Page 73 - Electrical Properties of Materials

56 The hydrogen atom and the periodic table

The negative sign of the energy which substituted in eqn (4.15) gives
means only that the energy of this
me 4
state is below our chosen zero E =– . (4.18)
2 2
point. [By writing the Coulomb 8 h
0
potential in the form of eqn (4.2)
Thus, the wave function assumed in eqn (4.13) is a solution of the differential
we tacitly took the potential en-
equation (4.12), provided that c 0 takes the value prescribed by eqn (4.17). Once
ergy as zero when the electron is
we have obtained the value of c 0 , the energy is determined as well. It can take
at infinity.]
only one single value satisfying eqn (4.18).
Let us work out now the energy obtained above numerically. Putting in the
constants, we get
)
(9.1 × 10 –31 )(1.6 × 10 –19 4 kg C 4
E =–
) (6.63 × 10
8(8.85 × 10 –12 2 –34 2 2 –2 2 2
) F m J s
= –2.18 × 10 –18 J. (4.19)
Expressed in joules, this number is rather small. Since in most of the sub-
sequent investigations this is the order of energy we shall be concerned with,
and since there is a strong human temptation to use numbers only between 0.01
and 100, we abandon with regret the SI unit of energy and use instead the elec-
tron volt, which is the energy of an electron when accelerated to 1 volt. Since

1eV = 1.6 × 10 –19 J, (4.20)
From experimental studies of the the above energy in the new unit comes to the more reasonable-looking
spectrum of hydrogen it was numerical value
known, well before the develop-
E = –13.6 eV. (4.21)
ment of quantum mechanics, that
the lowest energy level of hydro- What can we say about the electron’s position? As we have discussed many
gen must be –13.6 eV, and it was times before, the probability that an electron can be found in an elementary
a great success of Schrödinger’s volume (at the point r, θ, φ) is proportional to |ψ| —in the present case it is
2
theory that the same figure could proportional to exp(–2c 0 r). The highest probability is at the origin, and it de-
be deduced from a respectable- creases exponentially to zero as r tends to infinity. We could, however, ask a
looking differential equation. slightly different question: what is the probability that the electron can be found
in the spherical shell between r and r +dr? Then, the probability distribution
is proportional to
2
2 –2c 0 r
2
r |ψ| = r e , (4.22)
which has now a maximum, as can be seen in Fig. 4.2. The numerical value of
the maximum can be determined by differentiating eqn (4.22)
d 2 –2c 0 r –2c 0 r 2
(r e )=0=e (2r –2c 0 r ), (4.23)
dr
whence
2
1 4π 0
r = = 2 = 0.0528 nm. (4.24)
c 0 e m
This radius was again known in pre-quantum-mechanical times and was called
the radius of the first Bohr orbit, where electrons can orbit without radiating.
Niels Bohr, Nobel Prize, 1922.
Thus, in quantum theory, the Bohr orbit appears as the most probable position
of the electron.