2Ô0 
THE QUANTUM 
One first integral of the equations of motion expresses the fact 
that the angular momentum is constant. That is 
mr 2 p — constant = p (j) 
corresponding to the second of Kepler’s laws. The equation of the 
ellipse (eccentricity e, semi latus rectum l) may be written in this 
form 
me 
ÿ = -pi f 1 + eC0S ^) 
= |(l + ECOSp) . 
(2) 
an ordinary analytical expression for an ellipse in polar co-ordinates 
p being taken as zero at perihelion. 
The kinetic energy of the moving electron is 
But 
£ 
dr 
Tp 
T = ™(r* + 
P dr 
1 dp 
p da 
m dp 
-esin p 
and 
v P p 
rip — — =—o 
T mr m 
(i + £COS p) 
Hence 
The potential energy is 
me* 
T = ^2( T + 2£ COS p + £ 2 ) 
v — e — m& 
r p 
The total energy, H = T + V 
me 4 
and is constant as it should be. 
2 (i + £COS p) 
So we find 
W = - H = 
me- 
2p 2 
£ 2 ) 
(I - £ 2 ) 
(3) 
(4) 
(5) 
(6) 
(7) 
(8) 
(9) 
The Quantum Relations.—We now have to introduce the restrictions 
imposed by the quantum theory on the possible orbits. The expres 
sions obtained for the energy show that the polar co-ordinates p, r are 
canonical co-ordinates in the Hamiltonian sense, and may be regarded 
as equivalent to q x , q 2 in the equations of restriction (p. 22). 
We consider first the co-ordinate p, and “ quantize with respect 
to the angular motion ” only, that is we impose the restriction 
expressed by the equation 
Ç2 IT p2ir 
j p^dp = p J dp — 2np — n^h . . . (10) 
According to quantum mechanics the angular momentum must be 
an integral multiple of h/27i, which is the same condition as we