Responsible party: John Galt, Dark Helmet, Esquire

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We do it becuase it is a useful tool

Shroedinger's equation is the must fundamental tool for QM. We take solutions and eigenstates/eigenvectors to get energy levels

This is the simplest case single energy→single equation

<math> H_0|\psi_n^{(0)}>=E_n^{(0)}|\psi_n^{(0)}> </math>

<math> <\psi_m^{(0)}|\psi_n^{(0)}>=\delta_{mn} </math>

<math> H=H_0 +\lambda H' </math> ⇒ <math> H|\psi_n>=E_n|\psi_n> </math>

The goal is to seek an approximation of this new Hamiltonian expression. Specifically we want…

<math> E_n=E_n^{(0)}+\lambda E_n^{(1)}+\lambda^{2} E_n^{(2)} </math>

We define <math> <\psi_n^{(0)}|\psi_n^{(1)}>=<\psi_n^{(0)}|\psi_n^{(2)}>=0 </math>

A Fourier expansion can be used to express <math> |\psi_n^{(1)}>=\Sigma C_{mn}|\psi_n^{(0)}> </math> where m≠n

Plugging this into the new Hamiltion yields

<math> (H_0+\lambda H')(|\psi_n^{(0)}>+\lambda|\psi_n^{(1)}>)=(E_n^{(0)}+\lambda E_n^{(1)})(|\psi_n^{(0)}>+\lambda|\psi_n^(1)>) </math>

<math> H_0|\psi_n^{(0)}>+\lambda(H'|\psi_n^{(0)}>+H_0|\psi_n^{(1)}>)=E_n^{(0)}|\psi_n^{(0)}>+\lambda(E_n^{1}|\psi_n^{(0)}>+E_n^{(0)}|\psi_n^{(1)}>) </math>

<math> H'|\psi_n^{(0)}>+H_0|\psi_n^{(1)}>=E_n^{(1)}|\psi_n^{(0)}>+E_n^{(0)}|\psi_n^{(1)}> </math>

Now using the Fourier expansion expression

<math> H'|\psi_n^{(0)}>+\Sigma C_{nm}H_0|\psi_m^{(0)}>=E_n^{(1)}|\psi_n^{(0)}>+E_n^{(0)}\Sigma C_{nm}|\psi_m^{(0)}> </math>

<math> H'|\psi_n^{(0)}+\Sigma C_{nm}E_m^{(0)}|\psi_m^{(0)}>=E_n^{(1)}|\psi_n^{(0)}>+E_n^{(0)}\Sigma C_{nm}|\psi_m^{(0)}> </math>

Using this, one can find an expression for the expectation of the new Hamiltonian as follows

<math> <\psi_n^{(0)|H'|\psi_n^{(0)}>+\Sigma C_{nm}E_m^{(0)}<\psi_n^{(0)}|\psi_m^{(0)}>=E_n^{(1)}<\psi_n^{(0)}|\psi_n^{(0)}>+E_n^{(0)}\Sigma C_{nm}<\psi_n^{(0)}|\psi_m^{(0)}> </math>

<math><\psi_n^{(0)|H'|\psi_n^{(0)}>=E_n^{(1)} </math>

Now one can introduce a new parameter l≠n but l can equal m and show

<math><\psi_l^{(0)|H'|\psi_n^{(0)}>+\Sigma C_{nm}E_m^{(0)}<\psi_l^{(0)}|\psi_m^{(0)}>=E_n^{(1)}<\psi_l^{(0)}|\psi_n^{(0)}>+E_n^{(0)}\Sigma C_{nm}<\psi_l^{(0)}|\psi_m^{(0)}> </math>

<math><\psi_l^{(0)|H'|\psi_n^{(0)}>+C_{nl}E_l^{(0)}=C_{nl}E_n^{(0)} </math>

⇒<math>C_{nl}=<\psi_l^{(0)|H'|\psi_n^{(0)}>/(E_n^{(0)}-E_l^{(0)}) </math>

⇒<math>E_n^{(2)}=\Sigma|<\psi_l^{(0)|H'|\psi_n^{(0)}>|^2/(E_n^{(0)}-E_l^{(0)}) </math>

This was all i had for notes as well-Dark Helmet

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