Chap4_JQPMaster

Chapter 4

(Fairly) easy solution of the Schrodinger equationfor a three-dimensional rotationally invariant system

Version 8.35: August 17, 2010

Contents4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2654.2. The radial equation and mathematical properties of its solutions . . . . . . . 267

4.2.1. Bound and continuum states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2674.2.2. Radial equations and the radial quantum number . . . . . . . . . . . . . . . . . . . 2684.2.3. Energy ordering of bound states and binding energies . . . . . . . . . . . . . . . . 2694.2.4. Energies and the Virial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2694.2.5. Boundary conditions on the radial functions . . . . . . . . . . . . . . . . . . . . . . 2704.2.6. The radial quantum number and nodes in the radial function . . . . . . . . . . . . 2714.2.7. Orthogonality and nodes in the radial functions . . . . . . . . . . . . . . . . . . . . 2714.2.8. The essential degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

4.3. The radial probability density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2744.3.1. Bookkeeping, spectroscopic notation, and the principal quantum number . . . . . 2744.3.2. The position probability density according to Born . . . . . . . . . . . . . . . . . . 2764.3.3. Derivation and interpretation of the radial probability density . . . . . . . . . . . . 2764.3.4. Integrated probability densities: (finite) shell and sphere probabilities . . . . . . . 2794.3.5. Radial expectation values and uncertainties . . . . . . . . . . . . . . . . . . . . . . 280

4.4. The case of the disappearing derivative: The reduced radial equation . . . . . 2814.4.1. Introducing the reduced radial equation . . . . . . . . . . . . . . . . . . . . . . . . 2814.4.2. Boundary conditions on the reduced radial function . . . . . . . . . . . . . . . . . 2824.4.3. Subduing the reduced radial equation through pattern matching . . . . . . . . . . 283

4.4.3.1. The structure of the reduced radial equation . . . . . . . . . . . . . . . . 2834.4.3.2. The effective potential energy . . . . . . . . . . . . . . . . . . . . . . . . . 284

4.4.4. Orthonormality of the reduced radial function . . . . . . . . . . . . . . . . . . . . . 2844.4.5. Summary: Why bother with the reduced radial function? . . . . . . . . . . . . . . 285

4.5. The roles of the physical and effective potential energies . . . . . . . . . . . . 2864.5.1. The competition between the barrier term and the physical potential energy . . . . 2864.5.2. The effective potential energy and the number of bound states . . . . . . . . . . . 287

4.6. Generic properties of the reduced radial function . . . . . . . . . . . . . . . . . 2884.6.1. Limiting behavior of physically reasonable potential energies . . . . . . . . . . . . 2894.6.2. Cleaning up the reduced radial equation . . . . . . . . . . . . . . . . . . . . . . . . 2904.6.3. Generic behavior of reduced radial functions in the asymptotic limit . . . . . . . . 291

4.6.3.1. Derivation of the asymptotic reduced-radial equation . . . . . . . . . . . 2914.6.3.2. Solution of the asymptotic reduced-radial equation . . . . . . . . . . . . . 291

4.6.4. Generic behavior of reduced radial functions in the near-zero limit . . . . . . . . . 293

264 4. The central-force problem in three dimensions

4.6.5. Generic behavior between the near-zero and asymptotic limits . . . . . . . . . . . 2954.7. Qualitative solution of the radial equation . . . . . . . . . . . . . . . . . . . . . 297

4.7.1. Classically allowed and classically forbidden regions . . . . . . . . . . . . . . . . . 2984.7.2. Classical turning points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

4.8. Final thoughts: tips for potential modelers . . . . . . . . . . . . . . . . . . . . . 3024.9. Users guide to Chap. 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3034.10. Selected readings & references for Chap. 4 . . . . . . . . . . . . . . . . . . . . . 309

4.10.1. Numerical solution of the Schrodinger equation . . . . . . . . . . . . . . . . . . . . 3094.10.2. Programs for solving the radial equation . . . . . . . . . . . . . . . . . . . . . . . . 310

4.11. Exercises & problems for Chap. 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 310Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3294.A. Additional examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3304.B. Bound- and continuum-state radial functions . . . . . . . . . . . . . . . . . . . . 333

4.B.1. Bound versus continuum states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3334.B.2. Boundary conditions on bound- and continuum-state radial functions . . . . . . . 334

4.C. The number and energy-ordering of bound states . . . . . . . . . . . . . . . . . 3354.C.1. The number of bound states in a 3D attractive potential energy . . . . . . . . . . 3354.C.2. The order of bound-state energies of a rotationally invariant system . . . . . . . . 336

4.D. The ground state of the lithium atom: a tale of three models . . . . . . . . . . 3374.D.1. The principle of modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3384.D.2. Models of the ground state of lithium . . . . . . . . . . . . . . . . . . . . . . . . . 3384.D.3. Models for the valence electron in lithium . . . . . . . . . . . . . . . . . . . . . . . 3394.D.4. A radial function for the valence electron in Li . . . . . . . . . . . . . . . . . . . . 342

4.E. How to cope with an arbitrary wave function . . . . . . . . . . . . . . . . . . . 3454.E.1. The method of eigenfunction expansion . . . . . . . . . . . . . . . . . . . . . . . . 3454.E.2. Calculation of expectation values and related quantities . . . . . . . . . . . . . . . 346

Figures4.1. Bound and continuum state energies for a spherically symmetric potential energy. . . . . . 2674.2. Radial functions and the orthogonality integrand for s states of lithium. . . . . . . . . . . 2734.1. A spherical shell in 3D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2774.2. Radial probability densities for 1s and 2s electrons in atomic lithium. . . . . . . . . . . . 2784.3. Radial probability densities for 1s electrons in several atoms. . . . . . . . . . . . . . . . . 2794.4. Sphere probability for electrons in the ground state of atomic lithium.. . . . . . . . . . . . 2804.1. Radial and reduced radial functions for s states of atomic lithium. . . . . . . . . . . . . . 2834.1. Effects of the centrifugal barrier on the effective potential energy. . . . . . . . . . . . . . . 2874.1. Large r behavior of some typical physical potentials energies. . . . . . . . . . . . . . . . . 2904.2. Asymptotic behavior of the ` = 1 reduced radial equation for boron. . . . . . . . . . . . . 2914.3. Limiting behavior of bound-state reduced radial functions for scandium. . . . . . . . . . . 2934.4. Near-origin behavior of the reduced radial equation for boron. . . . . . . . . . . . . . . . . 2934.1. Comparison of a model Li potential energy to pure-Coulomb potential energies. . . . . . . 2984.2. Classically allowed and classically forbidden regions for a reduced radial function. . . . . . 3004.1.1.The Yukawa model of the potential energy of the deuteron. . . . . . . . . . . . . . . . . . 3114.2.1.Reduced radial functions for two bound states of a Yukawa potential energy. . . . . . . . . 3124.2.2.The reduced radial functions for a bound state of the Yukawa potential energy. . . . . . . 3134.1. The pure-Coulomb and Yukawa Potentials. . . . . . . . . . . . . . . . . . . . . . . . . . 3164.2.1.An effective potential energy and its reduced radial function. . . . . . . . . . . . . . . . . 3184.3.1.The reduced radial functions for Problem 4.3. . . . . . . . . . . . . . . . . . . . . . . . . . 3184.4.1.The reduced radial functions for Problem 4.4. . . . . . . . . . . . . . . . . . . . . . . . . . 3194.5.1.Three radial probability densities for Problem 4.5. . . . . . . . . . . . . . . . . . . . . . . 3194.6.1.Figure for Problem 4.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3204.7.1.The effective potential energy and two reduced radial functions for Problem 4.7. . . . . . 3204.9.1.Two effective potential energies for Problem 4.9. . . . . . . . . . . . . . . . . . . . . . . . 3214.12.1.A crude model of the nuclei of a diatomic molecule. . . . . . . . . . . . . . . . . . . . . . 3244.12.2.Absorption spectra for HCl. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3254.D.1.Spatial coordinates for the three electrons of atomic lithium. . . . . . . . . . . . . . . . . 3374.D.2. A model potential and screening function for the valence electron in an alkali atom. . . . 3404.D.3. The radial function and probability density for the 2s electron in Li. . . . . . . . . . . . 342

JQPMaster Version: 8.35 Printed: August 17, 2010

265

4.D.4. The radial probability density and shell probability for the 2s electron in Li. . . . . . . 3434.D.5. Integrands of radial integrals for the 2s electron in Li. . . . . . . . . . . . . . . . . . . . 344

Tables4.1. Radial functions labeled by the radial and principal quantum numbers. . . . . . . . . . . . 2754.2. Spectroscopic notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2754.1. Notation for solution of the central-force problem in 3D. . . . . . . . . . . . . . . . . . . . 3044.2. Symmetry and separation of variables for single-particle potentials. . . . . . . . . . . . . . 3044.3. Symmetries and angular momentum commutator relations. . . . . . . . . . . . . . . . . . 3044.4. The one- and three-dimensional time-independent Schrodinger equation. . . . . . . . . . . 3054.5. Behavior of the reduced radial function in classically allowed and forbidden regions. . . . 3064.6. The development of the stationary angular momentum eigenstates. . . . . . . . . . . . . . 3064.7. Properties of the reduced radial function for any spherically symmetric potential. . . . . . 3074.8. The structure of the reduced radial function. . . . . . . . . . . . . . . . . . . . . . . . . . 3074.9. Key properties of the reduced radial functions for bound and continuum SAMEs. . . . . 3074.10. Qualitative behavior of the radial function. . . . . . . . . . . . . . . . . . . . . . . . . . 3084.11. Properties in terms of the radial and reduced radial functions. . . . . . . . . . . . . . . 3084.2.1. The mean radius versus the radius of maximum radial probability density. . . . . . . . 3124.16.1. Data for the Yukawa potential energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3284.D.1. Radial mean values for the 2s state of atomic lithium. . . . . . . . . . . . . . . . . . . . 344

4.1. IntroductionYou know what happens next. The parable is perfectly transparent.But I have to tell you. I have to believe that my meaning resides not in the gross motion ofthe tale, but in the tics of syntax and cadence.

