Associated Legendre equation

This equation can be transformed into a famous equation called the associated Legendre equation by the change of variables ... 

We do not discuss the associated Legendre equation, but give its solutions in Appendix F. The solutions are called associated Legendre functions and are derivatives of polynomials known as Legendre polynomials. 

For a solution of the associated Legendre equation to exist that obeys the relevant boundary conditions, it turns out that the constant K must be equal to /(/ -1-1) where / is a positive integer at least as large as the magnitude of m. There is one solution for each set of values of the two quantum numbers I and m ... 

Equation (F-46) is the same as the associated Legendre equation if K = lit + 1), where I is an integer that must be at least as large as m. The set of solutions is known as the associated Legendre functions, given for non-negative values of m by ... 

This is an associated Legendre equation whose solution may be written... 

This equation is known as the associated Legendre equation, and the function Uy denoted by u = Pn( )j is called the associated Legendre polynomial of degree n and order m. From equations 4 36 and 4 37, we see that... 

The Linear Differential Equation of the Second Order, 48. The Legendre Polynomials, 62. The Associated Legendre Polynomials, 52. The General Solution of the Associated Legendre Equation, 53. The Functions 0j.r ( ) and 57. Recursion Formulae for the Legendre Polynomials, 59. The Hermite Polynomials, 60. The Laguerre Polynomials, 63. 

The differential equation (A.3) is known as the associated Legendre equation, while the solution to the differential equation (A.4) is fsiip) = exp(jmi ). 

With the substitution x = cos 9, the associated Legendre equation transforms to... 

Equation (E.13) relates the associated Legendre polynomial Pfip) to the (/ -I- w)th-order derivative in equation (5.58)... 

The generating functions g " p, s) for the associated Legendre polynomials may be found from equation (E.l) by letting... 

Equation (E.17) is the associated Legendre differential equation. Orthogonality... 

Formulae of this type are especially useful when looking for solutions of more complicated differential equations related to the simpler classical ones. Two important examples are the associated Legendre and associated Laguerre (18) equations, respectively... 

For m an integer the P/"(x) are polynomials. The polynomials P,°(x) are identical with the Legendre polynomials. The first few associated Legendre polynomials for x = cos 9 (a common form of Legendre s equation) are ... 

To normalize a function such as P m(cos0) it is necessary to equate the integral / [P,m( cos 9)]2dx = 1. Starting from the Rodriques formula and integrating by parts it can be shown that the normalized associated Legendre functions are ... 

Use of Associated Legendre Functions in Wave Mechanics. To illustrate the use of associated Legendre functions in wave mechanics, we shall consider one of the simplest problems in that subject — that of solving Sehrodinger s equation... 

Theoretical chemists learn about a number of special functions, the Hermite functions in connection with the quantisation of the harmonic oscillator, Legendre and associated Legendre functions in connection with multipole expansions, Bessel functions in connection with Coulomb Greens functions, the Coulomb wave functions and a few others. All these have in common that they are the solutions of second order linear equations with a parameter. It is usually the case that solutions of boundary value problems for these equations only exist for countable sets of values of the parameter. This is how quantisation crops up in the Schrddinger picture. Quantum chemists are very comfortable with this state of affairs, but rarely venture outside the linear world where everything seems to be ordered. 

Using these equations, we can write one term in the expansion of the wavepacket in terms of the associated Legendre polynomials [Eq. (4.60)] as... 

In these coordinates Laplace s equation separates such functions of the form P ( )P. (7l)(g )m(J satisfy it where are the associated Legendre functions. Thus for this interior problem (i.e. inside the spheroid, not outside it) general solution of the linear problem is of the form... 

The functions Qim(9) and consequently the spherical harmonics Yim(6, associated Legendre polynomials, whose definition and properties are presented in Appendix E. To show this relationship, we make the substitution of equation (5.42) for cos 6 in equation (5.51) and obtain... 

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