Computational Chemistry

Chapter 4, Harder Questions, Answers

Q2

What is the probability of finding a particle at a point?
The probability of finding a particle in a small region of space within a system
(say, a molecule) is proportional to the size of the region (assume the region is so
small that within it the probability per unit volume does not vary from one
infinitesimal volume to another). Then as the size of the region considered
approaches zero, the probability of finding a particle in it must approach zero.
The probability of finding a particleata point is zero.
More quantitatively: the probability of finding a particle in an infinitesimal
volume of spacedvin some system (e.g. a molecule) is given by

PðdvÞ¼rðx;y;zÞdv¼rðx;y;zÞdxdydz

in Cartesian coordinates, wherer(rho) is the probability distribution function
characteristic of that particle in that system. The probability is a pure number, so
rhas the units of reciprocal volume, volume#^1 , e.g. (m^3 )#^1 or in atomic units
(bohr^3 )#^1 .P(dv) generally varies from place to place in the system, as the coordi-
natesx,y,zare varied; referring to an “infinitesimal” volume is a shorthand way of
saying that

lim

Dv! 0

Pðx;y;zÞDv¼Pðx;y;zÞdv

The probability of finding the particle in a volumeVis

PðVÞ¼

ð

V

rðx;y;zÞdv

where the integration is carried out over the coordinates of the volume (in Cartesian
coordinates, over the values ofx,y,zwhich define the volume). For a point, the
volume is zero and the coordinates will vary from 0 to 0:

PðVÞ¼

ð^0

0

rðx;y;zÞdv¼½Fðx;y;zފ^00 ¼ 0

Note: this discussion applies to a point particle, such as an electron – unlike a
nucleus – is thought to be. For a particle of nonzero size we would have to define
what we mean by “at a point”; for example, we could say that a spherical particle is
at a point if its center is at the point.

Answers 611