In perturbation theory, the wavefunctions and energies 
for a system are expanded in terms of those for an unperturbed 
system having known solutions.

1. Non-perturbed Hamiltonian

Let's suppose that a quantum system has a Hamiltonian H and a 
wave function or the state ψ, with an eherhy E. The Schrodinger 
equation is then written as: H ψ = E ψ. If the system has "n" 
eigenstates |φ_n> with their corresponding "n" 
probilities P_n and "n" enrgies E_n, the state of the system 
can be written as |ψ > = Σ P_n|φ_n>, 
and the time-independendent Schrodinger equation for each 
eigenstate is rewritten as: H |φ_n> = E_n|φ_n>. 

At first, the system is not perturbed, we write then the 
time-independendent Schrodinger equation with "0" indexes as: 

H⁽⁰⁾ |φ⁽⁰⁾_n> = E⁽⁰⁾_n|φ_n>.

To write things simply, let's write |φ_n> = |n>.

H⁽⁰⁾ |n⁽⁰⁾> = E⁽⁰⁾_n|n⁽⁰⁾>

2. Perturbed Hamiltonian

Under some perturbation, the Hamiltian of the system takes 
the new following form:
H = H⁽⁰⁾ + λ H^(p), where λ is a dimensionless parameter chosen 
to maintain a small perturbation that has a small effect on the 
system. Now, the time-independendent Schrodinger equation is: 
H |n> = E_n|n>  or [H⁽⁰⁾ + λ H^(p)]|n> = E_n|n>

The equation to solve is then:

(H⁽⁰⁾ + λ H^(p))|n> = E_n|n>    (2.1)

The idea is to develop |n>  and its corresponding energy 
E_n as a power series in λ as follows:

E_n =  E⁽⁰⁾_n + λ E⁽¹⁾_n + λ² E⁽²⁾_n + ...    (2.2)

|n>  = |n⁽⁰⁾> + λ |n⁽¹⁾> + λ² |n⁽²⁾> + ...    (2.3)

The eigenstates |n> or orthogonal, so:

<n⁽⁰⁾|n> = <n⁽⁰⁾[|n⁽⁰⁾> + λ |n⁽¹⁾> + λ² |n⁽²⁾> + ... ]  = 
<n⁽⁰⁾|n⁽⁰⁾> + 0 + 0 + ... = 1 + 0 + 0 + ... = 1.
And
<n⁽¹⁾|n> = <n⁽¹⁾[|n⁽⁰⁾> + λ |n⁽¹⁾> + λ² |n⁽²⁾> + ... ]  = 
0 + <n⁽¹⁾|n⁽¹⁾> + 0 + ... =  + 0 + λ + ... = λ


<n⁽²⁾|n> = <n⁽²⁾[|n⁽⁰⁾> + λ |n⁽¹⁾> + λ² |n⁽²⁾> + ... ]  = 
0 + 0 + λ² <n⁽²⁾|n⁽²⁾> + 0 + ... =  λ².

Now let's insert the equations (2.2) and (2.3) in the 
equatio (2.1).It follows:
(H⁽⁰⁾ + λ H^(p))[|n⁽⁰⁾> + λ |n⁽¹⁾> + λ² |n⁽²⁾> + ...] = 
[E⁽⁰⁾_n + λ E⁽¹⁾_n + λ² E⁽²⁾_n + ...][|n⁽⁰⁾> + λ |n⁽¹⁾> + λ² |n⁽²⁾> + ...]

Developping yields:

H⁽⁰⁾|n⁽⁰⁾> + λ H⁽⁰⁾|n⁽¹⁾> + λ²H⁽⁰⁾ |n⁽²⁾> 
+ λ H^(p)|n⁽⁰⁾> + λ⁽²⁾ H^(p)|n⁽¹⁾> + λ⁽³⁾ H^(p)|n⁽²⁾> + λ² E⁽²⁾_n|n⁽⁰⁾> + ...
= 
E⁽⁰⁾_n|n⁽⁰⁾> + λE⁽⁰⁾_n |n⁽¹⁾> + λ² E⁽⁰⁾_n |n⁽²⁾>
+ λ E⁽¹⁾_n|n⁽⁰⁾> + λ² E⁽¹⁾_n|n⁽¹⁾> + λ³E⁽¹⁾_n|n⁽²⁾> + ...