A Moment at the Rivers Heart (Chiliad: A Meditation, Part Two), by Clive Barker

We now come to grips with the quantum-mechanical central-force problem in all its glory. In classical physics,

Definition (central force field) A particle is in a central force field provided the net force that actson the particle is directed radially (towards or away from the origin) with a magnitude that depends onlyon the radial distance of the particle from the origin. Such a force can be written F = F (r)er, where eris the radial unit vector of spherical coordinates (see Complement 1.D of Chap. 1). The magnitude F (r) ispositive for an attractive force (one directed towards the origin) and negative for a repulsive force (onedirected away from the origin).

In quantum mechanics, the concept of a force is a bit awkward, because we lack the concept of atrajectory. So its easier to discuss the central-force problem in terms of the corresponding potentialenergy. In classical physics, we can write any conservative force as F = V and thereby define a potentialenergy V . For a central force F = F (r)er, the function V is independent of the polar and azimuthal angles and : its spherically symmetric

V (r) = V (r, , ) = V (r),the potential energy corre-sponding to a central force.

(4.1.1)

The same symmetry property applies in quantum mechanics, where a central potential energy defines arotationally invariant system, the type we studied in Chap. 3.

In that chapter, we sought solutions to the time-independent Schrodinger equation (TISE)[Trad(r) +

1

2mr2L 2(, )

angular KE operator

+ V (r) E]E,`,m`(r, , ) = 0, (4.1.2)

where Trad is the radial kinetic-energy operator, and where the angular kinetic-energy operatorL 2(, )/2mr2, contains the square of the orbital angular momentum operator, L 2. Symmetry led us to


266 4.1. Introduction

seek separable solutions to Eq. (4.1.2):

E,`,m`(r, , ) = RE,`(r)Y`,m`(, ) (4.1.3)

We spent much of Chap. 3 learning about the angular factors in these solutions, the spherical harmon-ics Y`,m`(, ), and ended by deriving the radial equation[

Trad(r) +~2`(`+ 1)2mr2

+ V (r)

]RE,`(r) = ERE,`(r), ` = 0, 1, 2, . . . (4.1.4)

Solution of this equation gives the radial factors RE,`(r) in the SAME functions (4.1.3) along with thestationary-state energies E. These energies may correspond to bound states or to continuum (scattering)states.

The many results of Chap. 3 that deal with orbital angular momentumcommutation relations, eigen-value equations and their solutions, expectation values and uncertainties, and so onpertain to any single-particle system. In this chapter well use those results to learn as much as we can about solutions of theradial equation (4.1.4) without actually solving it. The heart of this chapter are generic properties ofradial functions for spherically symmetric potential energies (4.6). Generic properties are valu-able, because, like the spherical harmonics, they pertain to all rotationally invariant systems. These genericproperties lead to procedures for solving the 3D TISE qualitatively (4.7). In addition to all this stuff,a new theme (and problem-solving habit) appears in this chapter: modeling complicated systems. Likesymmetry, this theme will pervade the rest of this book. Our first direct mathematical assault on a specificradial equation comes in Chap. 5, where we engage one-electron atoms.

In this chapter, well follow a path we first trod in 2D. We begin in 4.2 with a close look at thestructure of the radial equation (4.1.4) and the mathematical properties of its solutions: boundary conditions,orthonormality, and nodes, properties we first encountered in Chap. 2. These properties establish thefoundation for understanding the physics inherent in radial functions via the radial probability densityof 4.3.

Once you qualitatively understand 3D radial functions, youll be ready to tackle the radial equationmathematically. In 4.4 well discover that introducing a related function, the reduced radial function,dramatically simplifies that mathematical chore. We further introduce the effective potential energy(4.5), which yields insights into the solutions of the radial equation without having to solve that equation.From these insights, we then develop generic mathematical and qualitative properties of all radial functionsfor all rotationally invariant single-particle systems (4.6).

By approaching the radial equation and its solution this way, rather than immediately diving into itsdetailed solution, youll learn how to brainstorm (Begin by brainstorming) central-force problems in advanceof solving them and how to pierce the inevitable veil of mathematics to see the underlying physics. Thatswhy most of this chapter emphasizes themathematical structure of the radial equation and how thatstructure influences radial functions and energies. Such insights are precious, because, with very fewexceptions, radial equations cannot be solved without approximations.


267

4.2. The radial equation and mathematical properties of its solutionsThere was mystery hereyes; but it might not be beyond human understanding. He hadlearned a lesson, though it was not one he could readily impart to others. At all costs, hemust not let Rama overwhelm him. That way lay failure, perhaps even madness.

Rendezvous with Rama, y Arthur C. Clarke

A lot of information about solutions of the 3D radial equation follows immediately by analogy to the solutionsof the 2D radial equation (Chap. 2). In this section well begin this generalization, remaining ever vigilantfor differences due to the additional dimension.

4.2.1 Bound and continuum states

A typical 3D potential energy supports bound states and continuum states.1 In terms of the systemsenergies, the difference between bound and continuum states is profound. The energies of bound states arequantized : they constitute a list of discrete valuesthe only values allowed by nature. In contrast, theenergies of continuum states can have any value above some minimum (see Complement 4.B.)

All potential energies well investigate will conform to a few limitations. These are not severe limitations:they characterize almost all potential energies used in quantum mechanics:

(1) Well consider only potential energies V (r) that decay to zero with increasing distance r from the forcecenter at the origin. If we choose the zero of energy as in Fig. 4.1, then we require

V (r) r 0

large-r behavior of atypical potential energy.

(4.2.1a)

With this choice of the zero of energy, all discrete bound-state energies are negative, while all continuum-state energies are positive.2

Figure 4.1. A generic sphericallysymmetric potential energy thatsupports two bound states. The en-ergies of these states are indicated bythe horizontal lines. Both energies arenegative because I chose the zero of en-ergy so the r limit of V (r) is zero.The continuum of allowed energies isthe shaded region, which includes allvalues E > 0. 0 1 2 3 4 5

-1.0

-0.5

0.0

0.5

r

continuum

bound states

(2) Well consider only potential energies that are sufficiently attractive to support at least one boundstationary state(Complement 4.C). With the zero of energy in Fig. 4.1, this means that the potentialenergy must be negative, V (r) < 0, at least in some finite region of r.

(3) Well consider only potential energies that near the origin behaves as

1Jargon: In one of the more quaint bits of jargon in quantum mechanics, a potential energy is said to support N boundstates if there exist N solution to its TISE with discrete energies.

2Commentary: Physicists sometimes use model potential energies that do not decay to zero asymptotically. A prime exampleis a 3D isotropic simple harmonic oscillator of frequency : this function, V (r) = m2r2/2, becomes infinite in the r limit. Such behavior is non-physical, so a model with this behavior is only useful for the study of states with energies deep inthe wellstates whose wave functions decay rapidly to zero with increasing r.


268 4.2.2 Radial equations and the radial quantum number

limr0

r2 V (r) = 0small-r behavior of atypical potential energy.

(4.2.1b)

The modest limitations in Eqs. (4.2.1) accommodate such physically important cases as potential energiesfor atoms, molecules, and solidsincluding the pure-Coulomb potential energy of an electron in ahydrogen atom (Chap. 5)but preclude certain mathematical pathologies that complicate solution of theradial equation.3

4.2.2 Radial equations and the radial quantum number

For a potential energy that obeys Eqs. (4.2.1), a separate radial equation [Eq. (4.1.4), p. 266] exists for eachorbital angular momentum quantum number : ` = 0, 1, . . . .

Solution of each of these equations may yield no, one, or several bound-state functions RE,`(r). So weneed an index to label bound-state radial functions for a particular `. A sensiblethough not uniqueindex isa radial quantum number nr. As in 2D (2.9.1), here nr is a non-negative integer : nr = 0, 1, 2, . . . , nmaxr .For any `, the upper limit nmaxr may be zero, a finite number, or even infinity. I will label each radialfunction and bound-state energy as Rnr,`(r) and Enr,`. We learned in Chap. 3 that for each nr and `, thereare (2` +1) SAMEs nr,`,m`(r, , ) of the form (4.1.3), because there are (2` +1) values of the projectionquantum number in ` m` `.

Here are the radial equations for the first three values of `; each is clearly a mathematically differentequation: [

Trad + V (r) Enr,0]Rnr,0(r) = 0, (` = 0), (4.2.2a)[

Trad +~2

mr2+ V (r) Enr,1

]Rnr,1(r) = 0, (` = 1), (4.2.2b)[

Trad +3~2

mr2+ V (r) Enr,2

]Rnr,2(r) = 0, (` = 2). (4.2.2c)

The difference arises from the term ~2`(`+1)/2mr2, which appeared when the angular kinetic-energy operatoracted on spherical harmonics in the derivation of the radial equation (3.12).

Rule: The total number of bound-state solutions of each radial equation depends on the value of ` in thatequation. The number of bound-states with a given ` that are supported by V (r) is nmaxr +1. The total numberof bound states supported by V (r) is the sum of the numbers of bound states for each `.4

Try This! 4.1. Dependence of SAME energies on quantum numbersr.

Use symmetry arguments to explain why 3D energies and radial functions do not depend on the pro-jection quantum number m`.

3Details: Physically reasonable spherically symmetric potential energies have the following properties: (1) In the near-zerolimit r 0 the magnitude of the potential energy, |V (r)|, either approaches a constant or approaches infinity less rapidlythan 1/r: that is, |V (r)| rp as r , where p 1. (2) In the asymptotic limit r , the magnitude |V (r)| goes tozero at least as fast than 1/r. (3) The potential energy is continuous for all 0 < r < except, perhaps, at a finite numberof points.

4Read on: Physicists have developed ways to predict the number of bound states supported by certain types of potentialenergies; see Complement 4.C.


