Potential barrier. Schroedinger equation.

[LagrangeEuler]

Homework Statement

Schroedinger equation for potential barrier.
What if $V_0=E$
First region. Particles are free.
$\psi_1(x)=Ae^{ikx}+Be^{-ikx}$
In third region
$\psi_3(x)=Ce^{ikx}$

Homework Equations

$\frac{d^2\psi}{dx^2}+\frac{2m}{\hbar^2}(V_0-E)\psi=0$
where $V_0$ is height of barrier.
For region II

The Attempt at a Solution

In second region
$\frac{d^2 \psi}{dx^2}=0$
from that
$\frac{d\psi}{dx}=C_1$
$\psi(x)=C_1x+C_2$
Boundary condition
$A+B=C_2$
$C_1a+C_2=Ce^{ika}$
$ikA-ikB=C_1$
$C_1=ikCe^{ika}$
System 4x4
Is this correct?
Could you tell me in this case do I have bond state?

 

[dauto]

Yes, it seems correct. Bond states have negative energy.

 

[LagrangeEuler]

Bound states have negative energy? Can you explain me this. In case of this problem.

 

[dauto]

The oscillatory solutions of regions 1 and 3 happen because energy is positive and the wave is free to propagate to infinity. (unbounded particle). If the energy is negative you get a exponentially decaying wave function and the wave does not propagate to infinity (bounded particle).