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).
Contents
- Structure of the eigenfunctions
- Derivation of the radial equation
- Relationship with 1 D Schrdinger equation
- Solutions for potentials of interest
- Vacuum case
- Sphere with square potential
- Sphere with infinite square potential
- 3D isotropic harmonic oscillator
- Derivation
- Hydrogen like atoms
- References
In the general case, the dynamics of a particle in a spherically symmetric potential are governed by a Hamiltonian of the following form:
where
Structure of the eigenfunctions
The eigenstates of the system have the form
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
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,
Relationship with 1-D Schrödinger equation
Note that the first term in the kinetic energy can be rewritten
If subsequently the substitution
the radial equation becomes
which is precisely a Schrödinger equation for the function u(r) with an effective potential given by
where the radial coordinate r ranges from 0 to
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.
-
V ( r ) = V 0 r < r 0 - 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.
Vacuum case
Let us now consider V(r) = 0 (if
the equation becomes a Bessel equation for J defined by
which regular solutions for positive energies are given by so-called Bessel functions of the first kind'
The solutions of Schrödinger equation in polar coordinates for a particle of mass
where
These solutions represent states of definite angular momentum, rather than of definite (linear) momentum, which are provided by plane waves
Sphere with "square" potential
Let us now consider the 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
Sphere with infinite "square" potential
In case where the potential well is infinitely deep, so that we can take
So that one is reduced to the computations of these zeros
In the special case
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,
Introducing
and recalling that
where the function
The normalization constant Nnl is,
The eigenfunction Rn,l(r) belongs to energy En and is to be multiplied by the spherical harmonic
This is the same result as given in the Harmonic Oscillator article, with the minor notational difference of
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 r2 dr.
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
Transformation to Laguerre polynomials
If the substitution
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
From the conditions on k follows: (i)
Recovery of the normalized radial wavefunction
Remembering that
The normalization condition for the radial wavefunction is
Substituting
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
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
The mass m0, 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 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
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
For
The equation for fl(x) becomes,
Provided
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
with normalization constant
which belongs to the energy
In the computation of the normalization constant use was made of the integral