4.2.3 Energy ordering of bound states and binding energies 269

4.2.3 Energy ordering of bound states and binding energies

Once we have solved the radial equation for a particular `, we can arrange the resulting bound-state ener-gies so5

E0,` E1,` E2,` Enmaxr ,` 0, (for the same `).ordering of bound-stateenergies for a given `

(4.2.3a)

Because each bound-state energy is negative, physicists often prefer to talk about the binding energy ,which is just the magnitude

nr,` |Enr,`| = Enr,` > 0 binding energy of a bound statewith quantum numbers nr and `. (4.2.3b)

The ground state , which is the most tightly bound state, has, sensibly, the largest binding energy. Allexcited states, which are less tightly bound than the ground state, have smaller binding energies. So theorder of the binding energies is

0 0,` 1,` nmaxr ,`ordering of bindingenergies for a given `

(4.2.3c)

I Warning: This ordering pertains only to bound states with the same orbital angular momentum quantumnumber `.

4.2.4 Energies and the Virial Theorem

The classical Virial Theorem. In classical physics, the Virial Theorem is a statistical statement abouttime-averaged values of the kinetic and potential energies of a system.6 If the potential energy has thepower-law form

V (r) rn+1, power-law potential energy, (4.2.4a)then according to classical Virial Theorem ,

T =1

2(n+ 1)V ,

classical Virial Theorem fora power-law potential energy,

(4.2.4b)

where an overbar signifies a classical time average (not a quantum-mechanical expectation value). For apotential energy V (r) 1/r, [n = 0 in (4.2.4a)], Eq. (4.2.4b) reads

T = 12V ,

classical Virial Theorem fora 1/r potential energy.

(4.2.4c)

The 1/r case includes the all-important pure-Coulomb potential energy ,

V (r) = Z e20

rpure-Coulomb potential energy, (4.2.5)

5Details: The use of less-than-or-equal-to () rather than less than (

270 4.2.5 Boundary conditions on the radial functions

where Z is the atomic number, and e0 is defined by e20 e2/4pi0.

The quantum Virial Theorem. The quantum counterpart of this classicalVirial Theorem is expressedin terms of expectation values rather than time averages:7

Theorem (The Virial Theorem.) If the potential energy of a system is spherically symmetric,V = V (r), then the expectation value of its kinetic energy is

T = 12

rdV

dr

(in any quantum state.) (4.2.6a)

If V is a 1/r potential energy, then

T = 12V quantum Virial Theorem for

a 1/r potential energy.(4.2.6b)

Equation (15.A.6b) leads to a useful relationship between the expectation value of the total energy E andthe expectation values of the kinetic and potential contributions to it:

T = E, and V = 2E (for a 1/r potential energy). (4.2.7)

Eqs. (4.2.6) and (4.2.7) pertain to any statestationary or nonstationaryof any system whose potentialenergy satisfies the stated criteria.8

Try This! 4.2. The Virial Theorem for a pure-Coulomb potential

Derive Eq. (15.A.6b) from Eq. (4.2.6a). Use hydrogen-atom wave functions from Tbl. 5.7, p. 401in Chap. 5 to verify this theorem for a few simple cases.

Try This! 4.3. The mean kinetic energy.

Explain why the quantum-mechanical Virial Theorem (4.2.6a) implies Eq. (4.2.7) for any state of apure-Coulomb potential.

4.2.5 Boundary conditions on the radial functions

In 3D, as in 2D (2.9.2), the number of nodes in the radial function Rnr,`(r) is related to the bound-state energy Enr,` which, in turn, is related to boundary conditions on Rnr,`(r). The 3D radial equation[Eq. (4.1.4), p. 266] is a second-order differential equation. So to specify a particular solution we need twoboundary conditions. One condition is the value of Rnr,`(r) at the origin r = 0. The other must ensurethat Rnr,`(r) can be normalized, which is possible only if it goes to zero in the asymptotic limit , r :9(1) A radial function must be bounded at the origin : that is, at r = 0, it must be finite:

|Rnr,`(0)|

4.2.6 The radial quantum number and nodes in the radial function 271

(2) A bound-state radial function must go to zero as r :

Rnr,`(r) r 0

bound-state boundary conditionin the asymptotic limit.

(4.2.8b)

If these conditions are met, then the radial normalization integral is finite: 0

Rnr,`(r)Rnr,`(r) r2 dr 0. According to our ordering scheme (4.2.3c), the ground state has theminimum allowed energy. So the ground-state radial function has the largest curvature that is consistentwith the boundary conditions (4.2.8) (2.9.2). This function, R0,`(r), must therefore have no nodes; sureenough, its radial quantum number is nr = 0. The radial function of next highest lying bound statethefirst excited state R1,`(r)has the next largest curvature consistent with the boundary conditions. Itsradial function therefore has one node, and its labeled by nr = 1. And so forth.

Key point! For a radial function Rnr,`(r), the radial quantum number nr equal the number of nodes inthe function:

number of nodes in Rnr,`(r) = nr (4.2.10)

4.2.7 Orthogonality and nodes in the radial functions

Two 3D radial functions with the same ` but different energiesthat is, different radial quantum numbersnr 6= nrmust be orthogonal (see Example 4.1):11

Rnr,`|Rnr,`r 0

Rnr,`(r)Rnr,`(r) same `

r2 dr = 0, if nr 6= nr, (4.2.11)

where the subscript r on the Dirac bracket Rnr,`|Rnr,`r reminds us to integrate only over r, and Iveassumed real radial functions (see the Aside on p. 271)

Rule: Radial functions with the same ` but different nr must be orthogonal.10Jargon: A node in a function f(r) is a positive value of r at which the function is zero, f(r) = 0. Even if f(r) = 0 at the

origin r = 0, that point is not considered a node.11Details: Equation (4.2.11) must hold because, for fixed `, the operator that acts on Rnr,`(r) in the radial equationthe

thing in square brackets in Eq. (4.1.4), p. 266is Hermitian.


272 4.2.7 Orthogonality and nodes in the radial functions

Orthogonality and radial nodes. The orthogonality requirement (4.2.11) explains why radial func-tions must have nodes. It also explains the positions of these nodes on the r > 0 axis. The radial func-tion Rnr+1,`(r) must have one more node than Rnr,`(r). This additional node must be located so Rnr+1,`(r)will be orthogonal, a la Eq. (4.2.11), to all radial functions with radial quantum numbers between 0 and nr(see Example 4.2):

Derivation of orthogonality condition. To understand the origin of this rule, we can reason by anal-ogy from the orthogonality requirement for SAMEs nr,`,m`(r, , ). This argument exploits the separableform of a SAME, nr,`,m`(r, , ) = Rnr,`(r)Y`,m`(, ), and the orthonormality of the spherical harmonics:

I Example 4.1 (Orthonormality of SAME radial functions.)To begin, Ill denote the orthonormality integral by the rather graceless symbol

I(nr, `,m` | nr, `,m`)

nr ,`,m`(r)nr,`,m`(r) d

3v (4.2.12a)

=

2pi0

pi0

0

nr ,`,m`(r)nr,`,m`(r) r2 dr sin d d, (4.2.12b)

where I used the infinitesimal volume element in spherical coordinates , d3v = r2 dr sin d d. Usingseparability of nr,`,m`(r, , ), I can (happily!) reduce the triple integral in Eq. (4.2.12b) to the product of apurely radial integral times a purely angular integral whose value I know:

I(nr, `,m` | nr, `,m`) =

0

Rnr,`(r)Rnr,`(r) r2 dr

Rnr,` | Rnr,`r

2pi0

pi0

Y `,m`(, )Y`,m`(, ) sin d d

Y`,m`| Y`,m`,

. (4.2.13a)

In Dirac notation (Appendix G) the structure of Eq. (4.2.13a) emerges clearly:

I(nr, `,m` | nr, `,m`) = Rnr,` | Rnr,`rY`,m` | Y`,m`,

= Rnr,` | Rnr,`r `,` m`,m` ,(4.2.13b)

In the second equality, I recognized that the angular integral equals the product of two delta functions for `and m` (orthogonality of spherical harmonics). Because of `,` , orthogonality of the functions {nr,`,m`(r) }does not require orthogonality of radial functions for different `; it requires orthogonality only of radial functionswith different radial quantum numbers nr and the same `.

We can interpret a SAME as a position probability density only if its radial function satisfies the radialnormalization condition

Rnr,`|Rnr,`r = 0

R2nr,`(r) r2 dr = 1 normalization of radial functions (4.2.14a)

Combining this condition with orthogonality [Eq. (4.2.11)], we get the radial orthonormality integral

Rnr,` | Rnr,`r = 0

Rnr,`(r)Rnr,`(r) r2 dr = nr,nr , (same `)

orthonormalityof radial func-tions.

(4.2.14b)

I Warning: Dont forget that the integrand of an integral that contains radial functions R(r) must havea factor of r2. This factor comes from the three-dimensional volume element in spherical coordinates,d3v = r2 dr sin d d. Its easy to forget this factor. Dont.

I Example 4.2 (The orthogonality requirement and nodes in the radial function.)The left panel in Fig. 4.2 shows R0,0(r) and R1,0(r) for a model of the three-electron lithium atom in its groundstate(Complement 4.D). For both states, ` = 0. The so-called 1s function R0,0(r) has no nodes (nr = 0);


4.2.8 The essential degeneracy 273

the 2s function R1,0(r) has one node (nr = 1). To illustrate how the orthogonality requirement determinesthe position of this node, the right panel shows the integrand R1s(r)R2s(r)r

2 of the radial orthonormalityintegral (4.2.14b). The node in R1,0(r) occurs at 0.802 a0, precisely so as to ensure that the net shaded areadefined by this integrand equals zero.12

0.0 0.5 1.0 1.5 2.0 2.5 3.0-2

-1

0

1

2

3

4

r Ha0L

Rn,{HrL

Radial functions: Li s states

1s

2s

0.0 0.5 1.0 1.5 2.0 2.5 3.0-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

r Ha0L

u1,

sHrL*

u2,

sHrL

Orthogonality integrand Li 1s and 2s states

Figure 4.2. Radial functions and the orthogonality integrand for s states of lithium. The minimum-energy (1s) radial function, R0,0(r), has no nodes, while the radial function with the next highest en-ergy, R1,0(r) (the 2s function), has one node. This node is located so the contributions to the orthogonalityintegral R0,0 | R1,0r above and below the r axis are equal. The r 0 limits of the radial functions areR1s(0) = 8.824 a

3/20 and R2s(0) = 1.204 a3/20 , where a0 is the first Bohr radius 1 a0 ~2/mee 20 = 0.5292A.

