Computational Chemistry

E¼total energy of the particle andV¼potential energy (the usual symbol), i.e.

1

2

mv^2 ¼E"V (4.25)

Substituting Eq.4.25formv^2 into Eq.4.23:

d^2 fðxÞ

dx^2

¼"

8 p^2 m

h^2

ðÞE"VfðxÞ (4.26)

f(x)¼amplitude of the particle/wave at a distancexfrom some chosen origin,
m¼mass of the particle,E¼total energy (kineticþpotential) of the particle,
V¼potential energy of the particle (possibly a function ofx).
This is the Schr€odinger equation for one-dimensional motion along the spatial
coordinatex. It is usually written

d^2 c

dx^2

þ

8 p^2 m

h^2

ðÞE"V c¼ 0 (4.27)

c¼amplitude of the particle/wave at a distancexfrom some chosen origin
The one-dimensional Schr€odinger equation is easily elevated to three-dimen-
sional status by replacing the one-dimensional operatord^2 /dx^2 by its three-dimen-
sional analogue

@^2

@x^2

þ

@^2

@y^2

þ

@^2

@z^2

¼r^2 (4.28)

r^2 is the Laplacian operator “del squared.” Replacingd^2 /dx^2 byr^2 , Eq.4.27
becomes

r^2 cþ

8 p^2 m

h^2

ðÞE"Vc¼ 0 '(4:29)

This is a common way of writing the Schr€odinger equation. It relates the
amplitudecof the particle/wave to the massmof the particle, its total energyE
and its potential energyV. The meaning ofcitself is, arguably, unknown [ 2 ] but the
currently popular interpretation ofc^2 , due to Born (Section 2.3) and Pauli^19 is that it
is proportional to the probability of finding the particle near a pointP(x, y, z) (recall
thatcis a function ofx, y, z):

Probðdx;dy;dzÞ¼c^2 dx dy dz '(4:30)

ProbðVÞ¼

Z

V

c^2 dx dy dz (4.31)

(^19) Wolfgang Pauli, born Vienna, 1900. Ph.D. Munich 1921. Professor Hamburg, Zurich, Princeton,
Zurich,. Best known for the Pauli exclusion principle. Nobel Prize 1945.Died Zurich 1958.
100 4 Introduction to Quantum Mechanics in Computational Chemistry