/ 
QUANTUM THEORY 
69 
i of the 
Lst one 
»mplete 
ociated 
? corre- 
or in tensor notation 
dH dq„ 
dq M ^ ds (49 
Where a., = J - v) if this is ± 1{ 
= 0 otherwise j ' ' 
4:8-2). 
:bit has 
case of 
d (unit) 
Consider a general transformation of coordinates (not confined to Hamilton 
ian coordinates) and let be a covariant tensor with the values (49-2) 
in our original system. Since H and s are invariants, dH/dq M and dq v jds 
are covariant and contravariant vectors respectively. Thus (49-1) is a 
tensor equation and holds in all coordinate systems. Of these the possible 
Hamiltonian systems are given by the condition that has transformed 
to its original values. If |a M „j denotes the determinant formed with the 
elements a M „, we have* 
1 V\ a nv\ dVis invariant for all coordinate systems. 
48-3). 
ites and 
■iodicity 
perfect 
And since |a M „| has the same value for all Hamiltonian systems 
\dV is invariant for all Hamiltonian coordinates, 
which proves the theorem. 
rdinates 
milton’s 
8-1) 
It may be noted that in mechanics Hamiltonian coordinates are dis 
tinguished from general coordinates in much the same way as in geometry 
Galilean coordinates (unaccelerated rectangular coordinates and time) are 
distinguished from general coordinates, viz. that a fundamental tensor 
characterising the continuum takes certain simple numerical values. 
48-4). 
The K and L levels. 
• type of 
umption 
ept that 
3 out the 
n. 
50. When the nucleus is attended by more than one electron mutual 
perturbations occur according to laws which have not yet been formulated. 
The atomic model cannot be worked out in detail, but a certain amount of 
knowledge of the arrangement of the electrons has been ascertained with 
the aid of experimental data. 
Considering normal atoms unexcited by high temperature, the first two 
mt, that 
irdinates 
vere not 
50 would 
ulus the 
electrons go into 1-quantum orbits and the next eight into 2-quantum 
orbits. These are called K and L electrons respectively. This structure is 
completed in Neon (Z = 10) and remains an undisturbed foundation in all 
higher elements. The M electrons in 3-quantum orbits start with Sodium 
(11) and reach a complement of 8 in Argon (18), after which 4-quantum 
orbits begin. But unlike the K, L structure the M structure is modified 
as (42-3) 
later and extended to 18 electrons in Copper (29) and all higher elements. 
Similarly the N electrons in 4-quantum orbits stop temporarily at 8, 
afterwards extended to 18, and then to 32. 
* Eddington, Mathematical Theory of Relativity , §§ 48, 49.