}^{o}}|\varphi_{p}^{i}\rangle[/tex]

see the book page 1101

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Perturbation theory using Cohen-Tannoudji

**Physics Forums | Science Articles, Homework Help, Discussion**