Logo
Mauro Murzi's pages on Philosophy of Science - Quantum mechanics
Commutative operators
 prev Alpha radioactivity. Non commutative operators next 

Index

Links

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:

Commutative operators

There is an important property which holds for a pair of commutative entities: Two commutative entities have the same set of eigenfunctions.

 prev Alpha radioactivity. Non commutative operators next