Physics 471 Problem Set 12 Fall 2003
44. The radial probability density P(r) for a hydrogenic state with quantum numbers n, is
given by r2|Rn(r)|2.
(a) From the general result for the form of the radial wave function,
Rn(r)=Anρe−ρv(ρ),ρ=r
na ,
determine An for the states with =n−1.
(b) Obtain P(r) for these states.
(c) Show that the most probable value of rfor the states with =n−1isgivenbythe
Bohr result
r=n2a.
45. An electron in a hydrogen atom is in a state given by the wave function
ψ(r)=α
√π3/2
e−α2r2/2.
(a) What is the value of is this state?
(b) Find an expression for the probability that a measurement of the energy of this
electron will give the value −13.6 eV. Assume α=1/a,whereais the Bohr radius.
46. The radius of a proton is about 1 femtometer (10−15 m). Calculate the probability that
the electron in the ground state of hydrogen will be found inside the proton. (Hint: Make
use of the fact that the Bohr radius a=0.53 ×10−10 m is very large compared to the
proton radius.)
47. Griffiths Problem 4.9.