Page 67 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

58 Schro Èdinger wave mechanics
p 2 1 2 2 2
H(p, r) ˆ ‡ V(r) ˆ (p ‡ p ‡ p ) ‡ V(r) (2:62)
z
y
x
2m 2m
When expressed in three dimensions, the de Broglie relation is
p ˆ "k (2:63)
where k is the vector wave number with components k x , k y , k z . The de Broglie
wavelength ë is still given by
2ð h
ë ˆ ˆ
k p
where now k and p are the magnitudes of the corresponding vectors. The wave
packet representing a particle in three dimensions is
1 i(p rÿEt)="
.
Ø(r, t) ˆ A(p, t)e dp (2:64)
(2ð") 3=2
As shown by equations (B.19), (B.20), and (B.27), the momentum-space wave
function A(p, t) is a generalized Fourier transform of Ø(r, t),

1 ÿi(p rÿEt)="
.
A(p, t) ˆ Ø(r, t)e dr (2:65)
(2ð") 3=2
The volume elements dr and dp are de®ned as
dr ˆ dx dy dz

dp ˆ dp x dp y dp z
and the integrations extend over the complete range of each variable.
For a particle moving in three-dimensional space, the quantity

Ø (r, t)Ø(r, t)dr ˆ Ø (x, y, z, t)Ø(x, y, z, t)dx dy dz
is the probability at time t of ®nding the particle with its x-coordinate between
x and x ‡ dx, its y-coordinate between y and y ‡ dy, and its z-coordinate

between z and z ‡ dz. The product Ø (r, t)Ø(r, t) is, then, the probability
density at the point r at time t. If the particle is under the in¯uence of an
external potential ®eld V(r), the wave function Ø(r, t) may be normalized

Ø (r, t)Ø(r, t)dr ˆ 1 (2:66)
The quantum-mechanical operators corresponding to the components of p
are
" @ " @ " @
^ p x ˆ , ^ p y ˆ , ^ p z ˆ
i @x i @ y i @z
or, in vector notation
"
^ p ˆ = (2:67)
i