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An important problem in quantum mechanics is that of a particle in a spherically symmetric potential, i.e., a potential that depends only on the distance between the particle and a defined center point. In particular, if the particle in question is an electron and the potential is derived from Coulomb's law, then the problem can be used to describe a hydrogen-like (one-electron) atom (or ion).
In the general case, the dynamics of a particle in a spherically symmetric potential are governed by a Hamiltonian of the following form:
where is the mass of the particle, is the momentum operator, and the potential depends only on , the modulus of the radius vector r. The quantum mechanical wavefunctions and energies (eigenvalues) are found by solving the Schrödinger equation with this Hamiltonian. Due to the spherical symmetry of the system, it is natural to use spherical coordinates , and . When this is done, the time-independent Schrödinger equation for the system is separable, allowing the angular problems to be dealt with easily, and leaving an ordinary differential equation in to determine the energies for the particular potential under discussion.
Structure of the eigenfunctions
in which the spherical polar angles θ and φ represent the colatitude and azimuthal angle, respectively. The last two factors of ψ are often grouped together as spherical harmonics, so that the eigenfunctions take the form
The differential equation which characterizes the function is called the radial equation.
Derivation of the radial equation
The kinetic energy operator in spherical polar coordinates is
The spherical harmonics satisfy
Substituting this into the Schrödinger equation we get a one-dimensional eigenvalue equation,
This equation can be reduced to an equivalent 1-D Schrödinger equation by substituting , where satisfies
which is precisely the one-dimensional Schrödinger equation with an effective potential given by
where the radial coordinate r ranges from 0 to . The correction to the potential V(r) is called the centrifugal barrier term.
If , then near the origin, .
Solutions for potentials of interest
Five special cases arise, of special importance:
- V(r) = 0, or solving the vacuum in the basis of spherical harmonics, which serves as the basis for other cases.
- (finite) for and infinite elsewhere, or a particle in the spherical equivalent of the square well, useful to describe bound states in a nucleus or quantum dot.
- As the previous case, but with an infinitely high jump in the potential on the surface of the sphere.
- V(r) ~ r2 for the three-dimensional isotropic harmonic oscillator.
- V(r) ~ 1/r to describe bound states of hydrogen-like atoms.
We outline the solutions in these cases, which should be compared to their counterparts in cartesian coordinates, cf. particle in a box. This article relies heavily on Bessel functions and Laguerre polynomials.
Let us now consider V(r) = 0 (if , replace everywhere E with ). Introducing the dimensionless variable
the equation becomes a Bessel equation for J defined by (whence the notational choice of J):
The solutions of Schrödinger equation in polar coordinates for a particle of mass in vacuum are labelled by three quantum numbers: discrete indices l and m, and k varying continuously in :
where , are the spherical Bessel functions and are the spherical harmonics.
These solutions represent states of definite angular momentum, rather than of definite (linear) momentum, which are provided by plane waves .
Sphere with finite "square" potential
Let us now consider the potential for and elsewhere. That is, inside a sphere of radius the potential is equal to V0 and it is zero outside the sphere. A potential with such a finite discontinuity is called a square potential.
We first consider bound states, i.e., states which display the particle mostly inside the box (confined states). Those have an energy E less than the potential outside the sphere, i.e., they have negative energy, and we shall see that there are a discrete number of such states, which we shall compare to positive energy with a continuous spectrum, describing scattering on the sphere (of unbound states). Also worth noticing is that unlike Coulomb potential, featuring an infinite number of discrete bound states, the spherical square well has only a finite (if any) number because of its finite range (if it has finite depth).
The resolution essentially follows that of the vacuum with normalization of the total wavefunction added, solving two Schrödinger equations — inside and outside the sphere — of the previous kind, i.e., with constant potential. Also the following constraints hold:
- The wavefunction must be regular at the origin.
- The wavefunction and its derivative must be continuous at the potential discontinuity.
- The wavefunction must converge at infinity.
The first constraint comes from the fact that Neumann N and Hankel H functions are singular at the origin. The physical argument that ψ must be defined everywhere selected Bessel function of the first kind J over the other possibilities in the vacuum case. For the same reason, the solution will be of this kind inside the sphere:
with A a constant to be determined later. Note that for bound states, .
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 kth zero of , we have:
So that one is reduced to the computations of these zeros , typically by using a table or calculator, as these zeros are not solvable for the general case.
In the special case (spherical symmetric orbitals), the spherical Bessel function is , which zeros can be easily given as . Their energy eigenvalues are thus:
3D isotropic harmonic oscillator
The potential of a 3D isotropic harmonic oscillator 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,
and recalling that , we will show that the radial Schrödinger equation has the normalized solution,
where the function is a generalized Laguerre polynomial in γr2 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 the Harmonic Oscillator article, with the minor notational difference of .
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 with the usual volume element r2 dr.
First we scale the radial coordinate
and then the equation becomes
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
Provided is a non-negative integral number, the solutions of this 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 m0, introduced above, is the reduced mass of the system. Because the electron mass is about 1836 times smaller than the mass of the lightest nucleus (the proton), the value of m0 is very close to the mass of the electron me for all hydrogenic atoms. In the remaining of the article we make the approximation m0 = me. Since me 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 xl+1. The other solution, x−l, is physically non-acceptable. Hence, to obtain a full range solution we substitute
The equation for fl(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 polynomials of Abramowitz and Stegun. 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 absorbs extra terms from
which belongs to the energy
In the computation of the normalization constant use was made of the integral 
- A. Messiah, Quantum Mechanics, vol. I, p. 78, North Holland Publishing Company, Amsterdam (1967). Translation from the French by G.M. Temmer
- Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 22". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 775. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
- 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 .
- Gerald Teschl (2009). Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators. American Mathematical Society. ISBN 9780821846605. CS1 maint: discouraged parameter (link)