10.4 Volume integrals 299
The physical interpretation of an orbital is in terms of an electron probability density;
for an electron in orbital ψthe quantity
|ψ(r, 1 θ, 1 φ)|
2dv (10.6)
is interpreted as the probability of finding the electron in the volume elementdvat
position (r, 1 θ, 1 φ). The square modulus,|ψ|
21 = 1 ψψ*, is used because wave functions are
in general complex functions. The probability of finding the electron in a region V
is then the volume integral.
10.4 Volume integrals
A triple, or three-fold, integral has the general form
(10.7)
where Vis a region in xyz-space. When the variables are the coordinates of a point in
ordinary space the integral is often called a volume integral. If the limits of integration
in (10.7) are constants then the region Vis a rectangular box of sidesx
21 − 1 x
1, y
21 − 1 y
1,
z
21 − 1 z
1.
EXAMPLE 10.5Evaluate the integral of the functionf(x, 1 y, 1 z) 1 = 111 + 1 xyzover the
rectangular box of sides a, b, cshown in Figure 10.5.
The integral (10.7) is
Then
(i)
This is a general result; the integral is the volume of the region V.
(ii)
ZZZZ ZZZ
Vxyz d xyz dx dy dz x dx y dy z
c b aab cv==
000 0 0 0ddz
Z
Vdv
ZZZZ ZZZ
Vd dxdydz dx dy dz
abc
c b aab cv==
=××=
000 0 0 0VV
ZZZ
VVVf x y z d(),, vv= d + xyz dv
ZZZZ
Vf x y z d f x y z dxdydz
zzyyxx(),, v= (),,
121212Z
V2||ψ dv
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...Figure 10.5