Question

Verify that solution (5-19) satisfies the Schrodinger equation in form (5.18).

Short answer

The solution of the Schrodinger equation is

$\begin{array}{c}\frac{{\text{d}}^{\text{2}}\text{ψ(x)}}{{\text{dx}}^{\text{2}}}{\text{=Aα}}^{\text{2}}{\text{exp}}^{\text{±αx}}\\ \frac{{\text{d}}^{\text{2}}\text{ψ(x)}}{{\text{dx}}^{\text{2}}}{\text{=α}}^{\text{2}}\text{ψ(x)}\end{array}$

Step-by-step solution

Time-dependent equation of Schrodinger.

Time-dependent equation of Schrodinger

$\frac{{\text{d}}^{\text{2}}\text{ψ(x)}}{\text{dx}}\text{=}\frac{\text{2m}\left({\text{U}}_{\text{o}}\text{(x)-E}\right)}{{\text{h}}^{\text{2}}}\text{ψ(x)………………(1)}$

and the wave function is provided

${\text{ψ(x)=Aexp}}^{\text{±αx}}\text{,α=}\sqrt{\frac{\text{2m}\left({\text{U}}_{\text{o}}\text{(x)-E}\right)}{{\text{h}}^{\text{2}}}}\text{………………(2)}$

Schrodinger equation

Differentiate the wave function twice and replace into the Schrodinger equation to ensure that it satisfies the Schrodinger equation (1).

$\begin{array}{c}\frac{\text{dψ(x)}}{\text{dx}}{\text{=±Aαexp}}^{\text{±αx}}\\ \text{=±αψ(x)}\\ \frac{{\text{d}}^{\text{2}}\text{ψ(x)}}{{\text{dx}}^{\text{2}}}{\text{=Aα}}^{\text{2}}{\text{exp}}^{\text{±αx}}\\ {\text{=α}}^{\text{2}}\text{ψ(x)}\end{array}$

So, the Schrodinger equation is$\begin{array}{l}\frac{{\text{d}}^{\text{2}}\text{ψ(x)}}{{\text{dx}}^{\text{2}}}{\text{=Aα}}^{\text{2}}{\text{exp}}^{\text{±αx}}\\ \frac{{\text{d}}^{\text{2}}\text{ψ(x)}}{{\text{dx}}^{\text{2}}}{\text{=α}}^{\text{2}}\text{ψ(x)}\end{array}$.