- #1

- 290

- 1

## Homework Statement

Given that

##T = \frac{4E(V-E)}{4E(V-E)+V^2\sinh^2 (ka)}##

Simplify the expression for T when a is large but not infinite, and again for the case when a tends to zero and the potential tends to infinity, such that ##d = Va## is a real constant. k is a constant, as is E (which is assumed to be less than V).

## Homework Equations

## The Attempt at a Solution

To try and simplify it for the first case, I used the exponential definition of sinh, squared it and got that

##\sinh^2(x) = \frac{1}{4}(e^{2x}+e^{-2x}-2)##

For large x the ##e^{-2x}## term would get small very quickly, especially compared to the other exponential term which would increase rapidly. So this becomes

##\sinh^2(x) = \frac{1}{4}(e^{2x}-2)##

I don't know if that's what the question meant. It doesn't seem massively simpler. Is there something else that can be done to simplify the expression?

And the second part, I don't know where to start! There should be some sort of constant ##d = Va##, but I thought ##sinh^2(0)=0##, which would mean that ##a## disappears. As far as I can see that expression simplifies to 1 for the second set of limits! Why is that wrong?