Quantum Mechanics Problem

I am asked to determine the expressions for coefficients of the solutions [tex]\psi_1[/tex] and [tex] \psi_2 [/tex] to the Schrodinger's Equation in a system where a particle travelling to the right encounters a potential step when [tex] E < V_0 [/tex], where E is the total energy.

What I was able to come up with is that
[tex]
\psi_1 = Ae^{kx} + Be^{-kx}, k = \frac{\sqrt{2mE}}{\hbar}
[/tex]

[tex]
\psi_2 = Ce^{qx} + De^{-qx}, q = \frac{\sqrt{2m(E-V_0)}}{\hbar}
[/tex]

What I also know is that [tex]Ce^{qx}[/tex] is unacceptable because the wave must decay exponentially when it hits the barrier.

My first question is whether or not this statement is true:

[tex]
\int_{-\infty}^{0}|\psi_1|^2dx + \int_{0}^{+\infty}|\psi_2|^2dx = 1
[/tex]

My second question is if [tex]Be^{-kx}[/tex] is an acceptable solution to [tex]\psi_1[/tex]. In my opinion i believe it is not acceptable because it diverges as x negative approaches infinity.

Thanks for any help.

 

I made a mistake in my [tex]\psi_1[/tex], i think the solution should be
[tex]
\psi_1 = Ae^{ikx} + B^{-ikx}
[/tex]
[tex]
\psi_2 = Ce^{-iqx}
[/tex]

Hmmm... If I remember correctly, in Eisberg's book he used [tex]\psi_2 = Ce^{-qx}[/tex] (a real function) and then equated it to [tex]\psi_1[/tex] (a complex function) at x=0, then applied the boundary conditions for continuity... is [tex]\psi_2 = Ce^{-qx}[/tex] really not normalizable?

Thanks again for all the help

 

Is my new [tex]\psi_1[/tex] normalizable now? i.e.,

[tex]
\psi_1 = Ae^{ipx} + Be^{-ipx}
[/tex]

Normalization is quite confusing to me sometimes...

Thanks again for the help

 

Um... is it necessary to convert [tex]Ae^{ipx} + Be^{-ipx}[/tex] to its ciskx and coskx - isinkx form before normalizing it? I trashed the idea of thinking that the wave function at the left had the form of either sine or cosine because they would definitely diverge when I integrate them from (-o0, 0)when they are normalized...