Three-Dimensional Quantum Mechanics: Bound States

Abstract

Figure 11.2 has already shown the radial wave function of bound states in a three-dimensional square-well potential. Now in Figure 13.1 we plot the radial wave function R _nℓ together with its square R ² _nℓ and the function r²R ² _nℓ for the low angular-momentum quantum numbers ℓ = 0, 1, 2. The reason for showing r²R ² _nℓ is that r²R ² _nℓ (r) dr represents the probability that a particle is within a spherical shell of radius r and thickness dr. Also shown in Figure 13.1 is the energy spectrum of the eigenvalues. We observe that the number of bound states is finite. The spacing between the different eigenvalues increases with increasing energy. For a given ℓ value the lowest-lying state has no node in r, the next one has one node, and so on. We can enumerate the eigenvalues E^nl, n = 1, 2,…, for a given ℓ by the number n − 1 of nodes they possess. In Figure 13.1 the square-well potential V(r) is drawn as a long-dash line, the effective potential as a short-dash line.