a)Write Complete wave function for an electron in a 2s orbital of hydrogen
b)find the probability that the electron is at a distance from the nucleus that is outside the radius of the node.
c)graph the radial distribution function for this system.
using quantum numbers of n=2 l=0 ml=0 ms=+/- (1/2)
Z = 1
eps = (8.854187816*10^(-12))/10000000000*Coulomb^2*Joule^(-1)*Angstrom^(-1)
me = 9.1093897*10^(-31) Kilogram
e = 1.602177338*10^(-19) Coulomb
h = 1.05457266*10^(-34) Joule*Second
a = 4*π*eps*h^2/(me*e^2)
R[r*Angstrom_] = (1\(2*2^(1/2)))*((Z/a)^(3/2))*(2 -
1/(2 p[r*Angstrom]))*Exp[-p[r*Angstrom]/4] (This is the Radial Wavefunc)
Y = (1/(2*π)^(1/2))*1*(2^(1/2)/2) (The complete roational wave func for theta and phi)
Λ = 1 (The spin wave function, is this right? since it has only one electron?)
The Attempt at a Solution
I did the solution in mathematica, what i did was set R=0 and solved for r which came out with units of (J*s^2)/(Kg*Angstrom^2) which doesnt seem right to me shouldnt r be a unit of distance in general? Ecspecially at the node right?.
Btw the node came out to be 6.614715537095822*10^-22 (J*s^2)/(Kg*Angstrom^2)
So i took that node and integrated r^2*R^2(r) from 0 to the node, times the integral from zero to 2pi dPhi times the integral of zero to pi of Y^2sin(theta)dTheat
∫(0 to node) r*r*R(r)*R(r) dr * ∫(0 to 2pi) dPhi *∫(0 to pi) Y*Y*sin(Theta) dTheta
This gave me 0.0003155974/Angstrom (Shouldn't it be dimensionless?)
Thus giving me what i think is the probablity (i took away the unit) of the elecron being form within the radius to the node so i took this answer and minused it from 1 to get the probability of the electron being beyond the node which was 0.999684 which is very but for a 2s orbital it is kinda reasonable.
but when it comes to graphing, the radial wavefunction times r^2 times 4 pi (the radial distrubution function) gives me non machine sized numbers, or it gives underflow (the number is to small for the machine to realize its not zero)
any suggestions or tips? And am I doing this right? Thank ya'll for takin the time to read it!