Bound states bring the novelty as compared to the vacuum case that "E" is now negative (in the vacuum it was to be positive). This, along with third constraint, selects Hankel function of the first kind as the only converging solution at infinity (the singularity at the origin of these functions does not matter since we are now outside the sphere):
Second constraint on continuity of ψ at along with normalization allows the determination of constants "A" and "B". Continuity of the derivative (or logarithmic derivative for convenience) requires quantization of energy.
Sphere with infinite square potential
In case where the potential well is infinitely deep, so that we can take inside the sphere and outside, the problem becomes that of matching the wavefunction inside the sphere (the spherical Bessel functions) with identically zero wavefunction outside the sphere. Allowed energies are those for which the radial wavefunction vanishes at the boundary. Thus, we use the zeros of the spherical Bessel functions to find the energy spectrum and wavefunctions. Calling the "k"th zero of , we have:
So that one is reduced to the computations of these zeros and to their ordering them (as illustrated graphically below) (note that zeros of "j" are the same as those of "J").
3D isotropic harmonic oscillator
The potential of a is: In this article it is shown that an "N"-dimensional isotropic harmonic oscillator has the energies:,i.e., "n" is a non-negative integral number; ω is the (same) fundamental frequency of the "N" modes of the oscillator. In this case "N = 3", so that the radial Schrödinger equation becomes,
:Introducing:and recalling that , we will show that the radial Schrödinger equation has the normalized solution,
:where the function is a generalized Laguerre polynomial in "γr ²" of order "k" (i.e., the highest power of the polynomial is proportional to "γkr2k").
The normalization constant "Nnl" is,
The eigenfunction "Rn,l(r)" belongs to energy "En" and is to be multiplied by the spherical harmonic , where:
This is the same result as given in if we realize that .
First we transform the radial equation by a few successive substitutions to the generalized Laguerre differential equation, which has known solutions: the generalized Laguerre functions.Then we normalize the generalized Laguerre functions to unity. This normalization is withthe usual volume element "r"² d"r".
First we scale the radial coordinate:and then the equation becomes:with .
Consideration of the limiting behaviour of "v(y)" at the origin and at infinity suggests the following substitution for "v(y)",:This substitution transforms the differential equation to:where we divided through with , which can be done so long as "y" is not zero.
Transformation to Laguerre polynomials
If the substitution is used, , and the differential operators become::
The expression between the square brackets multiplying "f(y)" becomes the differential equation characterizing the generalized Laguerre equation (see also Kummer's equation)::with .Provided is a non-negative integral number, the solutions ofthis equations are generalized (associated) Laguerre polynomials:.From the conditions on "k" follows: (i) and (ii) "n" and "l" are either both odd or both even. This leads to the condition on "l" given above.
Recovery of the normalized radial wavefunction
Remembering that , we get the normalized radial solution:
The normalization condition for the radial wavefunction is:Substituting , gives and the equation becomes
By making use of the orthogonality properties of the generalized Laguerre polynomials, this equation simplifies to:
Hence, the normalization constant can be expressed as:
Other forms of the normalization constant can be derived by using properties of the gamma function, while noting that "n" and "l" are both of the same parity. This means that "n+l" is always even, so that the gamma function becomes:
where we used the definition of the double factorial. Hence, the normalization constant is also given by:
A hydrogenic (hydrogen-like) atom is a two-particle system consisting of a nucleus and an electron. The two particles interact through the potential given by Coulomb's law:
* ε0 is the permittivity of the vacuum,
* "Z" is the atomic number ("eZ" is the charge of the nucleus),
* "e" is the elementary charge (charge of the electron),
* "r" is the distance between the electron and the nucleus.
The mass "m"0, introduced above, is the reduced mass of the system. Because the electron mass is about 1836 smaller than the mass of the lightest nucleus (the proton), the value of "m"0 is very close to the mass of the electron "m"e for all hydrogenic atoms. In the remaining of the article we make the approximation "m"0 = "m"e. Since "m"e will appear explicitly in the formulas it will be easy to correct for this approximation if necessary.
In order to simplify the Schrödinger equation, we introduce the following constants that define the atomic unit of energy and length, respectively,:.
Substitute and into the radial Schrödinger equation given above. This gives an equation in which all natural constants are hidden,:Two classes of solutions of this equation exist: (i) "W" is negative, the corresponding eigenfunctions are square integrable and the values of "W" are quantized (discrete spectrum).(ii) "W" is non-negative. Every real non-negative value of "W" is physically allowed (continuous spectrum), the corresponding eigenfunctions are non-square integrable. In the remaining part of this article only class (i) solutions will be considered. The wavefunctions are known as bound states, in contrast to the class (ii) solutions that are known as "scattering states".
For negative "W" the quantity is real and positive. The scaling of "y", i.e., substitution of gives the Schrödinger equation::For the inverse powers of "x" are negligible and a solution for large "x" is . The other solution, , is physically non-acceptable. For the inverse square power dominates and a solution for small "x" is "x""l"+1. The other solution, "x"-"l", is physically non-acceptable.Hence, to obtain a full range solution we substitute:The equation for "f""l"("x") becomes,:Provided is a non-negative integer, say "k", this equation has polynomial solutions written as:which are generalized Laguerre polynomials of order "k". We will take the convention for generalized Laguerre polynomialsof Abramowitz and Stegun. [Abramowitz_Stegun_ref|22|775] Note that the Laguerre polynomials given in many quantum mechanical textbooks, for instance the book of Messiah
, are those of Abramowitz and Stegun multiplied by a factor ("2l+1+k")! The definition given in this Wikipedia article coincides with the one of Abramowitz and Stegun.
The energy becomes:The principal quantum number "n" satisfies , or .Since , the total radial wavefunction is:
with normalization constant:
which belongs to the energy :
In the computation of the normalization constant use was made of the integral [ H. Margenau and G. M. Murphy, "The Mathematics of Physics and Chemistry", Van Nostrand, 2nd edition (1956), p. 130. Note that convention of the Laguerre polynomial in this book differs from the present one. If we indicate the Laguerre in the definition of Margenau and Murphy with a bar on top, we have .] :