# Thread: Find Simplified Expression for T

1. ## Find Simplified Expression for T

So this is a physics question (but the part to be asked is mathematical) from Quantum Mech.

In an earlier part of question, I obtain an expression for T:
$\displaystyle T=\frac{4\sqrt{E(E+V_0)}}{2E+2\sqrt{E(E+V_0)}+V_0}$

Then it asks, find a simplified expression for $\displaystyle T$ when $\displaystyle V_0 >> E$

The best way I could come up with to (attempt to) solve this, was to assume E = 0, which would make T=0

2. ## Re: Find Simplified Expression for T

Multiply top and bottom by $\displaystyle V_0^{-1}$:

$\displaystyle T = \dfrac{4\sqrt{E^2 + EV_0}}{2E + 2\sqrt{E^2+EV_0} + V_0}\left(\dfrac{V_0^{-1}}{V_0^{-1}}\right)$

Now, $\displaystyle V_0^{-1} = \sqrt{V_0^{-2}}$. So, you get:

$\displaystyle T = \dfrac{4\sqrt{E^2V_0^{-2} + EV_0^{-1}}}{2EV_0^{-1}+2\sqrt{E^2V_0^{-2}+EV_0^{-1}}+1}$

Each term with $\displaystyle V_0$ to a negative exponent approaches zero as $\displaystyle V_0 \gg E$. So, you get $\displaystyle T \to 0$, just as you thought. That is pretty simplified, isn't it?

Edit: But, if I had an idea of what $\displaystyle E$ and $\displaystyle V_0$ were, I might have come up with a different answer. If $\displaystyle V_0$ has an upper bound, and just appears large compared to $\displaystyle E$, that would change my answer significantly.

3. ## Re: Find Simplified Expression for T

So this is a physics question (but the part to be asked is mathematical) from Quantum Mech.

In an earlier part of question, I obtain an expression for T:
$\displaystyle T=\frac{4\sqrt{E(E+V_0)}}{2E+2\sqrt{E(E+V_0)}+V_0}$

Then it asks, find a simplified expression for $\displaystyle T$ when $\displaystyle V_0 >> E$

The best way I could come up with to (attempt to) solve this, was to assume E = 0, which would make T=0
If B is very small compared to A then AB is much less than A and much larger than B, but A+ B is very close to A.
If, in your equation, E is much smaller than $\displaystyle V_0$ then you can replace $\displaystyle V_0+ E$ with $\displaystyle V_0$ but cannot drop the E that is multiplied.

In other words T is close to $\displaystyle \dfrac{2\sqrt{E(V_0)}}{\sqrt{E(V_0)}+ V_0}$.

4. ## Re: Find Simplified Expression for T

Originally Posted by HallsofIvy
If B is very small compared to A then AB is much less than A and much larger than B, but A+ B is very close to A.
If, in your equation, E is much smaller than $\displaystyle V_0$ then you can replace $\displaystyle V_0+ E$ with $\displaystyle V_0$ but cannot drop the E that is multiplied.

In other words T is close to $\displaystyle \dfrac{2\sqrt{E(V_0)}}{\sqrt{E(V_0)}+ V_0}$.
That's a good way to look at it!

One minor correction: $\displaystyle \dfrac{4\sqrt{E(V_0)}}{2\sqrt{E(V_0)}+ V_0}$ $\displaystyle \left(\mbox{Or }\dfrac{2\sqrt{E(V_0)}}{\sqrt{E(V_0)}+ V_0/2}\right)$

Thank you