Infinitely large quantum number

[terp.asessed]

Hi--could someone explain how can one calculate for the root mean square fluctation in position when an electron (confined in a box) is quantum-mechanical and happens to be in a state (an infinitely large quantum number n) and why?
I do know how to calculate root mean square fluctation in position with the given function and n, as in (root-mean)² = <x²> - <x>² but am not sure how to do this with huge n value. Do I just use Schrödinger equation, where V (potential) is equal to 0?

 

[Nugatory]

You don't use Schrödinger's equation, you use a function that is a solution of Schrödinger's equation with the appropriate potential (which in this case is an infinite square well the width of the box). This is a common textbook exercise so googling for "infinite square well quantum" or the like will find you some worked examples.