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## Homework Statement

By substituting the wave function [itex]\psi (x) = Ax{e^{ - bx}}[/itex] into the Schoedinger equation for a 1-D atom, show that a solution can be obtained for [itex]b = 1/{a_0}[/itex], where [itex]{a_0}[/itex] is the Bohr radius.

## Homework Equations

[itex] - \frac{{{\hbar ^2}}}{{2m}}\frac{{{d^2}\psi (x)}}{{d{x^2}}} - \frac{{{e^2}}}{{4\pi {\varepsilon _0}x}}\psi (x) = E\psi (x)[/itex].

[itex]{a_0} = \frac{{4\pi {\varepsilon _0}{\hbar ^2}}}{{m{e^2}}}[/itex]

## The Attempt at a Solution

I get to the point where [itex]2b - x{b^2} = \frac{{2m}}{{{\hbar ^2}}}Ex + \frac{m}{{{\hbar ^2}}}\frac{{{e^2}}}{{2\pi {\varepsilon _0}}}[/itex]. If I let x = 0 I get the desired result. Can I do that?

Apparently the wave function solving the equation must satisfy two conditions:

[itex]\psi (x) \to 0[/itex] as [itex]x \to \infty[/itex]

[itex]\psi (0) = 0[/itex]

But why is that? Can anyone explain?

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