Schwarz Inequality | Quantum Science Philippines

## Schwarz Inequality

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