- #1

SoggyBottoms

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Consider the potential

[tex]

V(x) =

\begin{cases}

0, & x < -a & (I) \\

+W, & -a < x < a & (II) \\

0, & x > a & (III)

\end{cases}

[/tex]

for a particle coming in from the left ([itex]-\infty[/itex]) with energy E (0 < E < W). Give the solution to the Schrodinger equation for I, II and III and use these to calculate the reflection coefficient.

I have the answer to this problem in front of me, but I don't understand. First they calculate the solution to the Schrodinger equation for I, II and III:

[itex]\psi_I(x) = Ae^{ikx} + Be^{-ikx}, \ \mbox{with} \ k = \frac{\sqrt{2mE}}{\hbar}[/itex]

[itex]\psi_{II}(x) = Ce^{\kappa x} + De^{-\kappa x}, \ \mbox{with} \ \kappa = \frac{\sqrt{2m(E - W)}}{\hbar}[/itex]

[itex]\psi_{III}(x) = Fe^{i k x}, \ \mbox{with} \ k = \frac{\sqrt{2mE}}{\hbar}[/itex]

I understand [itex]\psi_I[/itex], but not [itex]\psi_{II}[/itex] and [itex]\psi_{III}[/itex]. Why is there no i in [itex]\psi_{II}[/itex]? And why is [itex]\psi_{III}[/itex] only a single term? I imagine it has something to do with the particle coming from the left?

[tex]

V(x) =

\begin{cases}

0, & x < -a & (I) \\

+W, & -a < x < a & (II) \\

0, & x > a & (III)

\end{cases}

[/tex]

for a particle coming in from the left ([itex]-\infty[/itex]) with energy E (0 < E < W). Give the solution to the Schrodinger equation for I, II and III and use these to calculate the reflection coefficient.

I have the answer to this problem in front of me, but I don't understand. First they calculate the solution to the Schrodinger equation for I, II and III:

[itex]\psi_I(x) = Ae^{ikx} + Be^{-ikx}, \ \mbox{with} \ k = \frac{\sqrt{2mE}}{\hbar}[/itex]

[itex]\psi_{II}(x) = Ce^{\kappa x} + De^{-\kappa x}, \ \mbox{with} \ \kappa = \frac{\sqrt{2m(E - W)}}{\hbar}[/itex]

[itex]\psi_{III}(x) = Fe^{i k x}, \ \mbox{with} \ k = \frac{\sqrt{2mE}}{\hbar}[/itex]

I understand [itex]\psi_I[/itex], but not [itex]\psi_{II}[/itex] and [itex]\psi_{III}[/itex]. Why is there no i in [itex]\psi_{II}[/itex]? And why is [itex]\psi_{III}[/itex] only a single term? I imagine it has something to do with the particle coming from the left?

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