Question

If the constant${\mathit{C}}_{\mathbf{x}\mathbf{}}\mathbf{in}\mathbf{}\mathbf{equation}\mathbf{}\mathbf{(}\mathbf{7}\mathbf{-}\mathbf{5}\mathbf{)}$were positive, the general mathematical solution would be

$\mathit{A}{\mathit{e}}^{\mathbf{+}\sqrt{{\mathbf{c}}_{\mathbf{x}}\mathbf{x}}}\mathbf{+}\mathit{B}{\mathit{e}}^{\mathbf{-}\sqrt{{\mathbf{c}}_{\mathbf{x}}\mathbf{x}}}$

Show that this function cannot be 0 at two points. This makes it an unacceptable solution for the infinite well, since it cannot be continuous with the wave functions outside the walls, which are 0.

Short answer

This is possible only if$x_1=x_2$ . (i.e., only one point) but this assumption does not happen in general cases. Thus, this is quite a contradiction to the assumption that the function is zero at two different points$x_1$ and $x_2$.

Step-by-step solution

Given data

The solution is:

$A{e}^{+\sqrt{{\mathrm{c}}_{\mathrm{x}}\mathrm{x}}}+B{e}^{-\sqrt{{\mathrm{c}}_{\mathrm{x}}\mathrm{x}}}$

To determine the function Ae+cxx+Be-cxx cannot be zero at two points

The infinity well problem is one of the important problems in quantum mechanics that help us to understand other phenomena.

The wave function outside the well is zero.

The given solution is:

$A{e}^{+\sqrt{{\mathrm{c}}_{\mathrm{x}}\mathrm{x}}}+B{e}^{-\sqrt{{\mathrm{c}}_{\mathrm{x}}\mathrm{x}}}$

It can also be written as:

$Aexp\left(+\sqrt{c_{x}x}\right)+Bexp\left(-\sqrt{c_{x}x}\right)$

These equations at two points $x_1,x_2$can be written as:

role="math" localid="1659763359765" $$\begin{gathered} Aexp\left(+\sqrt{c_x{x}_1}\right)+Bexp\left(-\sqrt{c_x{x}_1}\right) \\ Aexp\left(+\sqrt{c_x{x}_2}\right)+Bexp\left(-\sqrt{c_x{x}_2}\right) \end{gathered}$$

Solve each equation for B/A as:

role="math" localid="1659763407252" $$\begin{gathered} \frac{\mathrm{B}}{\mathrm{A}}=\exp \left(2\sqrt{{\mathrm{c}}_{\mathrm{x}}}{\mathrm{x}}_1\right) \\ \frac{\mathrm{B}}{\mathrm{A}}=\exp \left(2\sqrt{{\mathrm{c}}_{\mathrm{x}}}{\mathrm{x}}_2\right) \end{gathered}$$

From the above two expressions, we can conclude that,

$\exp \left(2\sqrt{{\mathrm{c}}_{\mathrm{x}}}{\mathrm{x}}_2\right)=\exp \left(2\sqrt{{\mathrm{c}}_{\mathrm{x}}}{\mathrm{x}}_1\right)$

Conclusion

This is possible only if$x_1=x_2$ . (i.e., only one point) but this assumption does not happen in general cases. Thus, this is quite a contradiction to the assumption that the function is zero at two different points$x_1$ and $x_2$.