Schwarz Inequality, also known as Cauchy–Schwarz inequality, Cauchy inequality, or the Cauchy–Schwarz–Bunyakovsky inequality, is useful in many Mathematical fields such as Linear Algebra. This Inequality was formulated by Augustin Cauchy (1821), Viktor Yakovlevich Bunyakovsky (1859) and Hermann Amandus Schwarz (1888).

The uncertainty principle of quantum mechanics, which relates the incompatibility of two operators, rests on this important theorem of Schwarz.

This is a theorem that arise from the inner product of two vectors which sates that the square magnitude of the inner product of two vectors is less than or equal to the product of the square magnitude of any vector, i. e.,

where and are any vectors which obey the four axioms of inner product. The four axioms are:

where α and β are scalar constants.

Exercise (1):

By going through the derivation of Schwarz Inequality, show that the inequality becomes an equality if

where μ is an arbitrary constant.

Solution:

Starting with the Schwarz Inequality

with the general equation

From the axiom;

we let the axiom equal to zero and substitute the value V so then we have,

Doing algebra and simple transformation we arrive to the equation

and from the general equation we have, we derived this

with the condition

About the Author

Debbie Claire R. Sanchez is currently a student of MSU-IIT pursuing her graduate study and hopefully will be graduating soon. She is very much interested in the field of Materials Science more specifically on Polymers. She plans to pursue her Ph. D in the United States and dreams on working in a well known company.

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:

;

and

,

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:

(i)

Our proof is as follows. Let and .

Then,

.

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:

(ii) .

To prove this in the case of the set , we let

Then,

.

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

(iii) .

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

(iv) .

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.