Equating by the order of λ (order of perturbation), we have:

H⁽⁰⁾|n⁽⁰⁾> =  E⁽⁰⁾_n|n⁽⁰⁾>    (2.4)
H⁽⁰⁾|n⁽¹⁾> + H^(p)|n⁽⁰⁾> = E⁽⁰⁾_n |n⁽¹⁾> + E⁽¹⁾_n|n⁽⁰⁾>   (2.5)
H⁽⁰⁾|n⁽²⁾> + H^(p)|n⁽¹⁾> = E⁽⁰⁾_n |n⁽²⁾> + E⁽¹⁾_n|n⁽¹⁾> + 
E⁽²⁾_n|n⁽⁰⁾>   (2.6)

2.1.First order of perturbation

From the equation (2.5) and using the bra <n⁽⁰⁾|, we get:
<n⁽⁰⁾|H⁽⁰⁾|n⁽¹⁾> + <n⁽⁰⁾|H^(p)|n⁽⁰⁾> = 
E⁽⁰⁾_n <n⁽⁰⁾|n⁽¹⁾> + E⁽¹⁾_n<n⁽⁰⁾|n⁽⁰⁾>
We find then:
0 + <n⁽⁰⁾|H^(p)|n⁽⁰⁾> = 0 + E⁽¹⁾_n
Therefore:
E⁽¹⁾_n = <n⁽⁰⁾|H^(p)|n⁽⁰⁾>

First order eigen value
E⁽¹⁾_n = <n⁽⁰⁾|H^(p)|n⁽⁰⁾>


From the equation (2.5) and using the bra <m⁽⁰⁾|, we get:
<m⁽⁰⁾|H⁽⁰⁾|n⁽¹⁾> + <m⁽⁰⁾|H^(p)|n⁽⁰⁾> = 
E⁽⁰⁾_n <m⁽⁰⁾|n⁽¹⁾> + E⁽¹⁾_n<m⁽⁰⁾|n^{(0)

We have:
<m⁽⁰⁾|H⁽⁰⁾|n⁽¹⁾> = E⁽⁰⁾_m<m⁽⁰⁾|n⁽¹⁾>, so:

(E⁽⁰⁾_m - E⁽⁰⁾_n) <m⁽⁰⁾|n⁽¹⁾> + <m⁽⁰⁾|H^(p)|n⁽⁰⁾> = 0, or: 
<m⁽⁰⁾|n⁽¹⁾> = <m⁽⁰⁾|H^(p)|n⁽⁰⁾> /( E⁽⁰⁾_n - E⁽⁰⁾_m)
Using the relation of the closure: Σ|m⁽⁰⁾><m⁽⁰⁾| = 1, we get: 
|n⁽¹⁾> = Σ <m⁽⁰⁾|H^(p)|n⁽⁰⁾>/(E⁽⁰⁾_n - E⁽⁰⁾_m) |m⁽⁰⁾>
(The sum is over m ≠n)

First order eigenstate
|n⁽¹⁾> = Σ (m ≠n)[<m⁽⁰⁾|H^(p)|n⁽⁰⁾>/
(E⁽⁰⁾_n - E⁽⁰⁾_m) ]|m⁽⁰⁾>}

2.2. Second order of perturbation

From the equation (2.6) and using the bra <n⁽⁰⁾|, we get:

<n⁽⁰⁾|H⁽⁰⁾|n⁽²⁾> + <n⁽⁰⁾|H^(p)|n⁽¹⁾> = 
E⁽⁰⁾_n <n⁽⁰⁾|n⁽²⁾> + E⁽¹⁾_n<n⁽⁰⁾|n⁽¹⁾> + E⁽²⁾_n<n⁽⁰⁾|n⁽⁰⁾>
That is:
0 + <n⁽⁰⁾|H^(p)|n⁽¹⁾> = 0 + 0 + E⁽²⁾_n, or:
E⁽²⁾_n = <n⁽⁰⁾|H^(p)|n⁽¹⁾>

Using the expression of |n⁽¹⁾>, we find: 
E⁽²⁾_n = <n⁽⁰⁾|H^(p)[Σ [<m⁽⁰⁾|H^(p)|n⁽⁰⁾> /(E⁽⁰⁾_n - E⁽⁰⁾_m) ]|m⁽⁰⁾>]

