Mathematical Methods for Physicists: A concise introduction - Site Map

FOURIER SERIES AND INTEGRALS
continuously and so the electron would lose energy continuously and it would
spiral into the nucleus after just a fraction of a second. This does not occur.
Furthermore, atoms do not radiate unless excited, and when radiation does
occur its spectrum consists of discrete frequencies rather than the continuum of
frequencies predicted by the classical electromagnetic theory. In 1913 Niels Bohr
(1885±1962) proposed a theory which successfully explained the radiation spectra
of the hydrogen atom. According to Bohr's postulates, an atom can exist in
certain allowed stationary states without radiation. Only when an electron
makes a transition between two allowed stationary states, does it emit or absorb
radiation. The possible stationary states are those in which the angular momentum
of the electron about the nucleus is quantized, that is, mvr ˆ np, where v is
the speed of the electron in the nth orbit and r is its radius. Bohr didn't clearly
describe this quantum condition. De Broglie attempted to explain it by ®tting a
standing wave around the circumference of each orbit. Thus de Broglie proposed
that n ˆ 2r, where is the wavelength associated with the nth orbit. Combining
this with Bohr's quantum condition we immediately obtain
ˆ h
mv ˆ h
p :
De Broglie proposed that any material particle of total energy E and momentum p
is accompanied by a wave whose wavelength is given by ˆ h=p and whose
frequency is given by the Planck formula ˆ E=h. Today we call these waves
de Broglie waves or matter waves. The physical nature of these matter waves was
not clearly described by de Broglie, we shall not ask what these matter waves are ±
this is addressed in most textbooks on quantum mechanics. Let us ask just one
question: what is the (phase) velocity of such a matter wave? If we denote this
velocity by u, then
u ˆ ˆ E
p ˆ 1 q
p 2 c 2 ‡ m 2 0
p
c4
q
ˆ c
1 ‡…m 0 c=p† 2
ˆ c2
v
!
m
p ˆ 0 v
p
;
1 v 2 =c 2
which shows that for a particle with m 0 > 0 the wave velocity u is always greater
than c, the speed of light in a vacuum. Instead of individual waves, de Broglie
suggested that we can think of particles inside a wave packet, synthesized from a
number of individual waves of di€erent frequencies, with the entire packet traveling
with the particle velocity v.
De Broglie's matter wave idea is one of the cornerstones of quantum
mechanics.
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