/
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.