Question

The wave function of 2s electron is given by `Psi_(2s) = (1)/(4 sqrt(2pi)) ((1)/(a_(0)))^(3//2) (2 - (r)/(a_(0))) e^(-r//a_(0))`
It has a node at `r = r_(0)`. Find the relation between `r_(0) and a_(0)`

Answer 1

The probability of finding 2s electron will be : `Psi_(2s)^(2) = (1)/(32pi) ((1)/(a_(0)))^(3) (2- (r_(0))/(a_(0)))^(2) e^(-2r//a_(0))`
Node is the point at which probability of finding electron is zero. Thus, `Psi_(2s)^(2) = 0` when `r = r_(0)`
`:. (1)/(32pi) ((1)/(a_(0)))^(2) (2 - (r_(0))/(a_(0)))^(2) e^(-2r_(0)//a_(0)) = 0`
In this expression, the only factor that can be zero is `(2 - (r_(0))/(a_(0)))`
Thus, `2 - (r_(0))/(a_(0)) = 0 or r_(0) = 2 a_(0)`