Question 39.51: What is the probability that in the ground state of the hydr......

According to Sample Problem — “ Probability of detection of the electron in a hydrogen atom,” the probability the electron in the ground state of a hydrogen atom can be found inside a sphere of radius r is given by

p(r)=1-e^{-2 x} ( 1+2 x+2 x^2 )

where x = r/a and a is the Bohr radius. We want r = a, so x = 1 and

p(a)=1-e^{-2}(1+2+2)=1-5 e^{-2}=0.323 .

The probability that the electron can be found outside this sphere is 1 – 0.323 = 0.677. It can be found outside about 68% of the time.