Physics 471 Problem Set 11 Fall 2006
45. Griffiths Problem 4.8. Solve part (b) using Mathematica to determine the zeros of j1(x).
Mathematica knows ordinary Bessel functions Jν(x) (BesselJ[ν,x]) and the spherical Bessel
function j`(x) is related to the ordinary Bessel function as
j`(x) = rπ
2xJ`+1
2(x).
The zeros of any Jν(x) can be found using the command BesselJZeros[ν,{m,n}] to get a
list of the mth through nth zeros. This command must be preceded by the command
<< NumericalMath‘BesselZeros‘.
46. (a) An electron in hydrogen is in the angular state Y0
0(θ, φ). Find the probability that it
is located in the region 0 ≤θ≤π/2, 0 ≤φ≤π/2.
(b) What is this probability if the electron is in the angular state Y1
1(θ, φ)?
47. The radial wave functions for the n= 2 level of hydrogen are
R20(r) = A0
2aµ1−r
2a¶e−r/2aR21(r) = A1
4a2re−r/2a,
where ais the Bohr radius. Normalize these functions by determining A0and A1.
48. Determine the probability that the electron in a hydrogen atom is located a distance r≤a
from the nucleus for the 2sand 2pstates in the previous problem.