A way out of this dilemma was suggested by Bohr^15 in 1913 [ 9 , 10 ]. He retained
the classical picture of electrons orbiting the nucleus in accord with Newton’s laws,
but subject to the constraint that the angular momentum of an electron must be an
integral multiple ofh/2p:
mvr¼nhðÞ= 2 p; n¼ 1 ; 2 ; 3 ; (4.7)m¼electron mass
v¼electron velocity
r¼radius of electron orbit
h¼Planck’s constant
Equation4.7is the Bohr postulate, that electrons can defy Maxwell’s laws
provided they occupy an orbit whose angular momentum (corresponding to an
orbit of appropriate radius) satisfies Eq.4.7. The Bohr postulate is not based on a
whim, as most textbooks imply, but rather follows from: (1) the Plank equation
Eq.4.3,DE¼hnand (2) starting with an orbit of large radius such that the motion
is essentially linear and classical physics applies, as no acceleration is involved,
then extrapolating to small-radius orbits. The fading of quantum-mechanical
equations into their classical analogues as macroscopic conditions are approached
is called the correspondence principle [ 11 ].
Using the postulate of Eq.4.7and classical physics, Bohr derived an equation
for the energy of an orbiting electron in a one-electron atom (a hydrogenlike atom,
H or Heþ, etc.) in terms of the charge on the nucleus and some constants of nature.
Starting with the total energy of the electron as the sum of its kinetic and potential
energies:
Et¼1
2
mv^2 "
Ze^2
4 pe 0 r(4.8)
Z¼nuclear charge (1 for H, 2 for He, etc.),e¼electron charge,e 0 ¼permitivity of
the vacuum.
Using force¼mass&acceleration:
Ze^2
4 pe 0 r^2¼
mv^2
r(4.9)
i.e.
Ze^2
4 pe 0 r¼mv^2 (4.10)(^15) Niels Bohr, born Copenhagen, 1885. Ph.D. University of Copenhagen. Professor, University of
Copenhagen. Nobel Prize in physics 1922. Founder of the “Copenhagen school” interpretation of
quantum theory. Died Copenhagen, 1962.
4.2 The Development of Quantum Mechanics. The Schr€odinger Equation 95