Hidden Treasures of the Schrödinger Equation

HIDDEN TREASURES OF THE SCHRÖDINGER EQUATION A New Year’s Gift for Students and Teachers of Quantum Mechanics from Chris Quigg December 2021 ·Revised February 2025 It is by solving the Schrödinger equation that aspiring physicists first learn to make predictions about the quantum landscape at scales a billion times smaller than our human scale of everyday experience. The traditional guided tour introduces us to the Laguerre and Hermite polynomials and Airy functions, among others, and trains us to visualize consequences of the spherical harmonics. Acquiring this arcane knowledge, we gain entry into the company of initiates. It is even good fun—if you are a physics student— to develop a mastery of the special functions that builds on what is learned from the study of classical electrodynamics. By learning to make numerical solutions we expand the domain of physical situations for which we can derive predictions beyond those we can treat in closed form. A complementary approach reveals interesting and useful relationships without the need for explicit solutions. In this holiday leaflet, I introduce a few of these Schrödinger Shortcuts, which I have occasionally presented in talks informally called “How to Cheat at Quantum Mechanics.” May they brighten the season and nourish your curiosity! © Chris Quigg 2021, 2025 Historical Preliminaries Johannes Kepler, born four-hundred-fifty years ago this week (*27 December 1571) and celebrated among physicists as discoverer of the three laws of planetary motion, was fascinated by Nature’s patterns. So taken was he with the snowflakes settling out of Prague’s winter skies that he began to ponder how, if every one was different from all the others, each was reliably hexagonal in form. Kepler documented his musings in a booklet of two dozen pages entitled Strena seu De Nive Sexangula, as the year 1610 turned to 1611. From his perch as Court Mathematician to Holy Roman Emperor Rudolf II, he offered his New Year’s gift, On the Six-Cornered Snowflake [1], to his friend and occasional patron, court counselor Johann Matthäus Wacker von Wackenfels. Wacker had brought Kepler the first news of Galileo’s discovery of Jupiter’s moons, the “Medicean stars.” Three centuries later, Kepler’s pamphlet remains a charming companion for a winter’s day. The end-of-year holidays also have a special significance in the history of quantum mechanics, for it was during a retreat to the mountain villageofArosainSwitzerlandattheendof1925thatErwinSchrödinger foundinspirationforhisinventionofwavemechanics[2]. Schrödinger’s equation is a remarkably powerful tool for investigating atomic, molecular, and condensed-matter systems. It has never gone out of style, and it took on a new life in particle physics with the discovery of “atoms” of heavy quarks and antiquarks, beginning with the charm–anticharm states of the J/ψ family in 1974 [3]. When the Υfamily of b¯ bstates was discovered in 1977, Jonathan Rosner and I joined the ranks of those who had been applying nonrelativistic quantum mechanics very insightfully to the charmonium states. Comparing the two quarkonium systems, we came upon many results about quantum bound states that we had not known before. We mused that either they were new, or they had been forgotten for forty years. We summarized much of what we and others had learned in a small treatise on quarkonium quantum mechanics [4]. Now that those (re)discoveries are themselves four decades old, I highlight here a few of them for students, teachers, and aficionados of quantum mechanics. The Schrödinger Equation In three spatial dimensions, the Schrödinger Equation has the form − ħh2 2µ∇2Ψ(r)+[V(r)−E]Ψ(r)=0, (1) where Ψ(r)is the Schrödinger wave function, µis the reduced mass of a two-body system, ris the relative coordinate, V(r)is the interaction 1 potential, Eis the energy eigenvalue, and ħhis Planck’s constant divided by 2π. For a central potential V(r) = V(r), it is appropriate to express the