Question

From the condition of the foregoing problem find:
(a) normalized eigenfunctions of the particle in the states for which `Psi(r )` depends only on `r` ,
(b) the most probable value `r_(pr)` for the ground state of the particle and probability of the particle to be in in the region `r lt r_(pr)`

Answer 1

(a) The nomalized wave functions are obtained from the normalization
`1=int|Psi|^(2)dV=int|Psi|^(2) 4pir^(2)dr`
`=int_(0)^(ro)A^(2)4pichi^(2)dr=4piA^(2)int_(0)^(ro)"sin"^(2)(npi r)/(r_(o))dr`
`=4piA^(2)(r_(0))/(npi)int_(0)^(bar(npi))sin^(2)xdr=4piA^(2)(r_(0))/(npi).(npi)/(2)=r_(0).2piA^(2)`
Hence `A=(1)/(sqrt(2pir_(0)))` and`Psi=(1)/(sqrt(2pi.r_(0)))("sin"(npir)/(r_(0)))/(r)`
(b) The radial probability distribution function is
`P_(n)(r )=4pir^(2)(Psi)^(2)=(2)/(r_(0)) "sin"^(2)(npir)/(r_(0))`
For the ground state `n=1`
so `P_(1)(r )=(2)/(r_(0))"sin"^(2)(pi r)/(r_(0))"sin"^(2)(pi r)/(r_(0))`
By inspection this is maximum for `r=(r_(0))/(2)`. Thus `r_(pr)=(r_(0))/(2)`
The probability for the particle to be found in the region `r lt r_(pr)` is clearly `50%` as one can immediately see from a graph of `sin^(2)x`.