Try This! 4.4. Normalization here, normalization there.

Derive Eq. (4.2.14a) from the normalization condition on the SAME function n,`,m`(r).

4.2.8 The essential degeneracy

The bound-state energies Enr,` in the radial equation appear in the TISE Hnr,`,m`(r) = Enr,` nr,`,m`(r)as eigenvalues of H. To each energy there corresponds one (or more) eigenfunctions nr,`,m`(r). Becausethese eigenfunctions depend on m`, each SAME energy En,` may be degenerate.

This so-called essential degeneracy arises from the angular momentum degeneracy we discussedin Chap. 3: each eigenvalue ~2`(`+ 1) of L 2 is (2`+1)fold degenerate, because for each ` there are (2`+1) lin-early independent spherical harmonics Y`,m`(, ) for integral values of m` between m` = ` and m` = +`:13

Rnr,`(r) Enr,`

nr,`,` (r)

nr,`,`1 (r)...

nr,`,`+1 (r)nr,`,` (r)

the essential degeneracy. (4.2.15)

For example, corresponding to E2,1 (nr = 2, ` = 1), there are three linearly independent Hamiltonianeigenfunctions, {2,1,1, 2,1,0, 2,1,+1 }. Ergo, this energy is three-fold degenerate.12Commentary: The symbol a0 stands for the first Bohr radius, 1 a0 ~2/mee 20 = 0.5292A, the characteristic size of

atomic and molecular systems and the atomic unit of length (Appendix F).13Read on: You can learn more about degeneracies in central force problems from Degeneracies of the spherical well,

harmonic oscillator, and hydrogen atom in arbitrary degeneracies by Richard W. Shea and P. K. Aravind, American Journalof Physics 64, 430, (1966).


274 4.3.1 Bookkeeping, spectroscopic notation, and the principal quantum number

Rule: The degree of degeneracy of each bound-state SAME energy Enr,` is at least (2`+ 1):

g (Enr,`) (2`+ 1) essential degeneracy of abound-state SAME energy. (4.2.16)

Notice that Eq. (4.2.16) is an inequality.14

Key Points

The structure of the 3D radial equation closely resembles that of the 2D radial equation. Moreover,3D radial functions obey orthogonality conditions (and, for bound states, normalization conditions)identical in form to those of 2D radial functions.

In 3D each value of ` spawns a different radial differential equation. Each such equation may have oneor more bound-state solutions, depending on the strength of the potential energy. Hence each energyand radial function must, in general, carry two labels, ` and nr, the former to signify a particular radialequation, the latter to signify a particular bound-state solution of that equation.

Orthogonality requirements impose a structure of nodes on all radial functions Rnr,`(r) for a particu-lar `: the number of nodes in Rnr,`(r) equals the radial quantum number nr.

Each bound-state energy Enr,` of a rotationally invariant system is at least (2`+ 1)-fold degenerate.The actual degree of degeneracy may be larger, depending on details of V (r).

4.3. The radial probability densityWhether we choose to credit the bizarre,to take it seriously,is finally irrelevant.The world does its work.

July, July, by Tim OBrien

As in 1D and 2D, Hamiltonian eigenfunctions in 3D contain information about the probability of finding aparticle at various locations in space. For a SAME, the angular part of that information is in its sphericalharmonic Y`,m`(, ); the radial part is in its radial function Rnr,`(r). Here well figure out how to mostinsightfully extract that radial information. But first, we have to deal with a slight but significant changein notation.

4.3.1 Bookkeeping, spectroscopic notation, and the principal quantum number

So far Ive labeled 3D SAMEs, radial functions, and energies by the radial quantum number nr, a non-negative integer that distinguishes various solutions of the radial equation for a particular `. Largely forhistorical reasons, its conventional to use instead the principal quantum number n, an integer that, fora particular `, has a minimum value `+ 1. So instead of Rnr,`(r), we write Rn,`(r).

15

Although little more than bookkeeping, this change in labeling can confuse you if youre unfamiliar withit. The key is the relationship between the radial quantum number nr and the principal quantum number nfor a particular `. Since nr = 0 corresponds to n = nmin = `+ 1, we have

14Commentary: The degree of degeneracy of an arbitrary SAME of a one-electron atom, for example, is greater than (2`+1).Ignoring spin, and treating the atom in the pure-Coluomb approximation (Chap. 5) gives a degree of degeneracy n2, where nis the so-called principal quantum number n = nr + `+ 1. Taking into account electron spin (Chap. 6) increases this degreeof degeneracy to 2n2.15A cautionary note: This convention is not applied to all rotationally invariant systems. An important exception is the 3D

isotropic harmonic oscillator (footnote 2). For convenience, SAMEs and radial functions for this system are labeled by nr,although (confusingly) this index is called the principal quantum number and is denoted by n. In any case, for this system theground-state radial function is labeled by n = 0 and has no nodes.


4.3.1 Bookkeeping, spectroscopic notation, and the principal quantum number 275

nr = n ` 1 n = nr + `+ 1 the radial and principalquantum numbers (3D). (4.3.1)

To illustrate, Ive tabulated various notations for a few radial functions in Tbl. 4.1. Ive also included theso-called spectroscopic notation for each state, a symbology you may have encountered previously; thisnotation is summarized in Tbl. 4.2.16

Key point! Accompanying this change in convention is a change in the rule in 4.2.6 about nodes in radialfunctions. This rule now reads: To ensure orthogonality of radial functions with different principle quantumnumbers n but the same `, each radial function Rn,`(r) must have nr nodes, where

number of nodes in Rn,`(r) = nr = n ` 1 (4.3.2)

These nodes must be positioned along the r axis, in such a way that the orthogonality condition Eq. (4.2.14b)will be satisfied for all radial functions Rn,`(r) for that `that is, for n = `+ 1, `+ 2, . . . .

Rnr,`(r) Rn,`(r) spectroscopic notation number of nodes

R0,0(r) R1,0(r) 1s 0

R1,0(r) R2,0(r) 2s 1

R0,1(r) R2,1(r) 2p 0

R2,0(r) R3,0(r) 3s 2

Table 4.1. Radial functions labeled by the radial quantum number and theprincipal quantum number. The third column shows conventional spectroscopic no-tation (Tbl. 4.2), while the fourth shows the number of radial nodes in Rn,`(r), whichis nr = n ` 1.

Table 4.2. Spectroscopic nota-tion. For states with ` > 3, thespectroscopic designations increasealphabetically, so ` = 4 states arelabeled g states and so forth.The names in the last column areof historical interest only; they re-fer to particular spectral series ob-served when atoms undergo transi-tions (Chap. 18) and are the originof the letters s, p, d, and f .

notation ` degree of degeneracy historical name

s state 0 0 sharpp state 1 3 principald state 2 5 diffusef state 3 7 fundamental

Try This! 4.5. Probably too much practice with notation for radial functions.

Extend Tbl. 4.1 to include functions with nr = 2 and ` = 1. Also include rows for n = 3, ` = 1 andn = 3, ` = 2. Finally include rows for states with spectroscopic notation 4s, 4p, 4d, and 4f . (If yourestill unsure that you understand how to deal with this change of notation, continue developing exerciseslike this until you become confident or just cant stand it anymore.)

16Commentary: Spectroscopists devised this notation in the early days of quantum physics. By convention, the spectroscopicnotation for a bound state with principal quantum number n, orbital angular momentum quantum number `, and projectionquantum number m` is nm` , where is s, p, d, f, g, . . . for ` = 0, 1, 2, 3, 4, . . . . Well use this notationthroughout our discussion of atoms, beginning in Chap. 5. (If you think this convention is arbitrary, wait till you see spectroscopicnotation for molecular states.)


276 4.3.3 Derivation and interpretation of the radial probability density

Try This! 4.6. The usefulness of convention.

Suppose we had defined the principal quantum number n in such a way that, for each set {un,`(r) }, thestate of lowest energy was labeled n = 1, the next lowest n = 2, and so forth. List the number of nodesin the first five functions un,`(r) for ` = 0, 1, 2, and 3. This should convince you that the admittedlyless intuitive convention nmin = `+ 1 leads to results that are easier to remember.

4.3.2 The position probability density according to Born

We ascribe physical significance to a wave function (r, t) via the Born interpretation (Appendix A):

Rule: The probability of finding a particle in state (r, t) in an infinitesimal volume element d3v at (vector)position r is the product of the position probability density P(r, t) times d3v, where

P(r, t) |(r, t)|2 = (r, t)(r, t) position probability density. (4.3.3)

If the state is stationary, so (r, t) = E(r) eiEt/~, then the 3D probability density (4.3.3) looses its time

dependence and becomes

PE(r) |E(r)|2 = E(r)E(r) (for a stationary state). (4.3.4)

For a bound-state SAME, with E(r) = n,`,m`(r), the probability density (4.3.4) looses its dependenceand becomes

Pn,`,m`(r, ) = n,`,m`(r)n,`,m`(r) (4.3.5a)= R2n,`(r)Y

`,m`

(, )Y`,m`(, ), by the separable form of n,`,m`(r), (4.3.5b)

=1

2piR2n,`(r) |`,m`()|2, using Y`,m`(, ) = 1

2pi`,m`() e

im`. (4.3.5c)

4.3.3 Derivation and interpretation of the radial probability density

The next step in simplifying the probability density is to change the question. Instead of asking for theprobability of finding a particle in d3v at a vector position r, we ask for the probability of finding theparticle in a spherical shell of thickness dr at radial position r. As you can see in Fig. 4.1, the answerrequires that we sum the probability density over all angles:

Rule: The probability of finding a particle with stationary-state wave function E(r) in a spherical shell ofthickness dr at a distance r from the origin is17 2pi

0

pi0

PE(r, ) d3v = 2pi0

pi0

E(r)E(r) r2 dr sin d d . (4.3.6)

I Warning: Even though we are integrating (summing) over all angles, as indicated by the under-brace in Eq. (4.3.6), the resulting integral must include the factor r2 dr from the 3D volume elementd3v = r2 dr sin d d.18

17Details: Notice that we do not integrate over r. Thats because we want the probability density at a particular value of r.18Commentary: This factor differs from the factor of that arises from the 2D volume element d2v in the 2D radial probability

density.


