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,
|
\hat{p} = ({ℏ\over
i} {∂\over
∂x}, {ℏ\over
i} {∂\over
∂y}, {ℏ\over
i} {∂\over
∂z}).
| (11.1) |
There exists a special notation for the vector of partial derivatives, which is usually called the gradient, and one writes
|
\hat{p} = {ℏ\over
i} \mathop{∇}.
| (11.2) |
We now that the energy, and Hamiltonian, can be written in classical mechanics as
|
E = {1\over
2}m{v}^{2} + V (x) = {1\over
2m}{p}^{2} + V (x),
| (11.3) |
where the square of a vector is defined as the sum of the squares of the components,
|
{({v}_{1},{v}_{2},{v}_{3})}^{2} = {v}_{
1}^{2} + {v}_{
2}^{2} + {v}_{
3}^{2}.
| (11.4) |
The Hamiltonian operator in quantum mechanics can now be read of from the classical one,
|
\hat{H} = {1\over
2m}\hat{{p}}^{2} + V (x) = −{{ℏ}^{2}\over
2m}\left ( {{∂}^{2}\over
∂{x}^{2}} + {{∂}^{2}\over
∂{y}^{2}} + {{∂}^{2}\over
∂{z}^{2}}\right ) + 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 Δ.