E⁽²⁾_n = Σ [<m⁽⁰⁾|H^(p)|n⁽⁰⁾> /( E⁽⁰⁾_n - E⁽⁰⁾_m) ] <n⁽⁰⁾|H^(p)|m⁽⁰⁾>

E⁽²⁾_n = Σ [<m⁽⁰⁾|H^(p)|n⁽⁰⁾> /( E⁽⁰⁾_n - E⁽⁰⁾_m) ] <n⁽⁰⁾|H^(p)|m⁽⁰⁾>
sum over m ≠n

Second order eigen value
E⁽²⁾_n = Σ (m≠n)[|<m⁽⁰⁾|H^(p)|n⁽⁰⁾>|²/
(E⁽⁰⁾_n - E⁽⁰⁾_m)] 


Now using the bra <m⁽⁰⁾| in the equation (2.6), we find:

<m⁽⁰⁾|H⁽⁰⁾|n⁽²⁾> + <m⁽⁰⁾|H^(p)|n⁽¹⁾> = 
E⁽⁰⁾_n <m⁽⁰⁾|n⁽²⁾> + E⁽¹⁾_n<m⁽⁰⁾|n⁽¹⁾> + E⁽²⁾_n<m⁽⁰⁾|n⁽⁰⁾>
Therefore:
[E⁽⁰⁾_m - E⁽⁰⁾_n]<m⁽⁰⁾|n⁽²⁾> + <m⁽⁰⁾|H^(p)|n⁽¹⁾> = 
E⁽¹⁾_n<m⁽⁰⁾|n⁽¹⁾> + E⁽²⁾_n<m⁽⁰⁾|n⁽⁰⁾>


[E⁽⁰⁾_n - E⁽⁰⁾_m ]<m⁽⁰⁾|n⁽²⁾> = 
<m⁽⁰⁾|H^(p)|n⁽¹⁾> - E⁽¹⁾_n<m⁽⁰⁾|n⁽¹⁾>  or:
<m⁽⁰⁾|n⁽²⁾> = [<m⁽⁰⁾|H^(p)|n⁽¹⁾> - E⁽¹⁾_n<m⁽⁰⁾|n⁽¹⁾>]/
[E⁽⁰⁾_n - E⁽⁰⁾_m]

Using the first order results and the relation of 
the cloture Σ<m⁽⁰⁾|m⁽⁰⁾> = 1, yields:

<m⁽⁰⁾|n⁽²⁾> = 
Σ(k≠n)[<m⁽⁰⁾|H^(p)|k⁽⁰⁾><k⁽⁰⁾|H^(p)|n⁽⁰⁾> /
( E⁽⁰⁾_n - E⁽⁰⁾_k)(E⁽⁰⁾_n - E⁽⁰⁾_m)
 - [<n⁽⁰⁾|H^(p)|n⁽⁰⁾><m⁽⁰⁾|H^(p)|n⁽⁰⁾>/(E⁽⁰⁾_n - E⁽⁰⁾_m)² 

Therefore:
|n⁽²⁾> = 
Σ(m≠n)Σ(k≠n)<m⁽⁰⁾|H^(p)|k⁽⁰⁾><k⁽⁰⁾|H^(p)|n⁽⁰⁾> /
( E⁽⁰⁾_n - E⁽⁰⁾_k)(E⁽⁰⁾_n - E⁽⁰⁾_m)|m⁽⁰⁾>
 - Σ(m≠n)<n⁽⁰⁾|H^(p)|n⁽⁰⁾><m⁽⁰⁾|H^(p)|n⁽⁰⁾> /
 (E⁽⁰⁾_n - E⁽⁰⁾_m)² |m⁽⁰⁾>
 
Second order eigenstate
|n⁽²⁾> = 
Σ(m≠n)Σ(k≠n)<m⁽⁰⁾|H^(p)|k⁽⁰⁾><k⁽⁰⁾|H^(p)|n⁽⁰⁾> /
( E⁽⁰⁾_n - E⁽⁰⁾_k)(E⁽⁰⁾_n - E⁽⁰⁾_m)|m⁽⁰⁾>
 - Σ(m≠n)<n⁽⁰⁾|H^(p)|n⁽⁰⁾><m⁽⁰⁾|H^(p)|n⁽⁰⁾> /
 ( E⁽⁰⁾_n - E⁽⁰⁾_m)² |m⁽⁰⁾>
 
 These standard equations are called Rayleigh-Schrödinger 
 perturbation theory (RSPT) equations.

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