QUANTUM THEORY 
59 
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where l 0 is the mean longitude of the electron regarded as a planet, e the 
eccentricity, a the semiaxis major, w the longitude of perihelion, i the 
inclination, O the longitude of the node. These variables are so chosen 
that , p 2 , p 3 are the momenta associated with the coordinates q x , q 2 , q 3 
by the Hamiltonian equations 
.(42-3), 
dq r dH dp r _ dH 
ds dp r ’ ds dq r 
the Hamiltonian function H being expressed as a function of these six 
variables and the time s. 
The principle of quantisation is that for variables satisfying (42-3) 
(42-4), 
where n r is an integer (or zero), h Planck’s constant, and the integral is 
taken over a complete period of the coordinate q r . 
With the variables (42-2) p 1} p 2 , p 3 are constants, and q 1} q 2 , q 3 are 
angular variables which accordingly have period 2tt. Hence the conditions 
(42-4) become 
27 Tin (¡xa)^ = nh 1 
2 t rm (fia)* (1 - e 2 )* = n'h f ( 42 ‘ 5 )> 
277W (fxa)^ (1 — e 2 )^ cos i = n"h 
where n, n', n" are integers. In order that e and i may be real we must 
have 
n > n' > n". 
We shall call n the principal quantum number of the orbit, and n ', n" 
subsidiary quantum numbers*. 
The negative energy of the system is (as in an astronomical orbit) 
Or by (42-1) and (42-5) 
2 a 
(42-61). 
- x = K/n 2 , K = 2 t r 2 mZ 2 e 4 //i 2 (42-62). 
Accordingly the energy is determined by the principal quantum number 
and is independent of n', n". 
The possible eccentricities and inclinations of the quantised orbits are 
given by 
„ n * . n 
e 2 = -o > COS % = —7 
n 2 n 
(42-7). 
* n' is also called the azimuthal quantum number, n - n' the radial quantum 
number and n the total quantum number. From the magnetic properties of the 
hydrogen atom it is known that a fourth quantum number must be involved which 
has no representation in the usual atomic model.