A hydrogen atom consists of a proton and an electron. According to the Bohr theory, the electron revolves about the proton in a circle of radius a (a = 5 · 10−9cm for the ground state). According to quantum mechanics, the electron may be at any distance r (from 0 to ∞) from the proton; for the ground state, the probability that the electron is in a volume element dV , at a distance r to r + dr from the proton, is proportional to e−2r/adV , where a is the Bohr radius. Write dV in spherical coordinates (see Chapter 5, Section 4) and find the density function f(r) so that f(r) dr is the probability that the electron is at a distance between r and r + drfrom the proton. (Remember that the probability for the electron to be somewhere must be 1.) Computer plot f(r) and show that its maximum value is at r = a; we then say that the most probable value of r is a. Also show that the average value of r−1 is a−1.
A hydrogen atom consists of a proton and an electron. According to the Bohr theory, the electron revolves about the proton in a circle of radius a (a = 5 · 10−9cm for the ground state). According to
from the proton. (Remember that the probability for the electron to be somewhere must be 1.) Computer plot f(r) and show that its maximum value is at r = a; we then say that the most probable value of r is a. Also show that the average value of r−1 is a−1.
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