Page 66 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics
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2.7 Particles in three dimensions 57
so that T and R become
16E(V 0 ÿ E) ÿ2a p
2m(V 0 ÿE)="
T e
V 2
0
16E(V 0 ÿ E) ÿ2a p
2m(V 0 ÿE)="
R 1 ÿ 2 e
V 0
In the limit as a !1,as V 0 !1,as m !1, or any combination, the
transmission coef®cient T approaches zero and the re¯ection coef®cient R
approaches unity, which are the classical-mechanical values. We also note that
in the limit " ! 0, the classical values for T and R are obtained.
Examples of tunneling in physical phenomena occur in the spontaneous
emission of an alpha particle by a nucleus, oxidation±reduction reactions,
electrode reactions, and the umbrella inversion of the ammonia molecule. For
these cases, the potential is not as simple as the one used here, but must be
selected to approximate as closely as possible the actual potential. However,
the basic qualitative results of the treatment here serve to explain the general
concept of tunneling.
2.7 Particles in three dimensions
Up to this point we have considered particle motion only in the x-direction.
The generalization of Schrodinger wave mechanics to three dimensions is
È
straightforward. In this section we summarize the basic ideas and equations of
wave mechanics as expressed in their three-dimensional form.
The position of any point in three-dimensional cartesian space is denoted by
the vector r with components x, y, z, so that
r ix jy kz (2:60)
where i, j, k are, respectively, the unit vectors along the x, y, z cartesian
coordinate axes. The linear momentum p of a particle of mass m is given by
dr dx dy dz
p m m i j k ip x jp y k p z (2:61)
dt dt dt dt
The x-component, p x , of the momentum now needs to carry a subscript,
whereas before it was denoted simply as p. The scalar or dot product of r and
p is
. .
r p p r xp x yp y zp z
and the magnitude p of the vector p is
2 1=2
2
.
2
p (p p) 1=2 ( p p p )
x y z
The classical Hamiltonian H(p, r) takes the form
