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fluidistic

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## Homework Statement

A particle of mass m goes toward the unidimensional barrier potential of the form [itex]V(x)=0[/itex] for [itex]x\leq 0[/itex] and [itex]a\leq x[/itex] and [itex]V(x)=V_0[/itex] for [itex]0<x<a[/itex].

1)Write the corresponding Schrödinger's equation.

2)Calculate the transmission coefficient for the cases [itex]0<E<V_0[/itex] and [itex]E>V_0[/itex]. Hint: Check out quantum mechanics books.

## Homework Equations

Schrödinger's equation.

## The Attempt at a Solution

I solved the Schrödinger's equation for Psi.

The result I have is

[itex]\Psi _I (x)=Ae^{ik_1 x}+Be^{-i _k1x}[/itex]

[itex]\Psi _{II}(x)=De^{-k_2x}[/itex]

[itex]\Psi _{III}(x)=Fe^{ik_1x}[/itex].

The continuity conditions give:

(1) [itex]A+B=D[/itex].

(2) [itex]i k_1 A-i k_1 B=-k_2 D[/itex].

(3) [itex]De^{-k_2a}=Fe^{ik_1a}[/itex]

(4) [itex]-k_2 De^{-ak_2}=ik_1 Fe^{ik_1a}[/itex].

I isolated B in function of A. I reached [itex]B=A \cdot \frac{\left (1 +\frac{ik_1}{k_2} \right ) }{ \left ( \frac{ik_1}{k_2} -1\right )}[/itex].

Since [itex]D=A+B[/itex], I got D in function of A only.

And since [itex]F=De^{-a(k_2+ik_1)}[/itex], I reached F in function of A only.

Now the coefficient of transmission is [itex]\frac{F}{A}[/itex]. This gives me [itex]\left [ 1+ \frac{\left ( 1+ \frac{ik_1}{k_2} \right ) }{\left ( \frac{ik_1}{k_2}-1 \right ) } \right ] e^{-a(k_2+i_k1)}[/itex].

I checked out in Cohen-Tanoudji's book about this and... he says that F/A represent the transmission coeffient just like me but then in the algebra, he does [itex]T= \big | \frac{F}{A} \big | ^2[/itex]. He reaches [itex]T=\frac{4k_1^2k_2^2}{4k_1^2k_2^2+(k_1^2-k_2^2)^2 \sin k_2 a}[/itex].

So I don't understand why he says F/A but does the modulus of it to the second power. And he seems to get a totally different answer from mine.

By the way my [itex]k_1=\sqrt {\frac{-2mE}{\hbar ^2}}[/itex] and my [itex]k_2=\sqrt {\frac{2m(V_0-E)}{\hbar ^2}}[/itex].

What am I doing wrong?!

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