#### fluidistic

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The corresponding Schrödinger's equation for the region greater than 0, is [tex]\Psi '' (x)+ \Psi (x) \frac{(E-V_0)2m}{\hbar ^2}=0[/tex].

Now if [tex]E=V_0[/tex], the solution to this equation is a straight line [tex]\Psi (x)=ax+b[/tex]. However since the region is infinitely large, there is no values for a and b other than 0 that will normalize [tex]\Psi[/tex], hence we must conclude that [tex]\Psi (x)=0[/tex] in this region.

Fine, that mean for [tex]E=V_0[/tex], there is a total reflection of a particle coming from the left (from negative x).

However if [tex]E\neq V_0[/tex] (for both [tex]E<V_0[/tex] and [tex]E>v_0[/tex]), there's no total reflection, quite strange. I'm trying to understand the physical meaning of that. It's like there's a well definite energy that isn't allowed for a particle to pass the potential.

If we now consider a region [tex]V(x)=\infty[/tex] for [tex]x<0[/tex], [tex]V(x)=V_0[/tex] for [tex]0 \leq x \leq a[/tex] and [tex]V(x)=\infty[/tex] for [tex]x>a[/tex], and if [tex]E=V_0[/tex], then I think there's an infinity of possible solutions. I mean there are an infinite number of constants A and B such that [tex]\int_0 ^a |Ax+B|^2 dx =1[/tex]. I don't think it's possible physically, so what's going on?

By the way I get the condition [tex]\frac{A^2 a^3}{3}+ a^2|AB| + B^2 a =1[/tex].

And also in the case of having a region of potential of the form [tex]V(x)=0[/tex] for [tex]x<0[/tex], [tex]V(x)=V_0[/tex] for [tex]0 \leq x \leq a[/tex] and [tex]V(x)=0[/tex] for [tex]x>a[/tex], the same problem appears in the central region. It seems like the particle can have infinitely many different [tex]\Psi (x)[/tex] which doesn't make sense to me.

What's happening?