91Ó°ÊÓ

The Schrodinger wave equation is a linear partial differential equation that governs the wave function of the Quantum Mechanical System.

The following can be interpreted from the equation.

$\hat{H}\mathrm{ψ}=E\mathrm{ψ}$and ${\hat{H}}^0{\mathrm{ψ}}^0=E^0{\mathrm{ψ}}^0$and$\hat{H}={\hat{H}}^0+\mathrm{λ}\hat{H}$

Write equation 6.17.

${\mathrm{ψ}}^0=\mathrm{Î\pm }{\mathrm{ψ}}_a^0+\mathrm{β}{\mathrm{ψ}}_b^0$

$\mathrm{Î\pm }{W}_{aa}+\mathrm{β}{W}_{ab}=\mathrm{Î\pm }{E}^1$

It is known that $E_n=E_n^6+\mathrm{λ}{E}_n^1+{\mathrm{λ}}^2{E}_n^2+...$and $\mathrm{ψ}={\mathrm{ψ}}^0+\mathrm{λ}{\mathrm{ψ}}^1+{\mathrm{λ}}^2{\mathrm{ψ}}^2+...$and ${\mathrm{ψ}}^0{=}_{j=1}^n{\mathrm{Î\pm }}_f{\mathrm{ψ}}_j^0$

Apply Schrodinger's equation.

$$\begin{gathered} \left({\hat{H}}^0+\mathrm{λ}\hat{H}\right)=\left[{\mathrm{ψ}}^0+\mathrm{λ}{\mathrm{ψ}}^1+{\mathrm{λ}}^2{\mathrm{ψ}}^2+...\right]=(E_n^0+\mathrm{λ}{E}_n^1+{\mathrm{λ}}^2{E}_n^2+...)\left[{\mathrm{ψ}}^0+\mathrm{λ}{\mathrm{ψ}}^1+{\mathrm{λ}}^2{\mathrm{ψ}}^2+...\right] \\ ={\hat{H}}^0{\mathrm{ψ}}^0+\mathrm{λ}\left[{\hat{H}}^0{\mathrm{ψ}}^1+{\hat{H}}^{\text{'}}{\mathrm{ψ}}^0\right]+{\mathrm{λ}}^2\left[{\hat{H}}^0{\mathrm{ψ}}^2+{\hat{H}}^{\text{'}}{\mathrm{ψ}}^1+...\right] \\ =E_n^0{\mathrm{ψ}}^0+\mathrm{λ}\left[E_n^0{\mathrm{ψ}}^0+E_n^1{\mathrm{ψ}}^0\right]+{\mathrm{λ}}^2\left[E_n^0{\mathrm{ψ}}^2+E_n^1{\mathrm{ψ}}^1+E_n^2{\mathrm{ψ}}^0\right]+... \end{gathered}$$

Obtain the first-order correction for the energy eigen values by equating the coefficients of the same order.

$=H^0{\mathrm{ψ}}^1+{\hat{H}}^{\text{'}}{\mathrm{ψ}}^0=E_n^0{\mathrm{ψ}}^1+E^1{\mathrm{ψ}}^0$

Multiply both sides by $\mathrm{ψ}\_j^{0*}$.

It is known that${\mathrm{ψ}}^0=\underset{j=1}{\overset{n}{∑}}{\mathrm{Î\pm }}_f{\mathrm{ψ}}_j^0$ and$\left({\mathrm{ψ}}^0\overline{){\hat{H}}^0=E_n^{0*}}\right({\mathrm{ψ}}^0\overline{),{\hat{H}}^{\text{'}}={\hat{H}}^{0*}}$ , . Then the

role="math" localid="1658211098621" $$\begin{gathered} E_n^0=E_n^0 \\ \left({\mathrm{ψ}}^0\overline{){\hat{H}}^0=E_n^{0*}}\right({\mathrm{ψ}}^0\overline{)} \end{gathered}$$

Write the generalized form of the equation.

Thus, the analog function can be interpreted as the eigenvalue equation for the matrix W.