Scalar product of two functions

Since the scalar product of two functions in direct space is equal to the scalar product of their Fourier transforms in reciprocal space, we have... 

It is also useful to define the transformed operator L whose operation on a function f is L f = L[Peqf). This operator coincides with the time reversed backward operator, further details on these relationships may be found in Refs. 43,44. L operates in the Hilbert space of phase space functions which have finite second moments with respect to the equilibrium distribution. The scalar product of two functions in this space is defined as (f, g) = (fgi q. It is the phase space integrated product of the two functions, weighted by the equilibrium distribution P The operator L is not Hermitian, its spectrum is in principle complex, contained in the left half of the complex plane. 

The operators just described will leave the scalar product of two functions of the function space unchanged (O,/, O,/,) = (/,-./ ). Such operators are said to be unitary and they can always be represented by unitary matrices (see 6-4). The proof that the 0M are unitary follows from considering... 

An important special case in such an expectation value is the scalar product of two functions, where the operator A is the identity operator / /f.r. x-,.) = l.Then... 

The operators just described will loave the scalar product of two functions of the function space unchanged 0Rff) = (/<>/ )-... 

The functional basis that supplies the global picture is connected through the expansion coefficients an to the spatial grid. This provides the ability to define the scalar product of two functions. If... 

The bra and the ket define a scalar (Vq l ) this is analogous to the scalar product of ordinary vectors, to the matrix product of row and column matrices, or to the scalar product of two functions defined through the integral... 

In order to complete the analogy between physical space and Hilbert space, we define the scalar product of two functions, fix) andg(jc), by... 

This relation is often used as the condition for completeness. The integral is analogous to the squared length of a vector, and a normalized function therefore has unit length . In such cases the sum of the squares of the expansion coefficients must approach unity as more terms are added. Generally the scalar product of two functions... 

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