4.3.3 Derivation and interpretation of the radial probability density 277

Figure 4.1. A spherical shellused to define the radial proba-bility density. The shell is locatedat radius r, and its thickness is dr.The radial probability density evalu-ated at r times dr is the probabilityof finding a particle in this shell, irre-spective of its angular position.

Deriving the radial probability density of a SAME. For a SAME, the separable form Eq. (4.3.5b)allows us to perform the angular integrals in Eq. (4.3.6) analytically: 2pi

0

pi0

PE(r, ) d3v = 2pi0

pi0

R2n,`(r)Y`,m`

(, )Y`,m`(, ) r2 dr sin d d, (4.3.7a)

= R2n,`(r) r2

2pi0

pi0

Y `,m`(, )Y`,m`(, ) sin d d, (4.3.7b)

= R2n,`(r) r2, (4.3.7c)

where the last step follows from orthonormality of the spherical harmonics. We have found a simpler versionof the Rule in 4.3.2:

Rule: The probability of finding a particle in a SAME n,`,m`(r) in a spherical shell of thickness dr at r isP radn,`(r) dr, where the radial probability density is

P radn,`(r) r2R2n,`(r) radial probability densityfor a 3D SAME (4.3.8)

Understanding the radial probability density of a SAME. Radial probability densities Pradn,` (r) areessential interpretive tools for central-force systems, especially atoms. Interpreting these functions correctlyrequires an appreciation of what they do not tell us. Neither Pradn,` (r) nor any quantity calculated from itcontains information about where inside the shell the particle might be: this angular information is washedout by the angular integration in Eqs. (4.3.7).

The radial probability density and normalization of the radial function. According to our inter-pretation of Pradn,` (r), if we add radial probability densities for shells of thickness dr at all possible radii,0 r

278 4.3.3 Derivation and interpretation of the radial probability density

I Example 4.3 (Radial probability densities for the ground state of atomic lithium.)Plots of radial probability densities for an atom are cornucopias of insights. For instance, Fig. 4.2 shows radialprobability densities for the 1s and 2s states of the model lithium atom of Complement 4.D [see also Fig. 4.2,p. 273].

I Noteworthy features of Fig. 4.2:(1) In the probabilistic sense typical of quantum mechanics, an electron in a 1s state is more localized near

the origin than one in a 2s state. (This observation exemplifies the shell model of the atom, which willassume center stage in Part III.)

(2) The maximum probability of finding a 2s electron, the peak of Prad2s (r), is much smaller than that offinding a 1s electron, the peak of Prad1s (r). [The peak value of Prad2s (r) is 0.267 at r = 3.10 a0, while thepeak value of Prad1s (r) is 1.456 at r = 0.372 a0.]

(3) There is a tiny but nonzero probability of finding a 2s electron very near the origin, at r = 0.305 a0(where the magnitude is 0.023). This little peak is due to the node in R2s(r) of Fig. 4.2 (4.2.7). J

Figure 4.2. Radial probability densitiesfor 1s and 2s electrons in atomic lithium.These radial probability densities were cal-culated as per Eq. (4.3.8) from the radialfunctions in Fig. 4.2, p. 273. They arebased on the model in Complement 4.D.The units of radius are bohr ( a0), where1 a0 ~2/mee 20 = 0.5292A. The radial prob-ability densities are in units of a30 , as re-quired to make the integrand in Eq. (4.3.9)dimensionless.

0 2 4 6 80.0

0.5

1.0

1.5

r Ha0L

Pn,{radHrL

Radial probability densities: Li s states

1s

2s

I Example 4.4 (Radial probability densities for 1s states of several atoms.)Plots of radial probability densities also facilitate comparison of properties of different atoms. Figure 4.3

compares Pradn,` (r) for 1s electrons (n = 1 and ` = 0) in several atoms: lithium, (Li: Z = 3); boron, (B: Z = 5);nitrogen, (N: Z = 7); and neon, (Ne: Z = 10).19

I Noteworthy features of Fig. 4.3:(1) The radius of the peak in Prad1s (r) decreases with increasing atomic number Zas does the average

radius where the electron would be found, from r1s = 0.558 a0 for Li to r1s = 0.155 a0 for Ne (4.3.5).Since the integral under Prad1s (r) must equal 1 for any atom, the magnitude of the peak in Prad1s (r) mustincrease with increasing Z.

(2) The spatial extent of the radial functionsthe range of r over which the radial probability densityis appreciabledecreases with increasing Z. Physically, therefore, we are more uncertain about theposition of a 1s electron in Li than in, say, Ne: the radial uncertainty for Li is (r)1s = 0.322 a0; whilefor Ne, its (r)1s = 0.090 a0 (4.3.5). J

19Jargon: The atomic number Z of an atom equals the number of protons in its nucleus. For a neutral atom, Z also equalsequals the number of bound electrons.


4.3.4 Integrated probability densities: (finite) shell and sphere probabilities 279

Figure 4.3. Radial probability densitiesfor 1s electrons in several atoms. Theseradial probability densities were calculated ac-cording to Eq. (4.3.8) from the radial func-tions of Morse et al. (1935). As in Fig. 4.2,the units of radius are bohr ( a0), where1 a0 ~2/mee 20 = 0.5292A. The radial prob-ability densities are in units of a30 .

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40

1

2

3

4

5

6

rHa0L

radi

alpr

obab

ilityd

ensit

y

1s radial probability densities

Li

B

N

Ne

4.3.4 Integrated probability densities: (finite) shell and sphere probabilities

No experiment can actually measure a particles position in an infinitesimal region of space. The closestmeasurable quantity to the radial probability density is the probability of finding the particle anywhere ina very thin spherical shell, of finite thickness r, at some radius r0. Heres a sensible way to define thisquantity, which Ill call a shell probability ,

Pshelln,` (r0) r0+ 12 rr0 12 r

Pradn,` (r) dr = r0+ 12 rr0 12 r

R2n,`(r) r2 dr, shell probability at r0. (4.3.10a)

This shell lies between two spheres, one of radius r0 r/2, the other of radius r0 + r/2 (Fig. 4.1, p. 277).A related quantity is the probability of finding the particle anywhere within a sphere of radius r0. The

corresponding sphere probability is the sum (integral) of shell probabilities for all shells with radii0 r r0:

Pspheren,` (r0) r00

Pradn,` (r) dr = r00

R2n,`(r) r2 dr, sphere probability at r0. (4.3.10b)

Like any quantity based on Pradn,` (r), the sphere probability contains no angular information. Moreover, ittells us nothing probability of finding the particle at a particular radius inside the sphere.

I Example 4.5 (The sphere probability for electrons in the ground state of atomic lithium.)In Fig. 4.4, I show sphere probabilities Pspheren,` (r0) for 1s and 2s electrons in the ground state of lithium.Unsurprisingly, Pspheren,` (r0) goes to zero as r 0. Equally unsurprisingly, Pspheren,` (r0) goes to 1 as r . [Inthat limit, the definition Eq. (4.3.10b) reduces to the normalization integral, Eq. (4.2.14a), p. 272]. Plots ofsphere probabilities Pspheren,` (r0) offer insights hard to see in plots of Pradn,` (r) like those in Fig. 4.2, p. 278. Forexample, the radius where Pspheren,` (r0) 1 is a reasonable measure of the size of the electrons state. Forthe ground state of lithium, the 1s electron, the sphere probability is 0.999 by r0 = 2.1 a0. For the much lesslocalized 2s electron, the sphere probability attains this value at r0 = 12.0 a0.


280 4.3.5 Radial expectation values and uncertainties

Figure 4.4. Sphere probability for 1sand 2s electrons in the ground state oflithium. The horizontal axis shows the ra-dius r0 in the definition Eq. (4.3.10b). Thethin horizontal line at 0.5 on the vertical axisseparates spheres that contain less than 50%of the position probability (below the line)from those that contain more than 50% (abovethe line). These calculations were based onthe radial functions of Morse et al. (1935).The sphere radii are in units of bohr ( a0), andthe sphere probabilities are dimensionless.

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

sphere radius Ha0L

sphe

repr

obab

ility

Sphere probability for Li

1s 2s

4.3.5 Radial expectation values and uncertainties

Plots of the radial probability density Pradn,` (r) and finite probabilities like Pshelln,` (r0) and Pspheren,` (r0) showthe radial distribution of the position probability of a particle. In many cases, however, we want one or twonumbers that convey the same type of information, albeit in less detail. The most useful such numbers arethe expectation value and uncertainty in the radial position of the particle.

The radial expectation value for a SAME. For a completely general state (r, t), expectation valuesand uncertainties depend on time, and in 3D, their evaluation requires doing triple integrals (ugh). Theradial expectation value , for example, is

r(t) (t) | r | (t) =

(r, t) r(r, t) d3v

=

2pi0

pi0

0

(r, t) r(r, t) r2 dr sin d d.

(4.3.11)

For a stationary state, the time dependence goes away, as usual, but were still confronted by a triple integral:

rE = E | r | E = 2pi0

pi0

0

E(r) r E(r) r2 dr sin d d. (4.3.12)

For a SAME, however, the separable structure n,`,m`(r, , ) = Rn,`(r)Y`,m`(, ) allows us to perform theangular integration analytically. As in our derivation of the radial probability density [Eqs. (4.3.7), p. 277],all that remains is a single radial integral :

rn,` n, `,m` | r | n, `,m` = 0

Rn,`(r) rRn,`(r) r2 dr (4.3.13a)

=

0

R2n,`(r) r3 dr =

0

rPradn,` (r) dr. (4.3.13b)

As the subscripts on rn,` indicate, the mean radius depends on the principal and orbital angular momentumquantum numbers but not on m`.

Now, look carefully at each step in Eqs. (4.3.13). In Eq. (4.3.13a), the underbrace emphasizes thecontribution r2 dr, all that remains of d3v = r2 dr sin d d after angular integration. This contributioncontains the familiar factor of r2. Thats why the power of r in the integrand in Eq. (4.3.13b) is r3, not r.The second equality in Eq. (4.3.13b) emphasizes the role of Pradn,` (r) = r2R2n,`(r):

Rule: The radial probability density P radn,`(r) acts as a weighting factor for each value of r where the particlemight be found. So the average radial position rn,`, is, sensibly, the weighted mean of all possible radii.


