In 1922 Sommerfeld asked me whether I would be willing to follow him to a meeting in G¨ottingen at which Bohr would present his theory. These days in G¨ottingen we now always refer to the “Bohr festival”. There for the first time I learned how a man like Bohr worked on problems of atomic physics.24
Werner Heisenberg, 1968 Many properties of atoms, molecules, stars, and galaxies can be detected by measuring the energy spectrum of the electromagnetic radiation emitted from such objects. In 1884, the physicist Balmer tried to find a mathematical rela- tion between the measured wave lengths λin the spectrum of the hydrogen atom. By trial and guesswork, he found out that
λ= constã m2
m2−4 for m= 3,4,5.
This is a discrete sequence for the wave length. A few years later, Rydberg and Ritz postulated that all the possible values of the wave length observed in the spectrum of the hydrogen atom are given by the following empirical formula
λnm= 2πc
ωnm, n < m, n, m= 1,2, . . . where
ωnm:=R 1 n2 − 1
m2
. (2.43)
The experiment yields the valueR= 2.07ã1016s−1; this is called the Rydberg frequency. In 1911, based on scattering experiments, Rutherford formulated the hypothesis that each atom consists of electrons which surround the nu- cleus like the planets surround the sun. In 1913 Bohr postulated the following for the hydrogen atom.
(i) The hydrogen atom consists of one electron which travels around the nucleus (one proton) in a circular orbit (Fig. 2.11(a)).
24Heisenberg’s lecture about the history of quantum mechanics can be found in the collection of lectures edited by A. Salam (1968).
(a) bound state p
e−
(b) scattering state p e− 1
Fig. 2.11.Hydrogen atom
(ii) The angular momentumaof the electron is quantized, that is, a=n, n= 1,2, . . .
where his Planck’s quantum of action and:=h/2π.
(iii) If the electron jumps from an orbit with energy En to an orbit with lower energyEm, the energy difference
∆E=En−Em
is emitted by electromagnetic radiation. According to Einstein’s light quanta hypothesis, the angular frequencyωof the emitted light quantum (photon) is given by
∆E=ω.
This implies the fundamental energy-frequency relation ωnm=En−Em
. (2.44)
From this we get the cocycle relation
ωnm+ωmk+ωkn= 0 for all n, m, k= 0,1,2, . . . .
From the mathematical point of view, the cocycle relation shows that the emitted frequencies of the hydrogen atom have a cohomological structure.
This observation was the crucial starting point of Heisenberg’s ingenious ap- proach to quantum mechanics in 1924. In fact, Heisenberg spoke about the Ritz combination principle, but not on cohomology which was unknown at that time. As a crucial topological tool, cohomology only emerged in the 1930s.
In order to discuss Bohr’s model, let En, rn, and vn denote energy, ra- dius, and velocity of thenth orbit of the electron. We will show below that postulates (i) through (iii) imply
En=− e2 8πε0r1 ã 1
n2, n= 1,2, . . . (2.45)
as well as
rn= λe
2παãn2 and vn=αc
n, n= 1,2, . . . Recall the definition of both the fine structure constant
α:= e2
4πε0c = 1 137.04 and the Compton wave length of the electron,λe := mh
ec. Here, we use the following notations: e charge of the proton, −e charge of the electron, mp
mass of the proton, me mass of the electron, ε0 electric field constant of vacuum.
From (2.45) along with (iii) we obtain immediately the Rydberg–Ritz formula (2.43). For the smallest orbit we get the energy
E1=−13.6eV.
The electron is bound to the nucleus with this energy. Note that this energy is needed in order to ionize the hydrogen atom, that is, to eject the electron.
Moreover, this gives the order of magnitude for the energies which occur in chemical reactions per atom. The radius of the smallest orbit
r1= 5ã10−11m
is called the Bohr radius of the hydrogen atom. Recall that 1 nm = 10−9m (nanometer) and 1 fm = 10−15m (femtometer). The radius of the proton
rproton= 1.4ã10−15m
is much less than the Bohr radius. If we represent the proton by a pea of radius 5mm, then the electron would surround the pea at a distance of 30m.
In addition, the mass quotient mp
me = 1836
shows that the proton is much heavier than the electron. The speed v1=αc= c
137.04
of the electron on the smallest orbit is much less than the velocity of light c. Therefore, relativistic effects do not play any role in Bohr’s atomic model.
Equivalently, formula (2.45) can be written as En=−mec2α2
2n2 , n= 1,2, . . . Here,mec2is the rest energy of the electron.
Let us now prove the claim (2.45). The motion of the electron around the proton on a circle of radiusris described by the equation
x=r(cosωti+ sinωtj)
wherei, jare fixed right-handed orthonormal vectors. Hence
˙
x(t) =ωr(−sinωti+ cosωtj) and
¨
x(t) =−ω2r(cosωti+ sinωtj).
As usual, the dot denotes the time derivative. For the angular momentum of the electron, we obtain
L=me(x×x) =˙ meωr2k
wherek:=i×j. There act two attractive forces between the proton and the electron, namely, Newton’s gravitational force
FN =−Gmpme r2 ã x
r
and the electrostatic Coulomb force FC=− e2
4πε0r2 ãx r.
Here r := ||x||. Since ||FN||/||FC|| = 5ã10−42, the gravitational force can be neglected. Thus, the motion of the electron is governed by the Newton equation
mex¨=FC. Hence
meω2r= e2
4πε0r2. (2.46)
The energy is equal to E= me
2 x˙2+U = meω2r2
2 − e2
4πε0r =− e2
8πε0r. (2.47) Bohr postulated that the angular momentum only attains the following quan- tized values
||L||=n, n= 1,2, . . .
This givesmeωr2=nforn= 1,2, . . .From (2.46) we obtain the orbit radii rn=r1n2, n= 1,2, . . .
withr1:= 4πε02/mee2=λe/2πα.The energy is given by
E =− e2
8πε0rn =− e2 8πε0r1n2, and the orbital velocity is equal tovn =||x˙||=ωrn =αc/n.
The Bohr model described the observed spectrum of the hydrogen atom in a perfect manner. However, physicists did not understand this model for the following reason. In terms of classical electrodynamics, the rotating elec- tron represents an accelerated electric charge. Such a charge steadily loses energy by emitting electromagnetic radiation. Therefore, the electron cannot move on stable orbits. In addition, classical physics cannot explain why the electron jumps spontaneously from one orbit to another one. This fundamen- tal problem was solved by Heisenberg in 1925 and Schr¨odinger in 1926, by introducing quantum mechanics, namely, Heisenberg’s matrix mechanics and Schr¨odinger’s wave mechanics, respectively. In terms of Schr¨odinger’s wave mechanics, the wave function of the electron of the hydrogen atom performs eigenoscillations which correspond to Bohr’s semiclassical circular orbits. In 1927 Heisenberg discovered the crucial quantum-mechanical uncertainty re- lation which tells us that
The electron is a quantum particle for which the classical notions of position and velocity do not make sense anymore.