In quantum mechanics, a particle in a spherically symmetric potential is a quantum system with a potential that depends only on the distance between the particle and a defined center point. One example of a spherically symmetric potential is the electron within a hydrogen atom. The electron's potential only depends on its distance from the proton in the atom's nucleus. This potential can be derived from Coulomb's law.
In the general case, the dynamics of a particle in a spherically symmetric potential are governed by a Hamiltonian of the following form:
To describe these spherically symmetric systems, it is natural to use spherical coordinates, , and . When this is done, the time-independent Schrödinger equation for the system is separable. This means solutions to the angular dimensions of the equation can be found independently of the radial dimension. This leaves an ordinary differential equation in terms only of the radius, , which determines the eigenstates for the particular potential, .
Structure of the eigenfunctions
The eigenstates of the system have 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:
Substituting this into the Schrödinger equation we get a one-dimensional eigenvalue equation,
If , then near the origin, .
Solutions for potentials of interest
Five special cases arise, of special importance:
- , or solving the vacuum in the basis of spherical harmonics, which serves as the basis for other cases.
- (finite) for and zero elsewhere.
- for and infinite elsewhere, the spherical equivalent of the square well, useful to describe bound states in a nucleus or quantum dot.
- for the three-dimensional isotropic harmonic oscillator.
- 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. The following derivations rely heavily on Bessel functions and Laguerre polynomials.
Vacuum case states
Let us now consider V(r) = 0 (if , replace everywhere E with ). Introducing the dimensionless variable
The solutions of the 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 :
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.[1]
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:
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 the third constraint, selects the Hankel functionof
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):
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:
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
Introducing
The normalization constant Nnℓ is,
The eigenfunction Rn,ℓ(r) belongs to energy En and is to be multiplied by the spherical harmonic , where
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
Consideration of the limiting behavior of v(y) at the origin and at infinity suggests the following substitution for v(y),
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):
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
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:
- 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,
For class (i) solutions with 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
The energy becomes
The principal quantum number n satisfies , or . Since , the total radial wavefunction is
In the computation of the normalization constant use was made of the integral[3]
References
- 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 .
https://en.wikipedia.org/wiki/Particle_in_a_spherically_symmetric_potential
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