What is the ratio of energy to barrier height for a 0.43 reflection coefficient?

[w3390]

Homework Statement

A particle of energy E approaches a step barrier of height U0. What should be the ratio E/U0 so that the reflection coefficient is 0.43?

Homework Equations

R=(k1-k2)^2/(k1+k2)^2 <--------reflection coefficient

The Attempt at a Solution

I am completely stumped at how to approach this problem. I cannot find how to get a ratio of E to U0 using any equations. Any help on how to approach this problem is much appreciated.

 

[gabbagabbahey]

Homework Statement

A particle of energy E approaches a step barrier of height U0. What should be the ratio E/U0 so that the reflection coefficient is 0.43?

Homework Equations

R=(k1-k2)^2/(k1+k2)^2 <--------reflection coefficient

The Attempt at a Solution

I am completely stumped at how to approach this problem. I cannot find how to get a ratio of E to U0 using any equations. Any help on how to approach this problem is much appreciated.

Well, how did you define $k_1$ and $k_2$ when deriving the above reflection coefficient equation?

 

[w3390]

I had k1=sqrt(2mE/(h-bar)^2) and k2=sqrt(2mK2/(h-bar)^2), where K2 is the final kinetic energy.

 

[gabbagabbahey]

Okay, so doesn't that mean K2=E-U0 and k2=sqrt(2m(E-U0)/(hbar)^2)?

Substitute k1 and k2 into your expression for R and simplify...

 

[w3390]

I don't know how the K2=E-U0 comes into play. However, if I substitute in k1 and k2 into the R expression, it looks like:

R=(k1^2-2k1k2+k2^2)/(k1^2+2k1k2+k2^2). This ends up being a complete mess and I have some E and U0 terms that stand alone and some E and U0 terms that are stuck inside square roots, so I can't get it to a ratio. I can't tell what I'm doing wrong.

 

[gabbagabbahey]

I don't know how the K2=E-U0 comes into play

A particle with energy E passes through a potential barrier of height U0...doesn't that mean its final energy is E-U0?

R=(k1^2-2k1k2+k2^2)/(k1^2+2k1k2+k2^2). This ends up being a complete mess and I have some E and U0 terms that stand alone and some E and U0 terms that are stuck inside square roots, so I can't get it to a ratio. I can't tell what I'm doing wrong.

Don't expand the squares...just divide everything by k1:

$$R=\left(\frac{k_1-k_2}{k_1+k_2}\right)^2=\left(\frac{1-\frac{k_2}{k_1}}{1+\frac{k_2}{k_1}}\right)^2$$

And $\frac{k_2}{k_1}=$____?

 

[w3390]

Okay thanks. That way really helps!