I have to post this question again as I did not get a satisfactory solution the
ID: 3280711 • Letter: I
Question
I have to post this question again as I did not get a satisfactory solution the last time... Please read the question carefully before answering. Also, it is preferred that you upload a photo of the handwritten, fully worked solution. Thanks.
In atoms there is a finite, though very small, probability that, at some instant, an orbital electron will actually be found inside the nucleus. In fact, some unstable nuclei use this occasional appearance of the electron to decay by electron capture. Assuming that the proton itself is a sphere of radius 1.1 × 10-15 m and that the wave function of the hydrogen atom's electron holds all the way to the proton's center, use the groundstate wave function to calculate the probability that the hydrogen atom's electron is inside its nucleus.
Explanation / Answer
Given,
Radius of hydrogen atom nucleus, a0 = 1.1 x 10-15 m
the ground state wave function of hydrogen atom
Y100 = A (1/ a03)1/2 2 e-r/a0
probability of finding the electron inside the nucleus is <Y100|Y100> = 1
A2 4x(1/ a03) e-2r/a0 r2 sin theta d theta d dr = 1
A2 4 x (1/pi a03) 4pi e-2r/a0 r2 dr =1
A2 4pi/a03 x (a0/2)3 e-x x2 dx = 1 x= 2r/a0
A2 x 4pi x 1/2 x 2! =1
A2 = 1 / 4pi
A2 = 0.079
the probability of finding the electron inside the nucleus is 0.079 is 7.9%
Y 100 = (1/pi a03)1/2 e-r/a0
we have to find the average value < r > for electron to be located
< r > = < Y100 | r | Y100>
= (1/pi a03) r e-2r/a0 r2 sin theta dr d d theta
= 1/pia03 x 4 pi e-2r/a0 r3 dr
= 4/a03 x ( a0/2)4 e-x x3 dx x = 2r/a0
= a0/4 x 3! integral of e-x xn = n! ( gamma function)
< r > = 3/2 a0
< r2> = <Y100 | r2 | Y100 >
= (1/pi a03) r2 e-2r/a0 r2 sin theta d theta d dr
= (1/pi a03) 4 pi e-2r/a0 r4 dr
= 4/a03 ( a0/2)5 e-x x5 dx x = 2r/a0
= a02/8 x 4!
< r2 > = 3a02
Uncertainity in position to finding the particle inside the nucleus
delta r = sqrt( < r2> - < r >2)
= sqrt ( 3a02 - (3a0/2 )2 )
= sqrt ( 3 a02 - 9a02/4 )
delta r = sqrt3 a0 /2
delta r = sqrt3 x 1.1 x 10-15 /2
delta r = 9.515 x10-16 m
uncetainity in postion to finding the particle inside the nucleus 9.515x10-16 m which less than the radius of nucleus.