Calculating Reflection Coefficient for a Potential Well

[captainjack2000]

Homework Statement

A potential well with V(x)=0 for x<0 and V(x) = Vo for x>0. The particle has energy E greater than Vo and is incident from the left side. Calculate the reflection coefficient.

Homework Equations

Relfection coefficient is given by R= (B*B)/(A*A)

The Attempt at a Solution

I think that the time independent Schrödinger equation for the two regions are
x<0 (-hbar/2m)dphisquared/dphi =Ephi(x)
and
x>0 ((-hbar/2m)dphisquared/dphi +Vo phi = Ephi(x)

the solutions to this
x<0 Aexp(ik1x) +Bexp(-ik1x) = phi(x)
x>0 Cexp(ik2x)

is this right
carrying this through and equating two solutions at x=0
A+b = C
and equating differentials at x=0 gives
ik1(A-B) = -ik2C/k1

but when I solve for A and B i get
A=(c/2)(1-ik2/k1)
B=(c/2)(1+ik2/k1)

when I find R i get R=1 which must wrong because particle is not always reflected it should pass over the potential barrier?

 

[Dick]

Things started to break down when you evaluated the derivatives. Why is there a k1 on the left side? And why didn't the i's cancel out in the final solution?

 

[siddharth]

ik1(A-B) = -ik2C/k1

Apart from the extra k1 on the left side which Dick pointed out, why is there a minus sign on the right side?

 

[captainjack2000]

sorry should the differentials be (A-B)=-ik2C/k1?
in which case i still get the same solutions for A and B and a coefficient of 1

 

[siddharth]

Why is there a minus sign on the left hand side?