InterviewSolution
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InterviewSolution
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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)` |
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Answer» (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`. |
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