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I Mz. Mo Perturbations of Supersymmetric Systems In 4r<?/? 00031 UWThPh-199ü-l «ANGWt January 30,1990 i mz. mo Perturbations of Supersymmetric Systems in Quantum Mechanics B. Baumgartner Institut für Theoretische Physik Universität Wien Abstract The methods of supersymmetry are extended to the factorization method. The degen­ eracy of levels in factorizable systems is broken under perturbations. With the methods of supersymmetry it is possible to state laws on the order of these perturbed energy levels. One proof of a confirmation of these laws has a more algebraic touch and works for first ordci perturbation theory. The proof of the laws beyond perturbation theory is hard, more of an analytic spirit and exploits convexity properties of the potentials. The convexity properties serve also for an intuitive argument. An important application is the law on the ordering of energy levels in atoms. 1 1 Introduction The main themes of this lecture can be traced back almost to the origin of quantum mechanics. We will not use any sophisticated new mathematics, so that all of the following could have been done, in principle, about 40 years ago, when Infeld and Hull revised Schrödinger's factorization method, [1,2,3], and recast it into a form which is now reborn with supersymmetric quantum mechanics [4]. Why then are the following results new? Probably because half a century ago everybody was interested in finding exact solutions or approximations. Now we are interested in finding inequalities, yielding certain laws, for example, for the ordering of those atomic energy levels which are degenerate in the hydrogen atom. (The law which governs this level ordering holds also for the bound states of electrons at donors in semiconductors [5].) The old intuitive argument to explain this level ordering is the remark that the clas­ sical orbits with lower angular momentum dive deeper into the charge cloud of the core electrons. There they feel a stronger attraction to the nucleus so that the effective po­ tential is lower than the Coulomb potential generated by the shielded nucleus, as it is experienced by the outermost electron when it has high angular momentum. For the out­ ermost electron this is the right argument, but it is not sufficient for an explanation of the ordering of Roentgen levels of the core electrons. So we complete this intuitive argument [6]: As the angular momentum decreases, the classically allowed region (also the region, i; where the quantum probability for the location of the electron is large) increases in all directions, both closer to the nucleus and to farther distances. Now the effective poten­ tial at intermedia e distances acts like a comparison potential Vc(r) = constant — Zejf/r. But everywhere in the newly acquired regions, the effective potential lies below this com­ parison potential. So the energies, which would be equal in Ve(r), drop with decreasing angular momentum. This is the result of a "concavity property" of the effective potential, stated as V"{r) + (2/r)V'{r) < 0 or (AV)(r) < 0, or (V'(r)/r2)' < 0. This concavity property gives not only an intuitive argument, it is also an essential ingredient in the proof of strict inequalities. This proof uses the tools of supersymmetric quantum mechanics, but it proceeds working with functional analysis. In first order perturbation theory one can prove this level ordering in a more algebraic spirit. There one needs not only supersymmetry, but its extension to the factorization method. By a certain transformation of coordinates one can exchange the roles of coupling con­ stants and eigenvalues. This trick transforms a supersymmetric pair of Hamiltonians into a new pair of isospectral operators. It is applicable to some factorizable systems, yielding a new class of factorization. Applied to the hydrogen atom, it established the relation­ ship between Coulomb potentials and harmonic oscillators. Not only the equations for the supersymmetric and factorizable systems, but also the inequalities for the perturbed systems and the strict inequalities beyond perturbation theory can be transformed in this way. Also these class-II-inequalities can be applied to a fundamental physical problem: The explanation of the level ordering of nucleons in atomic nuclei. Here one compares the Woods-Saxon potential with the potential of a harmonic oscillator [6]. 