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17 Laguerre Polynomials
all polynomials with degree smaller or equal n will be evaluated. For n noninteger or ... = laguerrex(alpha, n, x, ‘1F1’) the evaluation will
be based on the confluent hypergeometric function and only the polynomial with
degree n will be evaluated. The output variables are the class object “obj” and the
matrix of the polynomial values “Ln”. “obj” comes with the properties value, the
polynomial values. For n positive integer and obj = laguerrex(alpha, n,
x) obj.value is a matrix where each row belongs to a fixed degree running
through all polynomial values. The first row corresponds to degree 0, and the last
to n. In case the evaluation is based on the confluent hypergeometric function
obj.value will have the same structure as the input “x”. The property “z” is
the position at which the polynomial was evaluated, “degree” is the polynomial
degree and “alpha” its parameter, “info” are some general information. The output
argument “Ln” is identical to the property “value”. laguerrex comes in addition
with the method “plot”, with syntax obj = plot(obj, nl) and “obj” an
object of the class, “nl” an optional integer vector of the polynomial degrees which
should be plotted. Default is to plot all polynomials in quest. Example:
>> x = linspace(0,10); obj = laguerrex(1,3,x).plot; shg
The SPECFUNPHYS function laguerrezero with syntax lnaz = laguerrezero(alpha, n) returns the roots “lnaz” of the generalized Laguerre
polynomial L α
n with the input arguments “alpha” and “n”; “n” has to be an integer
scalar 0, 1, 2, · · · .
The radial wave function R n,l (r), Eq. (17.3), can be computed with the SPECFUNPHYS function Hrad: [Rnl, x, rho, n, l] = Hrad(n,l,rho,
Z). “n” is the principal quantum number, thus an integer starting with 1,
“l” the angular momentum, “rho” the radial coordinate scaled with the Bohr
radius a 0 and Z the atomic number (1 for the Hydrogen atom). “rho” and
“Z” are optional with default values 0 · · · 7 and Z = 1. The output arguments
are “Rnl” the values of the radial wave function, “n” and “l” the principal
quantum number and angular momentum, “x”=
2Zrho
n , and “rho” equals the
input argument “rho”. (The normalized hydrogen wave function is written as
ψ nlm (r, ϑ, φ) = R n,l (r)Y lm (ϑ, φ).)
References
1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1972)
2. Gradstein, I.S., Rhysik, I.M.: Tafeln · Tables II. Verlag Harry Deutsch Thun, Frankfurt A. M.
(1981)
17 Laguerre Polynomials
all polynomials with degree smaller or equal n will be evaluated. For n noninteger or ... = laguerrex(alpha, n, x, ‘1F1’) the evaluation will
be based on the confluent hypergeometric function and only the polynomial with
degree n will be evaluated. The output variables are the class object “obj” and the
matrix of the polynomial values “Ln”. “obj” comes with the properties value, the
polynomial values. For n positive integer and obj = laguerrex(alpha, n,
x) obj.value is a matrix where each row belongs to a fixed degree running
through all polynomial values. The first row corresponds to degree 0, and the last
to n. In case the evaluation is based on the confluent hypergeometric function
obj.value will have the same structure as the input “x”. The property “z” is
the position at which the polynomial was evaluated, “degree” is the polynomial
degree and “alpha” its parameter, “info” are some general information. The output
argument “Ln” is identical to the property “value”. laguerrex comes in addition
with the method “plot”, with syntax obj = plot(obj, nl) and “obj” an
object of the class, “nl” an optional integer vector of the polynomial degrees which
should be plotted. Default is to plot all polynomials in quest. Example:
>> x = linspace(0,10); obj = laguerrex(1,3,x).plot; shg
The SPECFUNPHYS function laguerrezero with syntax lnaz = laguerrezero(alpha, n) returns the roots “lnaz” of the generalized Laguerre
polynomial L α
n with the input arguments “alpha” and “n”; “n” has to be an integer
scalar 0, 1, 2, · · · .
The radial wave function R n,l (r), Eq. (17.3), can be computed with the SPECFUNPHYS function Hrad: [Rnl, x, rho, n, l] = Hrad(n,l,rho,
Z). “n” is the principal quantum number, thus an integer starting with 1,
“l” the angular momentum, “rho” the radial coordinate scaled with the Bohr
radius a 0 and Z the atomic number (1 for the Hydrogen atom). “rho” and
“Z” are optional with default values 0 · · · 7 and Z = 1. The output arguments
are “Rnl” the values of the radial wave function, “n” and “l” the principal
quantum number and angular momentum, “x”=
2Zrho
n , and “rho” equals the
input argument “rho”. (The normalized hydrogen wave function is written as
ψ nlm (r, ϑ, φ) = R n,l (r)Y lm (ϑ, φ).)
References
1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1972)
2. Gradstein, I.S., Rhysik, I.M.: Tafeln · Tables II. Verlag Harry Deutsch Thun, Frankfurt A. M.
(1981)