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
(11.3) |
where the square of a vector is defined as the sum of the squares of the components,
(11.4) |
The Hamiltonian operator in quantum mechanics can now be read of from the classical one,
(11.5) |
Let me introduce one more piece of notation: the square of the gradient operator is called the Laplacian, and is denoted by .