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Method of Normalization

Method of Normalization | Quantum Mechanics

Consider $\varphi$(x,t) is an unnormalized wave function. We can construct a normalized wave function as

$\psi\left(x,t\right)$=N$\varphi$(x,t)

where, N is the normalization constant.

Therefore,

$\int_{-\infty}^{\infty}\psi^\ast\left(x,t\right)\psi\left(x,t\right)$ = $|N^{2}| \intop_{-\infty}^{\infty}\varphi^{*}\left(x,t\right)\varphi\left(x,t\right)dx$=1

⟹$N=\frac{1}{\sqrt{\intop_{-\infty}^{\infty}\varphi^{*}\left(x,t\right)\varphi\left(x,t\right)dx}}$

Example: Normalize the wave function given by

$\psi\left(x\right)=Ne^{-\alpha\chi}\left(x>0\right)$

=$Ne^{\alpha\chi}\left(x<0\right)$

Solution. Normalization condition is

$\intop_{-\infty}^{\infty}\psi^{*}\psi dx$=1

⟹$|N^{2}| \left[\intop_{-\infty}^{0}e^{2\alpha\chi}dx+\intop_{0}^{\infty}e^{-2\alpha\chi}dx\right]=1$

⟹$|N^{2}|\left[\frac{1}{2\alpha}+\frac{1}{2\alpha}\right]=1⟹N=\sqrt{\alpha}$

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