First of all we know from classical mechanics that velocity and momentum,
as well as position, are represented by vectors. Thus we need to represent
the momentum operator by a vector of operators as well,
= (,,).
(11.1)
There exists a special notation for the vector of partial derivatives,
which is usually called the gradient, and one writes
= .
(11.2)
We now that the energy, and Hamiltonian, can be written in classical
mechanics as
E = mv2 + V(x) = p2 + V(x),
(11.3)
where the square of a vector is defined as the sum of the squares of
the components,
(v1, v2, v3)2 = v12 + v22 + v32.
(11.4)
The Hamiltonian operator in quantum mechanics can now be read
of from the classical one,
=
+ V(x) = -
+
+
+ V(x).
(11.5)
Let me introduce one more piece of notation: the square of the gradient
operator is called the Laplacian, and is denoted by .