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## Hermitian Inner Product

In lecture we defined the Hermitian inner product between two functions f(q) and g(q) to be given by

Note that this operation is very analogous to the familiar vector dot product

Demonstrate that the following properties hold for the Hermitian inner product and state the analogous property for the vector dot product. In the statements below f, g and h will be functions, whereas lower case Greek letters will be constants, so that, for instance, means .

Mathematicians call any operation ;SPMlt;|;SPMgt; holding the four properties listed below a Hermitian inner product. Because mathematicians take great care in their definitions, most of the properties of the operation we have defined for ;SPMlt;|;SPMgt; follow from just these four basic properties. You may use this analogy as a guide for any proofs you may need to give involving the Hermitian inner product. Because they are so analogous to the properties of the ordinary vector inner product (dot product), these relations should be easy to remember.

a)

b)

c)

d)

Note: c) above is redundant and not needed to define the Hermitian Inner Product (HIP). I include it to remind you that pulling a constant out of the left hand side of a HIP generates the complex conjugate of the constant.

Prof. Tomas Alberto Arias
Thu Mar 20 17:04:34 EST 1997