Particle in a spherically symmetric potential

From Academic Kids

In quantum mechanics, the particle in a spherically symmetric potential describes the dynamics of a particle in a central force field, i.e. with potential depending only on the distance of the particle to the center of force (radial dependency), having no angular dependency. In its quantum mechanical formulation, it amounts to solving the Schrödinger equation with potentials V(r) which depend only on r, the modulus of r.

Three special cases arise, of special importance:

  1. V(r)=0, or solving the vacuum in the basis of spherical harmonics, which serves as the basis for other cases.
  2. <math>V(r)=V_0<math> for <math>rsquare well, useful to describe scattering and bound states in a nucleus or quantum dot.
  3. V(r)~1/r to describe bound states of atoms, especially hydrogen.

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.


General considerations on the Schrödinger equation in a spherically symmetric potential

The time independent solution of 3D Schrödinger equation with hamiltonian <math>p^2/2m_0+V(r)<math> where <math>m_0<math> is the particle's mass, can be separated in the variables r, θ and φ so that the wavefunction <math>\psi<math> reads:


<math>Y_l^m<math> are the usual Spherical harmonics, while <math>R<math> needs be solved with the so-called radial equation:

<math>\left[-{\hbar^2\over 2m_0r^2}{d\over dr}(r^2{d\over dr})+\hbar^2{l(l+1)\over2m_0r^2}+V(r)\right]R(r)=ER(r)<math>

It has the shape of the 1D Schrödinger equation for the variable <math>u(r)\equiv rR(r)<math>, with a centrifugal term <math>\hbar^2l(l+1)/2m_0r^2<math> added to V, but r ranges from 0 to <math>\infty<math> rather than over R.

For more information about how one derive Spherical harmonics from spherical symmetry, see Angular momentum, since the spherical harmonics are the eigenstates of the operator L2.

Vacuum case

Let us now consider V(r)=0 (if <math>V_0<math>, replace everywhere E with <math>E-V_0<math>). Introducing the dimensionless variable

<math>\rho\equiv kr, \qquad k\equiv \sqrt{2m_0E\over\hbar^2}r<math>

the equation becomes a Bessel equation for J defined by <math>J(\rho)\equiv\sqrt\rho R(r)<math> (whence the notational choice of J):

<math>\rho^2{d^2J\over d\rho^2}+\rho{dJ\over d\rho}+(\rho^2-(l+1/2)^2)J=0<math>

which regular solutions for positive energies are given by so-called Bessel functions of the first kind <math>J_{l+1/2}(\rho)<math> so that the solutions written for R are the so-called Spherical Bessel function <math>R(r) = j_l(kr) \equiv \sqrt{\pi/(2kr)} J_{l+1/2}(kr)<math>.

The solutions of Schrödinger equation in polar coordinates for a particle of mass <math>m_0<math> in vacuum are labelled by three quantum numbers: discrete indices l and m, and k varying continuously in <math>[0,\infty]<math>:
where <math>k\equiv\sqrt{2m_0E}/\hbar<math>, <math>j_l<math> are the spherical Bessel function and <math>Y_l^m<math> are the spherical harmonics.

These solutions represent states of definite angular momentum, rather than of definite (linear) momentum, which are provided by plane waves <math>\exp(i \mathbf{k}\cdot\mathbf{r})<math>.

Spherical square well

Let us now consider the potential <math>V(r)=V_0<math> for <math>r

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 normalisation 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:

  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 N and Hankel H functions are nonsingular 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:

<math>R(r)=Aj_l\left(\sqrt{2m_0(E-V_0)\over\hbar^2}r\right),\qquad r

with A a constant to be determined later. Note that for bound states, <math>V_0

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):

<math>R(r)=Bh^{(1)}_l\left(i\sqrt{-2m_0E\over\hbar^2}r\right),\qquad r>r_0<math>

Second constraint on continuity of ψ at <math>r=r_0<math> along with normalization allows the determination of constants A and B. Continuity of the derivative (or logarithmic derivative for convenience) requires quantization of energy.

Infinite spherical square well

In case where the potential is infinitely deep, so that we can take <math>V_0=0<math> inside the sphere and <math>\infty<math> outside, the problem becomes that of matching the spherical Bessel function with identically zero wavefunction outside the sphere, thus providing a straightforward connection with the energy spectrum and the zero of spherical Bessel functions since—calling <math>u_{l,k}<math> the kth zero of <math>j_l<math>—allowed energies are those for which the radial wavefunction vanishes at the boundary. They go as the square of ordered Bessel J zeros:


So that one is reduced to the computations of these zeros <math>u_{l,k}<math> and to their ordering them (as illustrated graphically below) (note that zeros of j are the same as those of J).

Missing image
Zeros of Bessel j functions

Zeros of the first spherical Bessel equations

Calling s, p, d, f, g, h, etc., states with l=0, 1, 2, 3, 4, 5, etc., respectively, we obtain the following spectrum:

Missing image
Quantum states

Spectrum of the infinitely deep spherical square well

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