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[3. Schrödinger equations.]
Quantum operators.
An operator is a rule that transcripts (that is transforms) a given function
in another function. Quantum mechanics makes use of operator for translating
classical mechanical equations in quantum mechanical equations.
Let u be a classical mechanical quantity. Every classical mechanical
quantity is defined by position and momentum, so u is a function of
q and p :
u = u (q_{1} , q_{2} , q_{3} , p_{1} , p_{2} , p_{3})
In quantum mechanics, an operator u_{op} corresponding to u is
defined by means of the following two rules acting on the equation
u = u (q_{1} , q_{2} , q_{3} , p_{1} , p_{2} , p_{3}).
 Substitute every occurrences of q_{k} with the operator
"multiplication by q_{k}" and every occurrences of f (q_{k}) with
the operator "multiplication by f (q_{k})".
 Substitute every occurrences of p_{k}^{n} with the operator
.
For example, let u be the mechanical energy E given by (3.3).
The operator E_{op} associated to E is:
.
Note that in (3.4) the expression U(q_{1} , q_{2} , q_{3}) means
"multiply by U(q_{1} , q_{2} , q_{3}) ".
