298 Chapter 10Functions in 3 dimensions
and the value of this limit is the density at the point P. We note that although ρ
is defined as the limit of a ratio it is not a derivative in the normal sense. We can
however consider the differential massdm 1 = 1 ρdvin volume elementdv. The total
mass of the body is then the integral over the volume of the body,
(10.4)
Such volume integrals are triple integrals over the three coordinates of the space and
are discussed in Section 10.4.
The concept of density is applicable to any property that is distributed over
a region. One important example is the probability density of a distribution in
probability theory and statistics, discussed in Chapter 21. Probability densities are
also used in quantum mechanics for the interpretation of wave functions and for the
computation of the properties of atoms and molecules.
EXAMPLE 10.4Atomic orbitals and electron probability density
The solutions of the Schrödinger equation for an atom, the atomic orbitals, are nearly
always expressed in spherical polar coordinates. Some of the orbitals of the hydrogen
atom are listed in Table 10.1 (a
0is the bohr radius). Each orbital is the product of three
functions, one for each coordinate:
ψ(r, 1 θ, 1 φ) 1 = 1 R(r) 1 ⋅ 1 Θ(θ) 1 ⋅ 1 Φ(φ) (10.5)
The radial functionR(r)determines the size of the orbital, and the radial or ‘in-out’
motion of the electron in the orbital. The angular functionsΘ(θ) andΦ(φ) determine
the shape of the orbital and the angular motion of the electron (its angular momentum).
These functions are discussed in greater detail in Chapter 14.
Table 10.1 Atomic orbitals of the hydrogen atom
ψ θ
2
0
5
2
1
42
0praza
= re
−
π
2
cos
ψφθ
2
0
5
2
1
42
0praya
= re
−
π
2
sin sin
ψφθ
2
0
5
2
1
42
0praxa
= re
−
π
2
sin cos
ψ
2
0
3
0
2
1
42
2
0sraa
=−ra e
−
π
() 2
2
ψ
1
0
3
1
0sraa
= e
−
π
2
Mdm d==ZZ
VVρ v