281

The radial uncertainty for a SAME. Similar machinations yield the radial uncertainty:

(r)n,` =r2n,` r2n,`, (4.3.14a)

where r2n,` = n, `,m` | r2 | n, `,m` (4.3.14b)

=

0

R2n,`(r) r4 dr. (4.3.14c)

I used Eqs. (4.3.13b) and (4.3.14) to calculate the results in Examples 4.3 and 4.4.

I Warning: The one-variable integrals in Eqs. (4.3.13b) and (4.3.14) pertain only to SAMEs. For anarbitrary stationary state, you must evaluate three-variable integrals as in Eq. (4.3.12). For an arbitrarynon-stationary state, this integral will depend on time, as in Eq. (4.3.11).

Try This! 4.8. Labels galore.

Why does r for a SAME depend on n and ` but not on m`? For a potential energy V (r, ) that dependson as well as r, on which of the quantum numbers n, `,m` will r depend? Explain your answer.

4.4. The case of the disappearing derivative: The reduced radial equationOne of Sherlock Holmess defectsif, indeed, one may call ita defectwas that he was exceedingly loath to communicate his full plansto any other person until the instant of their fulfillment.

The Hound of the Baskervilles, by Sir Arthur Conan Doyle

The radial equation[Trad(r) +

~2`(`+ 1)2mr2

+ V (r)

]RE,`(r) = ERE,`(r), ` = 0, 1, 2, . . . (4.4.1a)

poses a couple of practical challenges. The first challenge is that its a differential equation. The secondcomes from the radial kinetic-energy operator,

Trad(r) = ~2

2m

1

r2d

dr

(r2

d

dr

)= ~

2

2m

(d2

dr2+2

r

d

dr

) .

radial kinetic-energyoperator

(4.4.1b)

As the underbrace emphasizes, Trad renders Eq. (4.4.1a) a second-order ordinary differential equation witha first-derivative term:[

~2

2m

(d2

dr2+2

r

d

dr

)+~2`(`+ 1)2mr2

+ V (r)

]RE,`(r) = ERE,`(r), ` = 0, 1, 2, . . . (4.4.1c)

The presence of the first-derivative, (2/r) dR(r)/dr, makes solving this equation more difficult than it needsto be. Moreover, it makes the mathematical structure of the 3D radial equation different from that of the1D TISEwhich precludes our adapting insights from the study of 1D quantum mechanics to the solutionof this equation.

4.4.1 Introducing the reduced radial equation

We can make these problems go away by getting rid of the first derivative term. Of course, I cant justerase the first derivative term because I dont like it. What I can do is define a new functionthe reducedradial function uE,`(r)that satisfies a second-order differential equation which has no first derivativeterm and from which I can determine RE,`(r). There exists such a function uE,`(r), and its very simply


282 4.4.2 Boundary conditions on the reduced radial function

related to RE,`(r):

un,`(r) r Rn,`(r) = Rn,`(r) = 1run,`(r) definition of the reduced radial function. (4.4.2)

Plugging Eq. (4.4.2) into Eq. (4.4.1c) and expanding out the derivatives leads to an equation for uE,`(r),the reduced radial equation

[ ~

2

2m

d2

dr2+~2`(`+ 1)2mr2

+ V (r) En,`]un,`(r) = 0 reduced radial equation. (4.4.4)

In terms of the reduced radial function un,`(r), the separable form a SAME becomes

n,`,m`(r, , ) =1

run,`(r)Y`,m`(, ) (4.4.5)

Whoa! Stop! Right now a question should be bothering youat least, this question bothered me when I first

encountered this step. How did he know that the definition in Eq. (4.4.2) would eliminate the first derivative

from Eq. (4.4.1c)more to the point, how am I supposed to figure out how to perform this trick when Im confronted

with a similar situation? There is, it turns out, a systematic procedurea cookbook for eliminating unwanted

derivative terms from a differential equation . You can apply this strategy to any differential equation involving

any variable or derivative. If youre interested, Ive developed and applied this strategy to the radial equation

in Example 4.12 in Complement 4.A.

4.4.2 Boundary conditions on the reduced radial function

Immediately upon deriving a differential equation, we should determine the boundary conditions its solutionsmust obey. The boundary conditions on un,`(r) follow directly from the conditions on the radial functionsRn,`(r) = un,`(r)/r (4.2.5). For bound states, [Eq. (4.2.8), p. 270],20

Rn,`(0) = a finite number = un,`(0) = 0, (4.4.6a)

Rn,`(r) r 0 = un,`(r) r 0. (4.4.6b)

To ensure normalizability of the radial function Rn,`(r) we need only require that its value at the origin be afinite number. But the reduced radial function un,`(r) is the radial function multiplied by 1/r, and 1/r blowsup as r 0. So un,`(r) must go to 0 as r 0: thats the point boundary condition in Eq. (4.4.6a). Thelimit boundary condition in (4.4.6b) for un,`(r) is the same as for Rn,`(r).

I Example 4.6 (Reduced radial functions for the ground state of atomic lithium.)To show you the effect of the modest little factor r in the definition Eq. (4.4.2), Ive plotted in Fig. 4.1the functions u1s(r) and u2s(r) that correspond to the radial functions R1s(r) and R2s(r) in the left panelin Fig. 4.1. The radial probability densities for these states are in Fig. 4.2, p. 278 and discussed in Example 4.3.

The contrasts between Rn,`(r) and un,`(r) are striking. The boundary condition that u1s(0) = 0 forcesthis function to turn over as r 0, resulting in a peak at r = 0.372 a0 thats not present in R1s(r). Sim-ilarly, u2s(r) has a minimum at r = 0.305 a0 and a maximum at r = 3.099 a0. These features appear in the

20Details: Although the boundary condition un,`(0) = 0 properly identifies most physically admissible radial functions, itis more stringent than absolutely necessary. The true origin of the boundary condition at the origin is the requirement thatthe Hamiltonian Trad + Tang + V (r) be Hermitian; this must be satisfied lest the eigenfunctions of this operator be physicallyinadmissible. As discussed in 12.4 of Merzbacher (1998), this Hermiticity requirement leads to a less stringent boundaryconditions at the origin which, in most cases, can be replaced by Eq. (4.4.6a). See also Chap. 6 of Galindo and Pascual (1991).


4.4.3 Subduing the reduced radial equation through pattern matching 283

0.0 0.5 1.0 1.5 2.0 2.5 3.0-2

-1

0

1

2

3

4

r Ha0L

Rn,{HrL

Radial functions: Li s states

1s

2s

0 2 4 6 8 10-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

r Ha0L

un,{HrL

Reduced radial Function: Li s states

1s

2s

Figure 4.1. Radial and reduced radial functions for 1s and 2s states of atomic lithium. Thereduced radial functions in the right panel were were calculated from the radial functions in the left panel[see Fig. 4.2, p. 273]. Both are based on the radial functions of Morse et al. (1935). The units of radius

are bohr ( a0), where 1 a0 ~2/mee 20 = 0.5292A. The units of the reduced radial functions are a1/20 .

radial probability densities, in Fig. 4.2, p. 278 and hence influence quantities like the sphere probabilities of Ex-ample 4.5. In fact, its much easier to predict and understand the radial distribution of position probabilitydensity from reduced radial functions than from the radial functions themselves. J

4.4.3 Subduing the reduced radial equation through pattern matching

4.4.3.1. The structure of the reduced radial equation

Ive already introduced The Principle of Pattern Matching : Pursue familiar patterns (Appendix Q).The key to pattern matching is mindful scrutiny : you must keep your brain engaged, not just your opticnerve. Look again at the reduced radial equation, this time focusing on its mathematical structure :{

~2

2m

d2

dr2+

[~2`(`+ 1)2mr2

+ V (r)

]

function of r

}un,`(r) = En,` un,`(r). (4.4.7)

We see a second-derivative of the unknown function un,`(r). Added to this is a function of r. One term in thisfunction, the barrier term came from action of the angular kinetic-energy operator on a spherical harmonic;the other is the physical potential energy V (r), a consequence of whatever forces act on the particle. On theright-hand side, we see the bound-state energy En,` times the unknown function.

21

21Jargon: The barrier term is usually called the centrifugal potential energy . This term is somewhat misleading, because,as weve seen, the term ~2`(`+ 1)/2mr2 comes from action of the angular kinetic-energy operator. This term centrifugalpotential energy comes from classical physics. Until about the last 2/3 of the 20th, textbooks in classical mechanics, whentalking about central-force systems, introduced a fictitious force they called the centrifugal force. (Nowadays, this force,which is still fictitious, is usually called the centripetal force.) Whateve you call it, the classical potential energy thatcorresponds to this fictitious force is L2/2mr2. In the quantum mechanical radial and reduced radial equations for a rotationallyinvariant system, the centrifugal potential energy looks just like the classical expression L2/2mr2 with the classical orbitalangular momentum L2 replaced by its eigenvalue in the SAME for these equations, ~2`(` + 1). But its actually an angular-kinetic-energy term.


284 4.4.4 Orthonormality of the reduced radial function

4.4.3.2. The effective potential energy

Now, where have you seen an equation like Eq. (4.6.3) before? Right: in your study of 1D quantum mechanics.The 1D TISE for a bound-state wave function E(x) of a particle of mass m with potential energy V (x) hasprecisely the same structure as Eq. (4.6.3):{

~2

2m

d2

dx2+ V (x)

}n(x) = En n(x), TISE (1D). (4.4.8)

The role of V (x) in the 1D TISE is played by the underbraced term in Eq. (4.6.3). This term is not thephysical potential energy: the physical potential energy is V (r). But structurally, this term plays the role ofa potential energy in the reduced radial equation. So Ill call it the effective potential energy :

V eff` (r) ~2`(`+ 1)2mr2

+ V (r) the effective potential energy. (4.4.9)

With this definition, we can make the reduced radial equation look even more like the 1D TISE:

[ ~

2

2m

d2

dr2+ V eff` (r)

]un,`(r) = En,` un,`(r)

reduced radial equation in termsof the effective potential energy.

