Complex function scalar product

In case the basis functions are complex, the scalar product is defined as the integral of the complex conjugate of the first function times the second function, as in Eq. 

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. 

One complex scalar product on C[ -1, 1], the complex vector space of continuous functions on [—1, 1], is... 

Complex scalar products arise naturally in quantum mechanics because there is an experimental interpretation for the complex scalar product of two wave functions (as we saw in Section 1.2). Students of physics should note that the traditional brac-ket notation is consistent with our complex scalar product notation—just put a bar in place of the comma. The physical importance of the bracket will allow us to apply our intuition about Euclidean geometry (such as orthogonality) to states of quantum systems. 

For example, the constant function 3 and the function cos rtx are perpendicular in the complex scalar product space C[—1, 1] since... 

If V is finite dimensional, then Definition 3.7 is consistent with Definition 2.2 (Exercise 3.13). In infinite-dimensional complex scalar product spaces. Definition 3.7 is usually simpler than an infinite-dimensional version of Definition 2.2. To make sense of an infinite linear combination of functions, one must address issues of convergence however, arguments involving perpendicular subspaces are often relatively simple. We can now define unitary bases. 

Eor example, if we consider C" with the standard complex scalar product, then the set ck k = 1,..., , where Ck denotes the vector whose kth entry is 1 and all of whose other entries are 0, is a unitary basis of V. A more sophisticated example (left to readers in Exercise 3.14) is that the set of functions... 

In this section we define distance in complex scalar product spaces and apply the idea to a space of functions. We show how distance lets us make precise statements about approximating functions by other functions. 

Exercise 3.19 (Used in Exercises 5.21 and 5.22) Suppose V is a finite-dimensional complex scalar product space. Recall the dual vector space V from Exercise 2.14. Consider the function t V V defined by... 

We will see below that the function F s, x, y), F t, x, y) has important properties. Specifically, F(s, x, yfi F t, x, y)) is a polynomial in 5 and t its coefficients contain complete information about the complex scalar product on P , and F s, x, y), F t, x, y) is invariant under the action of 51/(2) on C. Finally, we will show that these properties imply that the complex scalar product is invariant under the representation R , and thus the representation is unitary. 

Proposition 7.2 is crucial to our proof in Section 7.2 that the spherical harmonics span the complex scalar product space L (S ) of square-integrable functions on the two-sphere. 

On the unit sphere wq have x + y + z = 1. Because the complex scalar product in L (S ) depends only on the values of the functions on the sphere itself, we have... 

Proposition 10.10 Suppose n is a natural number and ( , ] is the standard complex scalar product on C . Suppose S P(C") —> P(C") is a physical symmetry. Then there is a unitary operator T C" C" a function k, equal to either the identity or the conjugation fi me lion, such that... 

T complex scalar product space of rotation-invariant functions in... 

C[—1, 1 ] complex scalar product space of continuous complex-valued functions on [—1, 11,45... 

These zeros uk of QK(u) coincide with the eigenvalues of both the evolution matrix U and the corresponding Hessenberg matrix H from Eqs. (131) and (130), respectively. The zeros of Qk(u) are called eigenzeros. The structure of CM is determined by its scalar product for analytic functions of complex variable z or u. For any two regular functions/(m) and g(u) from CM, the scalar product in CM is defined by the generalized Stieltjes integral ... 

Strictly (vF(r) vF(r)) should be used for the probability density but F(r) is used in place of its complex conjugate because the inner product of both real and complex functions give the same scalar.] E is not the exact energy but, by the variational principle (Me Quarrie, 1983), E is an upper limit on the energy. 

Because Y is a potentially complex function including an imaginary part, Y designates the complex conjugate wave function. The compact and quite famous "bracket" notation on the right-hand side of Equation (2.7) bears the name of Dirac, and the "bra" (Y and "ket" Y) symbols stand for Y and Y and their integration. Mathematically, an integral such as / Y Ydr has been re-written as a scalar product (Y Y) within a complex vector space. 

The (, ) denotes the scalar product of the function before and after the comma, i.e., the integral over the product of both. Examples will be given later which show the way to find the adjoint to a given M. If M is real, its charac-teristic values axe in general the same as those of If M is complex, the characteristic values of are conjugate complex to the characteristic values of M. In this case, with which we shall not be concerned, it is better to define the scalar product as the integral over the product of the second factor and the conjugate complex of the first factor. 

In the first chapter, we saw that if we wanted to rotate the 2px function, we automatically also needed its companion 2py function. If this is extended to out-of-plane rotations, the 2/ function will also be needed. The set of the three p-orbitals forms a prime example of what is called a linear vector space. In general, this is a space that consists of components that can be combined linearly using real or complex numbers as coefficients. An n-dimensional linear vector space consists of a set of n vectors that are linearly independent. The components or basis vectors will be denoted as fi, with I ranging from 1 to n. At this point we shall introduce the Dirac notation [1] and rewrite these functions as / >, which characterizes them as so-called kef-functions. Whenever we have such a set of vectors, we can set up a complementary set of so-called fera-functions, denoted as /t I The scalar product of a bra and a ket yields a number. It is denoted as the bracket fk fi). In other words, when a bra collides with a ket on its right, it yields a scalar number. A bra-vector is completely defined when its scalar product with every ket-vector of the vector space is given. 

In quantum mechanics, the bra-function of / is simply the complex-conjugate function, fk, and the bracket or scalar product is defined as the integral of the product of the functions over space ... 

It is important to check that the new object in Equation 1.57 has the same mathematical properties as an ordinary wave function. In the scalar product (v /i /2). we need to replace the complex conjugate of the simple function rgi by the transposed complex conjugate of the vector representing pi ... 

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