4.
THE WAVE
NATURE OF
PARTICLES
cou ld
not be
applied
to
ap eriodic
systems,
most
collision
and
sca ttering
problems were beyond its pale. Furthermore, it contained errors , contradictions,
and ambiguities."
It
did have some virtues, however.
predicted a large
body of experimental results from a few simple rules, and it set th e stage for the
new quantum mechanics which soon replaced it.
We will now proceed to
discuss the wave mechanics of de Broglie and Schrodinger.
121
c
THE
DIFFRACTION
OF
hand
the de Brogl ie frequency has not been a
very usefu l concept and it
com e~
into play only in th e calculation of the
p h a s ~
: el.ocity.
H ere a dis
tin ction ap pears betwee n the relativistic and nonr: latlvistic
c ~s~s.
If
the rest
mass energy is includ ed in the total energy
E
(as
In
de Broglie s tr eatment),
then the phase velocity of the wave becom es
U
=
AY
=
~
=
~
vi
p2
C2
+
(moc
2
)
2
=
c
J
I
or,
THE DEVELOPMENT
OF WAVE MECHANICS
120
h
P
holds for photons as well as for both relativistic and nonrelativistic materia l
particles, provided that the appropriate expression for
p
is used.
On the oth er
(4.15)
fik
v
2m
=
'2'
U
Whether or not particles of a given momentum will exhibit their
wa~(
characteristics will be determined by the relative magnitude oftheir de Broglie
wavelength in comparison with the physical dimensions of the environmen
in which they are found.
For wavelengths that are much smaller than the
dimensions of apertures and obstacles, diffraction and other wave effects
~ rc
not ordinarily observed.
I n such cases we can assume rectilinear
propag~~lOI
and problems can be treated by means of ray diagrams (for .example,
~ISlb l c
light in our everyday world).
However, for wavelengths which
a p ~ rmu m a t '
or exceed th e dimensions of objects, diffraction effects become qu ite importan
and ray diagrams become meaningless
(for example, audible sound
i~
ou
everyday world ).
In order to get some insigh t into th e kinds of behavior tc
5.
THE DIFFRACTION
OF PARTICLES
dos
I
dE
v
=
dk
=
Ii
dk .
that is onehalf of the particle velocity. Thus the phase velocity is not of any
physical significance. The group velocity of a particle wave, however, is given
by
In obtaining Equation 4.I5 we have expressed the relativistic momentum as
p
=
ymov,
wh ere
v
=
fJc
is the
par~icle
velocity and
y
~
(I
.(32)
to
From
our knowledge of waves" we identify the particle velocity
v
WIth the group
velocity of th e wave packet.
Since special relativity
requires that
v
be less
than
c,
we no te that Equation 4.15 calls for phase veloci ties greater than
c.
However, as no energy (that is, no signal or information) is transmitted at the
phase velocity, the fact that
u
>
c
constitutes no violation of the postulates of
special relativity.
.
.
In nonrelativistic quantum
mechamcs the rest mass term IS neglect ed
and the total energy
E
is merely the sum of the kinetic an d potential energies.
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 Spring '08
 Nelson
 mechanics, Magnetism, Diffraction, Bragg, th en th

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