# Quantum mechanics, potential barrier

1. Oct 22, 2011

### fluidistic

1. The problem statement, all variables and given/known data
A particle of mass m goes toward the unidimensional barrier potential of the form $V(x)=0$ for $x\leq 0$ and $a\leq x$ and $V(x)=V_0$ for $0<x<a$.
1)Write the corresponding Schrödinger's equation.
2)Calculate the transmission coefficient for the cases $0<E<V_0$ and $E>V_0$. Hint: Check out quantum mechanics books.

2. Relevant equations
Schrödinger's equation.

3. The attempt at a solution
I solved the Schrödinger's equation for Psi.
The result I have is
$\Psi _I (x)=Ae^{ik_1 x}+Be^{-i _k1x}$
$\Psi _{II}(x)=De^{-k_2x}$
$\Psi _{III}(x)=Fe^{ik_1x}$.
The continuity conditions give:
(1) $A+B=D$.
(2) $i k_1 A-i k_1 B=-k_2 D$.
(3) $De^{-k_2a}=Fe^{ik_1a}$
(4) $-k_2 De^{-ak_2}=ik_1 Fe^{ik_1a}$.

I isolated B in function of A. I reached $B=A \cdot \frac{\left (1 +\frac{ik_1}{k_2} \right ) }{ \left ( \frac{ik_1}{k_2} -1\right )}$.
Since $D=A+B$, I got D in function of A only.
And since $F=De^{-a(k_2+ik_1)}$, I reached F in function of A only.
Now the coefficient of transmission is $\frac{F}{A}$. This gives me $\left [ 1+ \frac{\left ( 1+ \frac{ik_1}{k_2} \right ) }{\left ( \frac{ik_1}{k_2}-1 \right ) } \right ] e^{-a(k_2+i_k1)}$.
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 $T= \big | \frac{F}{A} \big | ^2$. He reaches $T=\frac{4k_1^2k_2^2}{4k_1^2k_2^2+(k_1^2-k_2^2)^2 \sin k_2 a}$.
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 $k_1=\sqrt {\frac{-2mE}{\hbar ^2}}$ and my $k_2=\sqrt {\frac{2m(V_0-E)}{\hbar ^2}}$.
What am I doing wrong?!

Last edited: Oct 22, 2011
2. Oct 23, 2011

### vela

Staff Emeritus
It's possible to get reflections at x=a, so the region II wave function needs to be modified.

The transmission coefficient is |F/A|2.

3. Oct 23, 2011

### vela

Staff Emeritus
Where did he say this? I can never find anything in that encyclopedia of a textbook.

The closest thing I saw was that he said you could get the transmission coefficient from the ratio, but he never actually said the ratio itself was the transmission coefficient.

4. Oct 23, 2011

### fluidistic

Ah whoops, I totally missed this.
Again oops once again. You're right, he says that the ratio allows us to determine the coefficients.
Ok I understand now why it's the modulus to the second power. I'll rework on that.