PH253: hints on HW6-7, #1

There is one big issue with this problem: there are lots of shiny numbers that you're just dying to plug in to messy equations. Resist! It is a trap!

Seriously: doing algebra first to simplify the expressions is always, always, always the right way. Catching an algebra mistake is easy. Figuring out after the fact that your made a typing mistake or are experiencing floating point errors (relevant to this problem) is really hard. I realize you favor numbers over symbols, and it seems easier sometimes (what with the calculator and all) ... but in the long run, especially for messy problems, it is a trap.*

Anyway: specific hints for number 1 (be sure to check the link below):

(1) Find the simplest symbolic expression possible first before putting any numbers in. This has three advantages: first, you can more easily keep track of errors via dimensional analysis; second, you can easily check your work at a glance without re-typing all those numbers; third, plugging everything in at once will overload your poor little calculator (see link above, e.g.) and you will get "0" or "1" depending on the situation. If you are CS, you should understand why this happens ;-)
(2) Group the constants you do have left to form numbers you already know. For instance, the pre-factor outside the brackets in the given expression is just the Bohr ground state energy, 13.6eV.
(3) Be very, very careful with units. If you group everything correctly, you should just have a fraction times the Bohr ground state energy, and it will be a moot point.
(4) If it really isn't coming out correctly (both answers in the 1E-5 eV range), look here.

*Somewhere along the line, someone probably told you that as an engineer, "numbers are your friends." They were just being mean, like giving intentionally wrong driving directions. Funny for them, but just lost time for you.

When you think about it, symbols are your real friends, since they represent any number you like, not just one particular one. Symbols let you reduce the solution to its simplest form before having to fuss with all those messy numbers, reducing the button-pushing to a minimum. Really, numbers are your enemies. Use them as the last resort, and as few of them as possible - if you let too many accumulate, they gang up on you. Generally speaking, if the problem at hand can only be solved with a pile of numbers, you know you are in deep, deep trouble.