Laplacian operator in spherical coordinates, ∇2=1 r2 ∂ ∂rr2∂ ∂r+1 r2sinθ ∂ ∂θ sinθ∂ ∂θ+1 r2sin2θ ∂2 ∂φ2. (2) Now we may separate the radial and angular dependences as Ψ(r)=R(r)Y`m(θ,φ), (3) where the angular dependence of a solution resides in a spherical harmonic, Y`m(θ,φ). With the substitution of Eqn. 3, the radial wave function for a state with angular momentum `(in units of ħh) satisfies − ħh2 2µd2 dr2+2 r d dr R(r)−E−V(r)−`(`+1)ħh2 2µr2R(r)=0. (4) The radial equation simplifies if expressed in terms of a reduced radial wave function, u(r)≡rR(r), that satisfies the boundary conditions u(0)=0u0(0)=R(0). (5) A prime denotes the derivative of a function with respect to its argument. The normalization condition on the Schrödinger wave function, Rd3r|Ψ(r)|2=1, implies that R∞ 0dr u(r)2=1. The reduced radial equation, −u00(r)=2µ ħh2E−V(r)−`(`+1)ħh2 2µr2u(r), (6) is the starting point for the observations that follow. It has the same form as the one-dimensional Schrödinger equation for an effective potential given by [V(r)+ `(`+1)ħh2/2µr2], with the proviso that evenparity solutions in one (space) dimension do not satisfy the boundary condition u(0)=0. What can we learn from this equation without explicitly solving it? In this note, we will look at relationships that involve expectation values—averages—and we will see, by scaling the Schrödinger equation, how observables depend on the mass and coupling strength for powerlaw potentials. Many other insights, including semiclassical connections that link the level density with the shape of the potential or the probability density at the origin, and dualities between ionizing and confining power-law potentials, are introduced in [4]. 2 A Little Magic Involving Expectation Values Simple manipulations lead to numerous results of considerable general utility for the study of bound states in a central potential. Two informative connections illustrate the power of this approach. Probability density at the origin The absolute square of the s-wave (`=0) wave function at the origin enters the calculation of hyperfine splitting. It is readily determined for a solution in closed form, but may be challenging to fix precisely for a numerical solution. We may evaluate |Ψ(0)|2in a different way by acting on both sides of the Schrödinger equation with R∞ 0dr u0(r), −Z∞ 0 dr u0(r)u00(r) = Z∞ 0 dr 2µ ħh2[E−V(r)]u(r)u0(r). (7) Next, use the identities u0u00 =1 2(u02)0and uu0=1 2(u2)0to integrate by parts, obtaining −u0(r)2 2 ∞ 0 =2µ ħh2[E−V(r)]u(r)2 2 ∞ 0−1 2 2µ ħh2Z∞ 0 dr [E−V(r)]0u(r)2. (8) Evaluate and simplify to find u0(0)2=R(0)2=4π|Ψ(0)|2=−2µ ħh2 : 0 [E−V(0)]u(0)2 2+2µ ħh2dV d r ·. (9) where the factor of 4πarises from Y00 =1/p4π(cf. Eqn. 3). Our first hidden treasure of the Schrödinger equation is the connection |Ψ(0)|2=µ 2πħh2dV d r ·, (10) relating the wave function at the origin to the expectation value of the gradient of the potential. A particularly neat application is to the case of a linear potential V(r) = λrthat characterizes a particle tethered to an elastic string, the counterpart of Hooke’s law in classical physics. The wave function at the origin is the same for all s-wave states; it is |Ψ(0)|2=µ 2πħh2dV d r ·−→ µλ 2πħh2. (11) We obtain this result—which exhibits the dependence on mass and coupling strength—without solving the Schrödinger equation or working out properties of the Airy functions. 3 Similar derivations carry through for all the partial waves, leading to expressions for the first nonvanishing derivative of the wave function at the origin. Virial Theorem To derive the virial theorem relating kinetic, potential, and total energies, we act on both sides of the Schrödinger equation with R∞ 0dr r u0: −Z∞ 0 dr r u0(r)u00(r) = Z∞ 0 dr r 2µ ħh2[E−V(r)]u(r)u0(r)(12) Once again we integrate by parts. The left-hand side becomes −1 2r * 0 u0(r)2 ∞ 0 +1 2Z∞ 0 dr u0(r)2=1 2 : 0 u(r)u0(r) ∞ 0 =−1 2Z∞ 0 dr u(r)u00(r). (13) Using the Schrödinger equation to replace u00(r), we find −1 2Z∞ 0 dr 2µ ħh2[E−V(r)]u(r)2=1 2 