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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 uop and
vop be the associated quantum operators. Suppose that uop is
applied to a complex function φ to construct the function uop φ .
It is possible to apply vop to that function and obtain
vop uop φ ;
this expression indicates the application of operator uop to the function φ ,
followed by the application of operator vop to the resulting function.
The question arise whether the order of application of quantum operators is essential:
vop uop φ is the same
as uop vop φ ?
If you remember the meaning of uop vop 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
vop uop φ is not the same as
uop vop φ .
However, for some quantum operators uop and vop ,
the following relation holds:
(5.1) vop uop φ = uop vop 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 q1 and the
momentum p2 ; 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.
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