(4.4.10)

I Warning: Dont get carried away with the similarities between the reduced radial equation Eq. (4.4.10)and the 1D TISE Eq. (4.4.8). Two differences are paramount and should be memorized:

(1) The effective potential energy V eff` (r) in the reduced radial equation is not the physical potential energy.Rather, V eff` (r) is an artifice that appeared as a result of our derivation of this equation.

(2) The domain of the independent variable r in the reduced radial equation, 0 r

4.4.5 Summary: Why bother with the reduced radial function? 285

Inserting Rn,`(r) = un,`(r)/r into this equation yields the orthonormality relation for the reduced radialfunction:

un,` | un,`r = 0

un,`(r)un,`(r) dr = n,northonormality of bound-statereduced radial functions.

(4.4.11b)

Notice that the factor r2, which has been present in every radial integral so far, is gone. It was canceled bythe factor 1/r2 from the definition Rn,`(r) un,`(r)/r.

I Warning: Remember that Eqs. (4.4.11) apply only to radial functions with the same orbital quantumnumber `. No such relation pertain to functions with ` 6= `, whether or not the principal quantum numbers nand n in the functions are equal.

4.4.5 Summary: Why bother with the reduced radial function?

The structural similarity between the 1D TISE (4.4.8) and the reduced radial equation (4.4.10) allows us toadapt our understanding of bound-states in 1D to 3D rotationally invariant system:23

Skills for drawing qualitative sketches of the eigenfunctions n(x) [9.3]. Insights into the influence of symmetry properties on the eigenfunctions n(x) [9.3]. Known solutions for special cases such as the 1D infinite square well [7.4], the finite square well [8.8],and the simple harmonic oscillator [9.8].

Powerful problem-solving techniques such as power-series solution of differential equations [9.7], sep-aration of variables [7.1], and numerical methods for solving transcendental [8.8] and differentialequations.

Insights into the structure of Hamiltonian eigenfunctions and eigenvalues for a generic potential en-ergy, such as the nature of bound and continuum states, the effect of nodal structure on eigenfunctions,and so forth.

To show you how to perform manipulations using un,`(r), Ive included a non-trivial example, Example 4.13,in Complement 4.A.

Key Points

Introduction of the reduced radial function un,`(r) r Rn,`(r) eliminates the first-derivative termfrom the system-dependent differential equation we must solve to complete specification of a SAME[Eq. (4.4.5), p. 282].

The differential equation for the reduced radial function, Eq. (4.4.4), p. 282, contains, in addition to thephysical potential energy, V (r), a term ~2`(`+ 1)/2mr2 that arises from action of the angular kinetic-energy operator on a spherical harmonic in the separable form n,`,m`(r, , ) = Rn,`(r)Y`,m`(, ). Be-cause the sum of these two terms, the effective potential energy V eff` (r) ~2`(`+ 1)/2mr2 + V (r),appears in the reduced radial equation, this sum governs the mathematical behavior of the un,`(r).

The reduced radial functions satisfy boundary conditions at the origin and in the asymptotic limit[Eq. (4.4.6), p. 282] that follow from the boundary conditions on the radial function. They key differenceis that at the origin, Rn,`(r) must equal to a constant, while un,`(r) must equal zero.

Three-dimensional integrals written in terms of reduced radial functions un,`(r) will not contain theadditional factor of r2 that must be present when these integrals are written in terms of radial func-tions Rn,`(r).

23Read on: In this list, the parenthetical sections refer to Understanding Quantum Physics (UQP), which, of course, Iurge you to rush out and purchase.


286 4.5.1 The competition between the barrier term and the physical potential energy

I Aside. The radial momentum.It is often useful in problem solving to write the 3D radial kinetic-energy operator terms of the radialmomentum operator , which is defined as

pr i ~(

r+1

r

)= i ~1

r

(

rr

)radial momentum operator. (4.4.12a)

In terms of pr, the radial kinetic energy assumes the familiar form24

Trad =p2r2m

(4.4.12b)

Youll have a chance to properties of this operator in Problem 4.10.

Try This! 4.9. The radial kinetic-energy and radial-momentum operators.

Show (or otherwise convince yourself) that plugging Eq. (5.C.4) for the radial momentum

operator pr into Eq. (4.4.12b) for Trad gives Eq. (4.4.1b) for the radial kinetic-energy operator.

4.5. The roles of the physical and effective potential energiesIt is of the highest importance in the art of detection to be able to recognize, out of anumber of facts, which are incidental and which are vital. Otherwise your energy andattention must be dissipated instead of being concentrated.

Sherlock Holmes, in The Reigate Squires, by Sir Arthur Conan Doyle

The effective potential energy for a spherically symmetric physical potential energy, [Eq. (4.4.9), p. 284]

V eff` (r) V (r) +~2`(`+ 1)2mr2

, (4.5.1)

plays a vital role in bound states of a 3D rotationally invariant system. Indeed, it holds the key to thequalitative behavior of the reduced radial function un,`(r).

4.5.1 The competition between the barrier term and the physical potential energy

The definition Eq. (4.5.1) of the effective potential energy reveals two key features of the barrier term~2`(`+ 1)/2mr2:

(1) The barrier term is positive for all ` > 0 but equals zero if ` = 0.

(2) The barrier term is proportional to 1/r2, and therefore goes to zero as r increases from r = 1 to r .As r decreases from r = 1 to zero, however, this term grows very rapidly, and in the limit r 0, itgoes to +.25

In the next example, well examine the contributions to the effective potential energies for a typical system.

I Example 4.7 (The effective potential energy for s-state electrons in the ground state of lithium.)Figures 4.1 illustrate the effects of the barrier term on the effective potential energy for the model for thelithium in Complement 4.D. Scrutinize this figure to see how it illustrates the following rule:

Rule: For states with ` > 0, the effective potential energy is determined by a competition between the physicalpotential energy V (r), which must be attractive enough to support at least one bound state, and the barrier

24Read on: For mathematical reasons pr cannot represent an observable (it lacks self-adjoint extensionssee 6.2 of Galindoand Pascual (1991)). The square of this operator, however has no such deficiency and so can represent an observable.25Details: For most potential energies in nature, the asymptotic decrease to zero occurs as 1/rp for a constant p 1.


4.5.2 The effective potential energy and the number of bound states 287

term ~2`(`+ 1)/2mr2, which for ` > 0 is repulsive (positive) for all r. The magnitude of this term increases rapidlyas r 0.

0.0 0.5 1.0 1.5 2.0 2.5-4

-2

0

2

4

r Ha0L

pote

ntia

len

erg

y

VHrL

V{effHrL {H{+1L2r2

0.0 0.5 1.0 1.5 2.0 2.5-4

-2

0

2

4

r Ha0L

effe

ctive

pote

ntia

len

erg

y

{=0

{=1 {=2

{=3

Figure 4.1. Effects of the centrifugal barrier on the effective potential energy. Left panel: A physicalpotential energy V (r), the centrifugal barrier term ~2`(`+ 1)/2mr2, and the effective potential energy V eff` (r)for the model for lithium in Complement 4.D. The orbital angular momentum quantum number is ` = 1. Rightpanel: The effective potential energy for ` = 0, 1, 2, and 3.

The left panel in Fig. 4.1 shows this competition in action. The physical potential energy V (r) is purelyattractive. As r increases, the barrier term decreases: by about r & 2.0 a0 this term (for ` = 1 in this figure) haslittle effect on V (r), so here V eff1 (r) V (r). Going the other way, as r decreases to 0 the function V (r) keepsgetting stronger (more negative). But now the barrier term gets stronger more rapidly. By about r 1 a0, thebarrier term starts to win, and V eff1 (r) goes positive. The smaller r is, the more the barrier term dominates thephysical potential energy. By r = 0.01 a0, the physical potential energy equals 294.623E h, while the barrierterm is 104 E h, giving an effective potential energy of 9.7 103E h. Near the origin, the barrier term alwayswins.

The right panel shows this battle royal played out for three barriers of increasing strength. As ` increases,something important happens. Although V eff1 (r) remains weakly attractive for r . 1 a0, the barrier terms for` > 1 are so strong that V eff` (r) for ` > 1 are purely repulsive. So for this physical potential energy there are nobound-state solutions to radial equations for ` > 1.

4.5.2 The effective potential energy and the number of bound states

Figure 4.1 shows the great influence of V eff` (r) on the reduced radial functions. So great is this influence thatwe can use knowledge of V eff` (r) to answer three crucial questions:

(1) How many bound states of a particular ` does the physical potential energy V (r) support?

(2) For each such bound state, what is the qualitative behavior of un,`(r) for all r?

(3) Where are the classical turning pointsthose special values of r that separate each classicallyallowed region from the adjacent classically forbidden region?

Well discuss the qualitative behavior of the radial function in Secs. 4.6 and 4.7. Here Ill use Fig. 4.1 todiscuss the number of bound states (see also Complement 4.C).

According to the reduced radial equation Eq. (4.4.4), for each ` = 0, 1, 2, . . ., there may exist one ormore bound-states. Each bound-state has a discrete energy En,`, and a function un,`(r) that satisfies theboundary conditions un,`(r) 0 as r 0 and as r . What determines the number of bound statesfor a given ` is the effective potential energy V eff` (r). As illustrated in Fig. 4.1, no matter how strong the


288 4.6. Generic properties of the reduced radial function

physical potential energy, there (almost) always exists some `max such that such that V (r) supports nobound states with ` > `max.

26

Key Points

The barrier term results in an effective potential energy V eff` (r) that, for ` > 0, is weaker (less attractive)than the physical potential energy V (r) at all r.

The qualitative nature of un,`(r) (and hence of the radial probability density) is determined by the com-petition between the barrier term and V (r) and can depend strongly on the orbital angular momentumquantum number `.

The physical potential energy determines the total number of bound states of the system. But theeffective potential determines the number of bound states for each orbital angular momentum `.

Since V eff` (r) determines the number of bound states of orbital angular momentum ` for a system,the number of bound states decreases with increasing `. Typically, there is some `max such that V (r)supports no bound states for ` > `maxthe notable exception being the pure Coulomb potential energyof Chap. 5.