2µ ħh2〈E−V〉. (14) After integration by parts, the right-hand side of Eqn. 12 becomes 2µ ħh2: 0 [E−V(r)]u(r)2 2 ∞ 0−1 2 2µ ħh2Z∞ 0 dr (r[E−V(r)])0u(r)2. (15) Assembling the pieces, the kinetic energy 〈T〉of the bound particle is 〈E−V〉≡〈T〉=r 2 dV d r ·. (16) Power-law potentials offer interesting special cases of this second hidden treasure. For V(r) = λrν, with −2<ν<∞, we find 〈T〉=Dr 2νλrν−1E=ν 2〈V〉=ν (2+ν)E. (17) We recover at once the (perhaps) familiar relations for a Coulomb (or gravitational) potential, ν=−1 : 〈T〉=−E=−〈V〉/2, and for a harmonic oscillator, ν=2 : 〈T〉=〈V〉=E/2. The logarithmic potential, V(r) = Cln(r/r0), is a limiting case of the power laws as ν→0. For it, all s-wave levels have a common kinetic energy, 〈T〉=C/2. 4 Scaling the Schrödinger Equation For the special (but flexible) case of a power-law potential, V(r) = λrν, we may bring the Schrödinger equation to dimensionless form and infer directly how quantities with dimensions of energy or length depend on mass and coupling strength. Now the reduced radial wave equation is ħh2 2µu00(r) + E−λrν−`(`+1)ħh2/2µr2u(r) = 0, (18) in which the dimensionful parameters are 2µ/ħh2,E(with dimension [E] = µ), and λ, which has dimensions [λ] = [ħh−νµ1+ν]. We may bring the Schrödinger equation of Eqn. 18 to dimensionless form by introducing a scaled measure of length, ρ≡(ħh2/2µ|λ|)pr, (19) and expressing the energy eigenvalue as E=ħh2/2µ|λ|2p(ħh2/2µ)", (20) where "is dimensionless. The resulting equation becomes free of dimensionful quantities if we set p=−1/(2+ν)(21) and substitute w(ρ)for u(r). The scaled Schrödinger equation, w00(ρ) + ["−sgn(λ)ρν−`(`+1)/ρ2]w(ρ) = 0, (22) does not depend on either µor λ. We see (cf. Eqn. 20) that level spacings scale with mass and coupling strength as ∆E∼(2µ/ħh2)−ν/(2+ν)|λ|2/(2+ν), (23) with these notable consequences: Potential Power ν∆E∼ Coulomb −1µ|λ|2 r−1/2−1 2µ1/3|λ|4/3 Logarithmic →0µ0C Linear 1 µ−1/3|λ|2/3 Harmonic Oscillator 2 µ−1/2|λ|1/2 Infinite Square Well →∞ µ−1 5 Now looking back to Eqn. 19, we find that lengths scale as L∼(2µ|λ|/ħh2)−1/(2+ν), (24) which implies the following scaling laws for power-law potentials. Potential Power νL∼|Ψ(0)|2∼ Coulomb −1(µ|λ|)−1(µ|λ|)3 r−1/2−1 2(µ|λ|)−2/3(µ|λ|)2 Logarithmic →0(µC)−1/2(µ|λ|)3/2 Linear 1 (µ|λ|)−1/3(µ|λ|)1 Harmonic Oscillator 2 (µ|λ|)−1/4(µ|λ|)3/4 Infinite Square Well →∞ (µ0) (µ0) The hidden treasures of the Schrödinger equation are truly wondrous to behold. Happy exploring! Bibliography [1]A recent edition, with commentary, is Johannes Kepler, The SixCornered Snowflake, (Paul Dry Books, Philadelphia, 2010), ISBN 978-1-58988-053-5. [2]ThestoryofSchrödinger’swinterinspirationistoldinmanyplaces, including Walter J. Moore, Schrödinger: Life and Thought (Cambridge University Press, Cambridge, 1989) “Discovery of wave mechanics, pp. 191–229. See E. Schrödinger, “Quantisierung als Eigenwertproblem (Quantization as an eigenvalue problem),” Annalen der Physik 79, 361, 489 (1926); 80, 437 (1926); 81, 109 (1926). For English versions, see J. F. Shearer and W. M. Deans, translators, E. Schrödinger, Collected Papers on Wave Mechanics, (Blackie and Son, London, 1928). A short documentary film (in German), Thomas Gull & Stephan Läuppi, “Erwin Schrödinger—Eros und Atome” vimeo.com/279665873, follows Zürich physicist Laura Baudis on a winter pilgrimage to Arosa. [3]I recounted some of the history in C. Quigg, “Celebrating Quarkonium: The First Forty Years,” doi: 10.5281/zenodo.5812403 Seminar presented at CERN during the 2014 meeting of the Quarkonium Working Group, marking the fortieth anniversary of the discovery of the J/ψ (charmonium) family of heavy mesons. The slides containing many links to the original literature. I chose a different historical approach at the 2024 SLAC Summer Institute, describing 6 the world into which J/ψ was born: “The Night before Charmonium,” doi: 10.5281/zenodo.14844786. [4]C.QuiggandJ.L.Rosner,“QuantumMechanicswithApplications to Quarkonium,” Phys. Rept. 56, 167-235 (1979), FERMILAB– PUB–79–022–T. 7