Can this ratio be written any better?

Question

This topic is closely related to previous topic where we were to calculate ratio $\lambda_e/\lambda_p$ for proton and electron with same velocities.

This time we I want to know if it is possible to derive a ratio $\lambda_e/\lambda_p$ for proton and electron who have same kinetic energies $(E_{ke} = E_{kp})$. So i write this like:

\begin{align} \frac{\lambda_e}{\lambda_p} = \frac{\tfrac{h}{p_e}}{\tfrac{h}{p_p}} = \frac{p_p}{p_e} = \frac{\sqrt{{E_{kp}}^2 + 2E_{kp}E_{0p}}}{\sqrt{{E_{ke}}^2 + 2E_{ke}E_{0e}}} \longleftarrow\substack{\text{Here i know that kinetic}\\\text{energies are the same so}\\\text{i have to use $E_{kp}=E_{ke}$}} \end{align}

I want to know if this can be reduced a bit more or is this the best possible result. If anyone has any idea please do tell.

If $E_{ke}=E_{kp}$, than it would simply turn into $\lambda_e=\lambda_p$. — fibonatic, Jul 23 '13 at 15:30
I guess that in the denominator you have written $E_{0p}$ instead of $E_{0e}$, am I right? — neutrino, Jul 23 '13 at 15:38

neutrino · Answer 1 · 2013-07-23T22:35:37.853

It's difficult to simplify much more your last expression, but:

\begin{align} \frac{\lambda_e}{\lambda_p} = \frac{\tfrac{h}{p_e}}{\tfrac{h}{p_p}} = \frac{p_p}{p_e} = \frac{\sqrt{1 + 2\frac{E_{0p}}{E_{kp}}}}{\sqrt{1 + 2\frac{E_{0e}}{E_{ke}} }} =\sqrt{\frac{1 + 2\frac{E_{0p}}{E_{kp}}}{1 + 2\frac{E_{0e}}{E_{ke}}}} \end{align}

Now, you can perform some approximations. For example, if $E_{0e} >> {E_{ke}}$:

\begin{align} \frac{\lambda_e}{\lambda_p} = \sqrt{\frac{1 + 2\frac{E_{0p}}{E_{kp}}}{1 + 2\frac{E_{0e}}{E_{ke}}}} \approx \sqrt{\frac{E_{0p}}{E_{0e}}} \end{align}

Can this ratio be written any better?

1 Answers1