by CARIEL O. MONTALBAN
In quantum mechanics, I have learned that the wavefunctions, , reside in Hilbert’s space. What is Hilbert’s space? I guess to answer this question requires exploring the basic properties of Hilbert’s space.
Hilbert’s space is a linear vector space whose elements, entities or components obey certain rules or axioms. This means firstly than you can add these elements and the resulting sum is also as a member or entity of that space. Secondly, you can multiply the elements with any arbitrary scalar and the product yields something which is also a component of that same space. Additionally, the operations of addition and multiplication obey definite rules. These rules are called axioms for addition and multiplication.
By means of simple problems discussed below, I illustrate these axioms which are obeyed by a linear vector space and to which the wavefunctions, , of quantum mechanics belongs.
As a simple example, let us consider the set of all entities of the form where are real numbers. Do these form a linear vector space? First, we have know how these elements are added and how they multiply with scalars. If their addition and multiplication are defined respectively as follows:
we can then verify that the axioms required for a linear vector space are satisfied in this case.
From the addition operation, we can write the null vector of the set as:
Also from the multiplication operation, we can then write down the inverse of simply as .
We can now verify that all four axioms for addition of elements of the set are satisfied.
First Axiom: Commutativity Property
The operation of addition in a linear vector space is commutative; which means that we don’t care about the order in which the elements are added because we always get the same result. This axiom is written as:
Our proof is as follows. Let and .
Thus in a linear vector, the addition of vectors is commutative.
Second Axiom: Associative Property
The operation of addition in a linear vector space is associative which means that we don’t care about the order in which two elements are added to the third one because we always get the same result. This axiom is expressed as:
To prove this in the case of the set , we let
Therefore the addition of vectors in a linear vector space is associative.
Third Axiom: Existence of an identity element
The third requirement for a set to be a linear vector space is that the identity element exists. The identity element is defined as
The identity element of the set is therefore none other than the null vector
To show this property, we just apply the definition of addition hence
Fourth Axiom: Existence of an inverse
The inverse of a vector should exist in a linear vector space. The inverse is defined by the statement
For the set we can then verify the existence of an inverse as follows:
Examples of non-vector spaces
From the four axioms of addition of linear vector space, we can further make the following observations.
(1) If are required to be positive numbers, we can’t construct a vector space because Axiom (iv) will not be satisfied.
(2) The vectors of the form do not form a linear vector space. To show this, we let
where are all real numbers.
Then by Axiom (i),
Thus, does not form a linear vector space. The closure property is clearly violated since
About the Author:
CARIEL O. MONTALBAN finished his B.S. in Physics from Mindanao State University-Iligan Institute of Technology (MSU-IIT), Iligan City, Philippines in March 2008 and is now a graduate student of the same university. He hopes to become an active researcher in the field of experimental physics in the future.