Question

Under what circumstance does the integral ${\mathbf{\int }}_{{\mathbf{x}}_{\mathbf{0}}}^{\mathbf{\infty }}{\mathit{x}}^{\mathbf{b}}\mathit{d}\mathit{x}$diverge? Use this to argue that a physically acceptable wave function must fall to 0 faster than$\mathbf{|}\mathit{x}{\mathbf{|}}^{\mathbf{-}\mathbf{1}\mathbf{/}\mathbf{2}}$ does as $\mathit{x}$gets large.

Short answer

The function will keep on diverging unless $b$ is less than hence we can say the integral will diverge at the given infinite limit (upper limit). The value of b equal to $-1$ cannot be possible because it would make the entire function infinite. Now, for the total integral to not diverge, it is necessary for the${\psi }^2$ to fall off more rapidly or at least as rapid as $|x{|}^{-1}$ and$\psi$ to fall off more rapidly or at least as rapid as $|\text{x}{|}^{-1/2}$

Step-by-step solution

Given expression 

The Integral is given as,

${\int }_{x_0}^{\infty }{x}^{b}dx$.

Concept of Integration 

Integration concept used

${\mathbf{\int }}_{\mathbf{a}}^{\mathbf{b}}\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathit{d}\mathit{x}\mathbf{=}\mathit{F}\mathbf{\left(}\mathit{b}\mathbf{\right)}\mathbf{-}\mathit{F}\mathbf{\left(}\mathit{a}\mathbf{\right)}$

Integration concept used

In order to calculate the conditions under which the given integral may diverge, start by writing the integral as follows:

${\int }_{x_0}^{\infty }{x}^{b}dx$

Here, $\text{dx}$ is called the differential of the variable$x$.

The upper limit of the Integral is 8.

Integrating the given function

Integrating the above-given function under the given limits we will get

${\int }_{x_0}^{\infty }{x}^{b}dx={\left[\frac{x^{b+1}}{b+1}\right]}_{x_0}^{\infty }$

The function will keep on diverging unless $b$ is less than $-1$ hence we can say the integral will diverge at the given infinite limit (upper limit). The value of b equal to $-1$ cannot be possible because it would make the entire function infinite.

Now, for the total integral to not diverge, it is necessary for the ${\psi }^2$ to fall off more rapidly or at least as rapid as $|x{|}^{-1}$ and $\psi$to fall off more rapidly or at least as rapid as$|\text{x}{|}^{-1/2}$ .