Biomedical Engineering Reference
In-Depth Information
giving γ = 20.6. We can compute
v
directly from γ. In this example, however, we
know that
v
is very nearly equal to
c
.Using
v
=
c
in Eq. (2.22), we therefore write
6.63 × 10
-34
h
γ
mc
=
10
-13
m.
λ =
10
8
=
1.18
×
(2.28)
20.6
×
9.11
×
10
-31
×
3
×
For the last example one can, alternatively, derive the relativistic form of Eq. (2.25)
for electrons. The result is (Problem 42)
12.264
T
eV
1+
λ
Å
=
.
(Relativistic electrons)
(2.29)
10
6
T
eV
1.022
×
The last term in the denominator is
T
eV
/(2
mc
2
) and is, therefore, not important
when the electron's kinetic energy can be neglected compared with its rest energy.
Equation (2.29) then becomes identical with Eq. (2.25).
Also, in the period just before the discovery of quantummechanics, Pauli formu-
lated his famous exclusion principle. This rule can be expressed by stating that no
two electrons in an atom can have the same set of four quantum numbers. We shall
discuss the Pauli principle in connection with the periodic system of the elements
in Section 2.6.
2.5
Quantum Mechanics
Quantum mechanics was discovered by Heisenberg in 1925 and, from a com-
pletely different point of view, independently by Schroedinger at about the same
time. Heisenberg's formulation is termed matrix mechanics and Schroedinger's
is called wave mechanics. Although they are entirely different in their mathemati-
cal formulation, Schroedinger showed in 1926 that the two systems are completely
equivalent and lead to the same results. We shall discuss each in turn.
Heisenberg associated the failure of the Bohr theory with the fact that it was
based on quantities that are not directly observable, like the classical position and
speed of an electron in orbit about the nucleus. He proposed a system of mechanics
based on observable quantities, notably the frequencies and intensities of the lines
in the emission spectrum of atoms and molecules. He then represented dynamical
variables (e.g., the position
x
of an electron) in terms of observables and worked
out rules for representing
x
2
when the representation for
x
is given. In so doing,
Heisenberg found that certain pairs of variables did not commute multiplicatively
(i.e.,
xp
px
when
x
and
p
represent position and momentum in the direction of
x
),
a mathematical property of matrices recognized by others after Heisenberg's orig-
inal formulation. Heisenberg's matrix mechanics was applied to various systems
and gave results that agreed with those predicted by Bohr's theory where the latter
was consistent with experiment. In other instances it gave new theoretical predic-
tions that also agreed with observations. For example, Heisenberg explained the
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