4.6. Generic properties of the reduced radial functionThere is a thread here which we have not yet graspedand which might lead us through the tangle.

Sherlock Holmes in The Adventure of the Devils Foot by Sir Arthur Conan Doyle

Nature is staggeringly diverse. Much of what makes it tractable is that we need not start anew withevery system, because systems share generic properties. For single-particle systems, for instance, everystationary-state wave function (r, t) has the form

(r, t) = E(r) eiEt/~, where HE(r) = EE(r). (4.6.1)

In the study of rotationally invariant 3D systems, every stationary angular momentum eigenstate (SAME)has the form

E,`,m`(r, , ) = RE,`(r)Y`,m`(, ) =1

ruE,`(r)Y`,m`(, ) (4.6.2)

where Y`,m`(, ) is a spherical harmonica function thats known, system-independent, and hence generic.Moreover, every reduced radial function satisfies the reduced radial equation,

~2

2m

d2

dr2un,`(r)

radial KE

+~2`(`+ 1)2mr2

un,`(r) angular KE

+ V (r)un,`(r) physical potential energy

En,` un,`(r) energy

= 0 (4.6.3)

Not only equations, such as the TISE and the radial equation, and functional forms, such as the timefactor eiEt/~ and spherical harmonics, are generic. So are boundary conditions: every bound-state SAMEradial function Rn,`(r) must be constant at r = 0 and go to zero as r .

Because generic properties must hold for large classes of similar physical systems, knowing them greatlyempowers a physicist. Generic properties

(1) simplify the algebraic and/or numeric work required to calculate properties of a system;

(2) guide solution of specific equations, such as the reduced radial equation;

26Details: An exception to this statement is the pure-Coulomb potential energy of Eq. (4.2.5), p. 269, a very special case forwhich, as well see in Chap. 5, an infinite number of bound stationary states exist for all `.


4.6.1 Limiting behavior of physically reasonable potential energies 289

(3) enable (almost) fool-proof checks on results of numerical solutions of equations;27 and

(4) develop insight and the ability to deduce, with little or no algebra, qualitative information aboutproperties of a system.

The tactic Im advocating combines two of our habits for effective problem-solving (Appendix Q): Begin bybrainstorming and Pursue familiar patterns:

Tip. Before trying to solve an equation, either on paper or on a computer, list every property you know its solutionsmust satisfy by virtue of the nature of the system the equation describes. Then use these generic properties tosimplify the equations you must solve and the calculations you must perform, and to deduce as much as possibleabout the results you expect. After youve solved the equation, carefully compare your answers to the genericproperties in your list and resolve any discrepancies.

For radial functions, some generic properties follow from requirements such as orthogonality and boundaryconditions. Others follow from the mathematical structure of the reduced radial equation, Eq. (4.6.3).Because this equation contains V (r), its solutionsthe functions un,`(r) and energies En,`are system-dependent, not generic. To find generic properties, we must look to regions of r where the potential-energyterm V (r)un,`(r) is negligible compared to other terms in this equation. There are two such regions: verysmall r and very large r.

4.6.1 Limiting behavior of physically reasonable potential energies

Potential energies in nature are finite and continuous. A typical attractive (negative) spherically symmetricpotential energy has the qualitative properties illustrated in Fig. 4.1, p. 267:28

In the asymptotic limit r , the interaction that gives rise to the potential energy vanishes, soV (r) approaches zero:29

V (r) r 0, asymptotic limit of a physically reasonable potential energy. (4.6.4)

In the near-zero limit r 0, the potential energy goes to some negative value. (Some modelpotential energies, such as the pure Coulomb potential energy well study in Chap. 5, go to in thislimit.)

To illustrate these behaviors, Ive plotted model potential energies for lithium, helium, and the molecule C60(fullerene) in Fig. 4.1. All these potential energies go to zero (at quite different rates) as r and to somelarge negative value (or to ) as r 0.27Commentary: For instance, if your computer generates a reduced radial function that violates one of its generic properties,

such as the asymptotic boundary condition un,`(r) 0 or some of the algebraic conditions well discuss in this section, thenyou know your answer is wrong.28Memory Jog: We choose the zero of energy at the top of the potential well. With this definition an attractive potential

energy is one thats negative everywhere. With this choice, all bound states have negative energies, En,` < 0, and the continuumconsists of all positive energies E > 0. Any potential energy that supports one or more bound states must be attractive forsome range of r. Most potential energies in nature are attractive for all r, with the possible exception of a small region nearr = 0 and/or at very large r.29Commentary: Although the interaction V (r) between a particle and a force center may be of infinite range, it must diminish

as the separation between the particle and the force center increases (as r ). For instance, the pure Coulomb potentialenergy, which varies with r as 1/r, never equals zero. But as r , the Coulomb potential energy goes to zero: V (r) 0.Note that some widely-used models display unphysical behavior at large r. For instance, the potential energy of an isotropic3D SHO, V (r) = m2r2/2 blows up as r . Such potential energies can be used to model interactions for small r only,because their large-r limiting behavior is unphysical.


290 4.6.2 Cleaning up the reduced radial equation

Figure 4.1. Large r behavior of some typ-ical physical potential energies. The po-tential energies shown (in atomic units) area model potential for the fullerene C60 (solidcurve), the long-range behavior of the electron-atom interaction potential energy for a He atomin the 2 1S excited state with polarizability = 803 a30 (dash curve), and the model Lipotential energy developed in Complement 4.D(short dash curve). Notice that the horizontalaxis starts at r = 9.0 a0.

C60

-r4

Li

9 10 11 12 13 14 15

0.

-0.05

-0.1

-0.15

-0.2

r

pote

ntia

len

erg

y

4.6.2 Cleaning up the reduced radial equation

Before analyzing the radial equation (4.6.3) at small and large r, I want to eliminate some clutter. First, Illtemporarily omit the usual subscripts from u(r) and E. Second, Ill multiply everything by 2m/~2, therebyisolating the second derivative, which will play a primary role in the analysis. Heres whats left:

d2

dr2u(r) +

2m

~2V (r)

U(r)

u(r) +`(`+ 1)

r2u(r) +

2mE~2 2

u(r) = 0. (4.6.5a)

With the indicated definitions,

U(r) 2m~2

V (r), and 2 2mE~2

, (4.6.5b)

this equation simplifies to the clean, convenient form30

d2

dr2u(r) + U(r)u(r) +

`(`+ 1)

r2u(r) + 2u(r) = 0 (4.6.6)

where is called the decay constant . I can now generate an asymptotic equationthe reduced radialequation in the asymptotic limit r .

I Warning: Dont forget that the decay constant depends on n and `that is, each radial function has adifferent decay constant. Restoring the subscripts, the definition of the decay constant reads31

n,` 2m(En,`)

~2= En,` =

~22n,`2m

(4.6.7a)

n,` 2mn,`~2

= n,` =~22n,`2m

(4.6.7b)

30Details: Its always a good idea, when manipulating equations, to check that the dimensions are treated correctly. Thedimensions of potential energy, of course, are [V (r)] = E . Since

[2m/~2

]= E1 L2 , the dimensions of U(r) are [U(r)] = L2 .

This is consistent with the other terms in Eq. (4.6.6). Note also that [] = L1 .31Jargon: The decay constant is related to the more familiar wave number kn,`

2mEn,`/~2 by kn,` = i n,`. The wave

number is inconvenient for discussing bound states because its imaginary. (Bound-state energies En,` are negative.) But for a

continuum state with energy E > 0, the wave number kn,` 2mE/~2 is real. So scattering theories are usually formulated

in terms of kn,`. The main advantage of the binding energy is that its positive: this feature makes it easier to keep track ofthe energy ordering of bound states and eliminates the need to keep track of pesky minus signs.


4.6.3 Generic behavior of reduced radial functions in the asymptotic limit 291

The expression in terms of the binding energy E [Eq. (4.2.3b), p. 269] emphasizes that, since E < 0for all bound states, the decay constant is real.

4.6.3 Generic behavior of reduced radial functions in the asymptotic limit

4.6.3.1. Derivation of the asymptotic reduced-radial equation

As r , the physical potential energy V (r) goes to zero according toV (r) 0 as r [Eq. (4.6.4),p. 289]. Similarly, the barrier term ~2`(` + 1)/2mr2 goes to zero. So for any V (r) is, there is some r largeenough that we can approximate the radial equation (4.6.6), to any desired accuracy, by

d2

dr2u(r) + 2 u(r) = 0,

asymptotic reducedradial equation.

(4.6.8)

To see this, look at Fig. 4.2. The solid curve shows the terms retained in the approximation Eq. (4.6.8);the dashed curve shows all four terms in complete reduced radial equation (4.6.6). As r increases, terms otherthan those in Eq. (4.6.8) cease to be important: to graphical accuracy, the two curves are indistinguishablebeyond about 14 a0.

Figure 4.2. Asymptotic behavior of the the` = 1 reduced radial equation for boron.This figure shows large-r behavior of terms inthe reduced radial equation for the 2p electronin the ground state of boron. The solid curve isthe sum of the second-derivative d2u2p(r)/ dr2and 2pu2p(r); the dashed curve is the sum of allterms in the 2p reduced radial equation. [Calcu-lations based on the model-based valence-electron(2p) radial functions for boron of Morse et al.(1935).]

6 8 10 12 14 16-0.010

-0.008

-0.006

-0.004

-0.002

0.000

r Ha0L

BH2pL

4.6.3.2. Solution of the asymptotic reduced-radial equation

What kind of a function solves Eq. (4.6.8)? It must be a function whose second derivative equals 2 timesthe function itself. One of the derivatives you learn in elementary calculus does the job:32

d

drer = er = d

2

dr2er = 2 er. (4.6.9)

So the asymptotic equation (4.6.8) is solved by functions whose radial dependence is exponential, either er

or er. Since Eq. (4.6.8) is a second-order differential equation, its most general solution is an arbitrarylinear combination of these two linearly independent functions:33

32Details: Any linear combination of e+r and er yields a new set of (two) linearly independent solutions of the asymptoticradial equation. The most common such set consist of the hyperbolic sine

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