Fitts D.D. - Principles of Quantum Mechanics[c] As Applied to Chemistry and Chemical Physics (1999)(en)

Appendix C

Dirac delta function

The Dirac delta function ä(x) is de®ned by the conditions

As a consequence of this de®nition, if f (x) is an arbitrary function which is wellde®ned at x 0, then integration of f (x) with the delta function selects out the value

of f (x) at the origin

…

f (x)ä(x) dx f (0) (C:3)

The integration is taken over the range of x for which f (x) is de®ned, provided that the

range includes the origin. It also follows that

…

f (x)ä(x ÿ x0) dx f (x0) (C:4)

since ä(x ÿ x0) 0 except when x x0. The range of integration in equation (C.4) must include the point x x0.

The following properties of the Dirac delta function can be demonstrated by multiplying both sides of each expression by f (x) and observing that, on integration, each side gives the same result

As de®ned above, the delta function by itself lacks mathematical rigor and has no meaning. Only when it appears in an integral does it have an operational meaning. That two integrals are equal does not imply that the integrands are equal. However, for the sake of convenience we often write mathematical expressions involving ä(x) such

292

as those in equations (C.5a±e). Thus, the expressions (C.5a±e) and similar ones involving ä(x) are not to be taken as mathematical identities, but rather as operational identities. One side can replace the other within an integral that includes the origin, for ä(0), or the point x0 for ä(x ÿ x0).

The concept of the Dirac delta function can be made more mathematically rigorous by regarding ä(x) as the limit of a function which becomes successively more peaked at the origin when a parameter approaches zero. One such function is

Other expressions which can be used to de®ne ä(x) include

The delta function is the derivative of the Heaviside unit step function H(x), de®ned

and d H=dx, which equals ä(x), satis®es equation (C.1). The differential d H(x, å) equals dx=å for x between ÿå=2 and å=2 and is zero otherwise. If we take the integral

of ä(x) from ÿ1 to 1, we have

…1 …1 …1 …å=2

ä(x) dx d H lim dH(x, å) 1

Appendix D

Hermite polynomials

The Hermite polynomials Hn(î) are de®ned by means of an in®nite series expansion of the generating function g(î, s),

Applying this property with x s and y ÿî to the nth-order partial derivative in

and g(î, s) takes the form

296

For convenience, we have abbreviated the integral with the symbol I. To evaluate the left integral, we substitute the analytical forms for the generating functions from equation (D.1) to give

where equation (A.5) has been used. We next expand e2st in the power series (A.1) to obtain

Completeness

If we de®ne the set of functions ön(î) as

then equation (D.14) shows that the members of this set are orthonormal with weighting factor unity. We can also demonstrate1 that this set is complete.

We begin with the integral formula (A.8) which, with suitable de®nitions for the parameters, may be written as

1 See D. Park (1992) Introduction to the Quantum Theory, 3rd edition (McGraw-Hill, New York), p. 565.

Appendix E

Legendre and associated Legendre polynomials

Legendre polynomials

The Legendre polynomials Pl(ì) may be de®ned as the coef®cients of sl in an in®nite series expansion of a generating function g(ì, s)

so that

g(ì, s) X1 Xn (ÿ1)á(2n)!ìnÿá s n á

n 0 á 0 2n á n!á!(n ÿ á)!

We next collect all the coef®cients of sl for some arbitrary l and replace the summation over n with a summation over l. Since n á l, when n l, we have á 0; when n l ÿ 1, we have á 1; and so on until n l ÿ M, á M, where M < l ÿ M or

M< l=2. The summation over á terminates at á M, with M l=2 for l even and

M(l ÿ 1)=2 for l odd, because á cannot be greater than n. The result is

301