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5. Heisenberg indeterminacy principle.
The section begins with the distinction between commutative and non commutative
operators. The role of this distinction, which is a main feature of quantum mechanics, is illustrated by the
analysis of measurement in quantum mechanics.
A brief explanation of Ehrenfest theorem on mean values precedes the formulation
of Heisenberg indeterminacy principle.
The section ends with some physical applications of Heisenberg indeterminacy principle.
Commutative operators.
Let u and v be two physical quantities, and let u_{op} and
v_{op} be the associated quantum operators. Suppose that u_{op} is
applied to a complex function φ to construct the function u_{op} φ .
It is possible to apply v_{op} to that function and obtain
v_{op} u_{op} φ ;
this expression indicates the application of operator u_{op} to the function φ ,
followed by the application of operator v_{op} to the resulting function.
The question arise whether the order of application of quantum operators is essential:
v_{op} u_{op} φ is the same
as u_{op} v_{op} φ ?
If you remember the meaning of u_{op} v_{op} φ ,
which is not a multiplication of number, but an application of quantum operators to transform a
function in another function, you can guess that in general
v_{op} u_{op} φ is not the same as
u_{op} v_{op} φ .
However, for some quantum operators u_{op} and v_{op} ,
the following relation holds:
(5.1) v_{op} u_{op} φ = u_{op} v_{op} φ
Two particular quantum operators for which the relation (5.1) is true for every function φ are called
commutative operators, and the corresponding physical quantities are called commutative entities.
Note that the notion of commutative operators is applicable only to a pair of quantum operators:
An operator is commutative with respect to another operators.
An as example of a pair of commutative quantum operators, consider the position q_{1} and the
momentum p_{2} ; note that they are referred to two different axes. We have:
There is an important property which holds for a pair of commutative entities:
Two commutative entities have the same set of eigenfunctions.