2 2 Factorizable Systems 2.1 The Necessary Tools from Supersymmetric Quantum Me­ chanics We work on C7(a, 6), where a may be finite or — oo, also b may be finite or +oo. One can reconstruct a Hamiltonian out of <f^x), the positive ground state wave function, or another positive, not necessarily square integrable solution of the differential equation (-£i + V(x)-T,)<Kx) = 0- (2.1) from the theorem on nodes (the Sturm oscillation theorem) we know that ^(x) can be chosen positive, only if 17 is not above the ground state energy. The negative logarithmic derivative of <f> G(x) := -flx)/tfx), (2.2) (by some authors called the "superpotential") serves to define the operator A = ± + G(x), (2.3) (first in Cl(a, b), then taking its closure) and its adjoint A' = ~ + G{x). (2.4) Taking the Friedrichs extension of A* A, we get the Hamiltonian H = A'A + T, = --£i + G»(«) - G\x) + r, (2.5) (with Dirichlet boundary conditions in case of finite a or 6). With the relations A^ = 0, (H- 9)4 «0, (2.6) one observes that the potential is reconstructed by V(X) = G2(X)-G'(X) + TI, (2.7) which is the Riccati equation. Now one constructs a supersymmetric partner Hx = AA' +»/, (2.8) with 2 Vl(x) = G (x) + G'(x) + r,. (2.9) 3 By use of intertwining relations AH = HXA, HA* = A'Hly (2.10) it is easily confirmed that A maps bound states and scattering states tpE of H to bound states and scattering states fitE of H\. <plJS{x) = (E-V)-V*(A<pE)(x), (2.11) <pE(x) = {E-r>)-W(A*v>iJS){x), HipE = Bps, Hiffiß = E<p1JS, (2.12) \\<PiM\ = WM- (2.13) So H and H\ are isospectral, or, if ^ € L\a, 6), essentially isospectral [7], since <f> is annihilated by A, so 17 is not an element of the spectrum of /fx. 2.2 Factorizable Systems, Class I The construction of a supersymmetric partner may be iterated. This gives a family Gt(x), ty, such that for £ = 1,2,3... with At=± + Gt(x)t (2.14) the Hamiltonian Hi = At^A^ +17,_, <2.15) is also represented as Ht = A'Ae + Tii. (2.16) The function, number and operators G, 17, A, H we started with are now denoted as Go, 170, A)» Ho- The eigenfunctions {<ptlE} to {Hi} with energies £ are related by 1,7 H>i+\£ ~ (E-ru)~ Ae(piiE, (2.17) i/2 Vtj " (£ - Vt)~ A}(pt+hE. As an abstract construction, this procedure is always possible. If the functions Gt(x) can be written down explicitly, one speaks of a "factorizable system". Infeld and Hull worked out a classification into types A to F, which is represented here in a table. (Since we use only real parameters, we split their types A and E into three subtypes.) 4 Eactorizable Systems of Class I Type Domain -£<(*) % convertible F(x) Applications (a,6) 0* = < + e) into class II 2 A a fO,xx ) /icotx + -— 4K/I + /i no cotx Harmonic ' ' sin« oscillator in 2 /J R+ Mcothx + -r^- 4Kfl-(l no cothx carved space sirnx 2 no tanx coshx -P B R fi + e~* -f yes 1 Harmonic C R+ fi/x — KX 4nfi no 1/x oscillator in more than 1 dimension 1-dimensional D R -x 2p no 0 harmonic oscillator E a (0,T) ficotx-q/ft no 2 cotx Hydrogen atom ß R+ ficothx-q/ft -f-q2lf yes 2 cothx in curved space 7 R ntaahx - q/fi -f-q*lf yes 2tanhx F R+ /i/x - g//i -92/M2 yes 2/x Hydrogen atom In the most important applications Tft is chosen as the ground state energy of H(. Because of the isospectral property i\t gives then 'he energy of the first excited state of ///_!, of the second excited state of ///_? ... and l'nally of the J-tn excited state of H0. So the energies of all the H( can be labeled by a main quantum number N. In the case of the hydrogen atom we have a = 0, i = oo, i 6 R+, Z G(x) = -^ + (2.18) ( 2(^+1)' z* (2.19) m = -4(€+l)2' <p e(i+i) z "<=-3?+ (2.20) 5 The energy of states with the main quantum number N is equal to the ground state energy i// with I = N — 1: £*=~ (2.21) I remark that, in order to comply with the usual convention, one could change the ap­ pearance of indices, writing (fsj instead of <pt£ for E = En. Also the sign of every second wave functions could be changed. Then (2.17) would read as VN/+1 = -(Efr-m)~1/2At<pN4i (2.22) <PN4 = -(Es-m)~1/2Ae<pN4+t- This would of course have no effect on the considerations of the energies. 2.3 Exchange of Coupling Constants and Eigenvalues Any pair (i/o, H\) of supersymmetric essentially isospectral Hamiltonians can be consid­ ered as an element of a family of supersymmetric partners (HQ(£), Hi(Q), £ € R, each pair connected by similar intertwining operators Ai0=-£+G<x) + t/2, (2.22) so that, with TJ = — f 2/4, Hott) = ~ + G?{z)-G'(x) + tG(x), HM) = ~+G2(x) + G'(x)+(G(x). With MO«*. = (e + ^'W (2.23) the equations (ffo(O-e)Vo^ (2.24c) are related to (fWO-Ovu,,.
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