In quantum mechanics, a particle in a spherically symmetric potential is a system where a particle's potential energy depends only on its distance from a central point, not on the direction. This model is fundamental to physics because it can be used to describe a wide range of real-world phenomena, from the behavior of a single electron in a hydrogen atom to the approximate structure of atomic nuclei.

The particle's behavior is described by the Time-independent Schrödinger equation. Because of the spherical symmetry, the problem can be greatly simplified by using spherical coordinates) and a mathematical technique called separation of variables. This allows the solution (the wavefunction, and an angular part. The angular solutions are universal for all spherically symmetric potentials and are known as spherical harmonics and determines the allowed energy levels of the system.

In the general time-independent case, the dynamics of a particle in a spherically symmetric potential are governed by a Hamiltonian is the momentum operator depends only on the radial distance from the origin. This mathematical setup leads to an ordinary differential equation for the radial part of the wavefunction, which can be solved for important potentials like the Coulomb potential (for atoms) and the spherical square well (for nuclei).

Structure of the eigenfunctions

If solved by separation of variables, the eigenstates of the systemin which the spherical angles represent the polar and azimuthal are often grouped together as spherical harmonics, so that the eigenfunctions is called the radial equation.

Derivation of the radial equation

The kinetic energy operator in spherical polar coordinates is the angular momentum operatorThe spherical harmonicsSubstituting this into the Schrödinger equationwhich is precisely the one-dimensional Schrödinger equation with an effective potential. The correction to the potential V(r) is called the centrifugal barrier term.

.

Spherically symmetric Hamiltonians

Since angular momentum operators are generators of rotation, applying the Baker-Campbell-Hausdorff Lemma

, or that every angular momentum component commutes with the Hamiltonian.

is used to label different wavefunctions.

values. Thus the wavefunction is expressed as:

Solutions for potentials of interest

There are five cases of special importance:

1. , or solving the vacuum in the basis of spherical harmonics, which serves as the basis for other cases.

2. and zero elsewhere.

3. and infinite elsewhere, the spherical equivalent of the square well, useful to describe bound states in a nucleus or quantum dot.

4. for the three-dimensional isotropic harmonic oscillator.

5. to describe bound states of hydrogen-like atoms.

The solutions are outlined in these cases, which should be compared to their counterparts in cartesian coordinates, cf. particle in a box.

Vacuum case states

where regular solutions for positive energies are given by so-called Bessel functions of the first kind are the so-called spherical Bessel function

The solutions of the Schrödinger equation in polar coordinates in vacuum are thus labelled by three quantum numbers: discrete indices ℓ and m, and k.

Sphere with finite "square" potential

and it is zero outside the sphere. A potential with such a finite discontinuity is called a square potential.

less than the potential outside the sphere, i.e., they have negative energy. 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.

The resolution essentially follows that of the vacuum case above 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. The following constraints must hold for a normalizable, physical wavefunction:

1. The wavefunction must be regular at the origin.

2. The wavefunction and its derivative must be continuous at the potential discontinuity.

3. The wavefunction must converge at infinity.

The first constraint comes from the fact that Neumann and Hankel functions must be defined everywhere selected Bessel function of the first kind. This, along with the third constraint, selects the Hankel function. Continuity of the derivative (or logarithmic derivative for convenience) requires quantization of energy.

Sphere with infinite "square" potential

outside, the problem becomes that of matching the wavefunction inside the sphere (the spherical Bessel functions the kth, typically by using a table or calculator, as these zeros are not solvable for the general case.

===3D isotropic harmonic oscillator=== The potential of a 3D isotropic harmonic oscillator An N-dimensional isotropic harmonic oscillator is the (same) fundamental frequency

is a generalized Laguerre polynomial.

This is the same result as the quantum harmonic oscillator.

Derivation

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 .

First we scale.

Consideration of the limiting behavior of at the origin and at infinity suggests the following substitution for , which can be done so long as y is not zero.

Transformation to Laguerre polynomials

becomes the differential equation characterizing the generalized Laguerre equation (see also Kummer's equation.

is a non-negative integral number, the solutions of this equations are generalized (associated) Laguerre polynomials

given above.

Recovery of the normalized radial wavefunction

The normalization condition for the radial wave function

By making use of the orthogonality properties of the generalized Laguerre polynomials

Hence, the normalization constant

Other forms of the normalization constant can be derived by using properties of the gamma function where we used the definition of the double factorial

Hydrogen-like atoms

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 where

• 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.

In order to simplify the Schrödinger equation, we introduce the following constants that define the atomic unit of energy and length:

is the reduced mass Two classes of solutions of this equation exist:

For class (i) solutions with negative W

the inverse powers of x is xℓ+1

is a non-negative integer which are generalized Laguerre polynomials

The principal quantum number via

References