introduction-weak-interaction-volume-one

CHAPTER ONE : QUANTUM MEaiANICS .
1 .1 Matter Waves .
In 1923 Lade Broglie suggested (1) that radiation and matter might i n
some way be both wave-like and corpuscular . He was led to this idea by the fac t
that in some experiments, such as that of Young's slits (2), light behaved as if
it were a wave, whereas in others, such as Compton scattering (3),
it behaved
like a corpuscle . He postulated that electrons might also possess a dual nature ,
and should thus be able to display wave-like characteristics .
Let us now assume that matter or de Broglie waves exist, and attemp t
to find their form for a single particle moving uniformly in the absence of
any external force' (4) . Let the mass of the particle be m, its momentum p
and its energy be E . We should expect the de Broglie wave of the particle to b e
longditudinal and hence we represent it in the standard manner by the wave
function (5) :
1J1' (x, t) = A exp (jx . k - j f t) , (1 .1 .1 )
where x and t are the position and time co-ordinates of points on the wave ,
A is the amplitude of the wave,
f is its frequency, and k is its wave o r
propagation vector . De Broglie' s problem was to find a formula for 16. in terms
of the kinematical and dynamical variables of the particle . The wave describe d
by (1 .1 .1) is a plane wave, whose planes of constant phase,
4 , are given by
(x . Is - ft) _ (1 .1 .2 )
These planes, and hence the whole wave, propagate with the phase velocit y
f'k ,~k2 . (1 .1
of
.3 )
However, in the light of later developments, we find that we must not equate
the phase, but the group velocity of the de Broglie wave to the velocity of th e
particle . Group velocity is the velocity with which a signal or 'packet' o f
energy may be propagated on the wave in a dispersive medium, and it is given
by the formula (6) :
v = (df /dv) (dv/dk) .
A further postulate is that the relation