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UWThPh-199ü-l «ANGWt January 30,1990 i mz. mo

Perturbations of Supersymmetric Systems in

B. Baumgartner Institut für Theoretische Physik Universität Wien

Abstract

The methods of 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 levels. One proof of a confirmation of these laws has a more algebraic touch and works for first ordci perturbation . 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 , 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 . (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 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 , 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 . 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 [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

(-£i + V(x)-T,)

(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\.

)-W(A*v>iJS){x),

HipE = Bps, Hiffiß = E 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 {

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'

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

VN/+1 = -(Efr-m)~1/2At

Ai0=-£+G

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,,. (2.246) The parameter f enters as a for the superpotential G(x). If G{x) does not change sign and is negative, the eigenvalues e(£) are decreasing functions of the coupling constant £. These functions may be inverted to £(e). So we pose the problem to find a family of operators Lo{s), L\(t), such that the equations (2.24) may be transformed into me) - OtfW = 0. (2.25) 6

The solution to this problem is found in two steps. The firstste p is to define

He) -1 := |G(x)r/J(ff,(0 - e)\G(x)\-l<*, (2.26)

0l>iC(x) := |C?(x)|»/V.^(x). (2.27) So (i,(e)-O^,>4 = 0 (2.28) holds, L,(e) are symmetric operators and do not depend on (. In the second step one makes a coordinate transformation x —• y, connected with a unitary transformation T: C2(a,6),dx) -* C2(c,d),dy), such that -\G\-ll2-r^\G\-112 is _ ox d2 mapped to —-J-T + #(y)> and

ay*

T: Li(e) - £,(e) = -~ + U(y) + (&-&- e)/\G\. (2.29)

The appropriate transformation is ^ = \G{x)\1'2, y(x) = c + £ |G(x')|1/2oV, (2.30)

, T : iKx) - V(y) = |0(»)r 'V(x) = \G{x)\^V{x), (2.31) tf(y) = -C'/iG2 + 5G'2/16G3 = G/4G - (3/16)(G/G)2, (2.32) where we denote — = {x) -* tj>(y) is unitary, but defined in (2.31) is not normalized, when

(2.33) with B"(0 = |ö|,/2^ - C/4G + C + e/2, (2.34)

B+(0 = -|G|1/2 j- + C/4C +

Htfwll = R*M||. (2.36) 7

2.4 Factorizable Systems. Class II This class of factorizations emerges by applying the change of coupling constants and eigenvalues to each supersymmetric pair of a class I factorizable system. Because of the conditions that all Gt(x) + £ should appear in the facorization, Gi(x) purely negative, not all types of class I are amenable to such a change, only the types B, Ep, E, and F. Type B is transformed to Schrödinger's factorization of the harmonic oscillator. The most important example involves again the hydrogen atom, type F, with Gt(x) = -(*+l)/x, i = Z/(t+l). With y = 2((*+l)x)1'3 the corresponding Schrodinger equation WO ~ «W«) = (-£ + ^F21" f ~ «Ort«) = 0 (2.37) is transformed into the Schrodingsr equation for the harmonic oscillator («.) - 0*) = $$&-£ + (2< + 1/y + 3/2) + «V - 0HV) - 0 (2.38) where K2 = -e/4 (2.39) is positive, since the disjoint eigenvalues e of the hydrogen atom are negative. The type F factorization for the hydrogen atom works also for non-integer values of L We have to choose to get for the oscillator

IHO = UHA + ^ = 0,1,2,3,.... (2.41) The operators At, A}, which shift IHA by one unit, are transformed to operators B~(Q, B+((), which shift tjjo by two units:

here I = tuo. In subsection 3.3 we will extend this relation between the unperturbed Hydrogen atom and the unperturbed harmonic oscillator to the perturbed systems.

3 First Order Perturbation Theory for Factorizable Systems 3.1 Level Splittings of a Perturbed Supersymmetric Pair

We consider a pair of Hamiltonians (//*,///+J), each one perturbed by XV. We want to compare the energies in the perturbed system, in first order perturbation theory:

E(N,l, X) = EN + %«*|V|v>«). (3.1) 8

The

{EN-m)SNjt = \{vtjS\{AiA,y + VA'tAi-2A\VAt)WtjS) =

3 3 = ^iAAMt,V\ + lV,A})At)\

= H

(EN-m)6NJt = ^(e+1^) = 2 (3-4)

= M

3.2 Level Splittings of the Perturbed Hydrogen Atom

Combining the formula (3.3), expressing SNft+i, and (3.4) as it stands, we can get rid of the ^-dependence of the different operators acting on V. It depends on the detailed form of Gt, hence on the type of factorization, how to combine them. In the case of the hydrogen •.torn we have to multiply the equations containing Gt by £ + 1:

(e+l)(EN-m)SN4-^+2)(EN-Vw)SNMi - A(/+l/2)(^+liF|(K"+(2/x)V')|^+,,£). (3.5) This recmrence relation for 6/tj can be completed by

(EN - T]N-I)6N^I-I = 0, (3.6) so we get

(t + 1)(EN - Vt)6N4 = A £ V + v - l/2){, (3-7) and conclude that we can state

Theorem 1: For a hydrogen atom, perturbed by a spherically symmetric potential XV, there appears an ordering of the levels £(JV, £, A) > E{N, £ +1, A) if AV = V" + {2/x)V > 0, E{N, £, A) < E{N,t +1, A) if AV = V" + (2/x)V < 0. For the other factorizable systems there hold similar theorems. The factor in front of V depends on the type of factorization. It is listed as F(x) in the table of factorizations. 9

3.3 Level Splittings of the Perturbed Three-Dimensional Har­ monic Oscillator It is an elementary but space-consuming procedure to perform similar calculations as above, with the 2?-operators replacing the A-operators. The result is.

E{N, /, A) - E{N, I + 2, A) = (3.8)

jfl, (E/«-3-M)/4

1 = A(^3 - *'(2* + 3))" £ (21 + AV + lK^w^KV - (1/r) V) W*»*>. so we have Theorem 2: For a three-dimensional harmonic oscillator, perturbed by a spherically symmetric potential AV, there emerges an ordering of the energy levels:

E(N, t, A) > E(N, * + 2, A) if V" - (l/y)V > 0,' E(N, /, A) < E(N, i + 2, A) if V" - (1/y) V < 0.

3.4 Harmonic Oscillator in One Dimension: Level Spacings This is the simplest example of a factorizable system. It allows moreover an extension of our methods to find inequalities of higher order. Apart from a factor, the intertwinors are the usual annihilation and creation operators:

Ai = fa-X> A< = ~5ä:~x' fc = 2m. (3-9)

2 Hi = ~ + x + 2t. (3.10) So the sequence of supersymmetric partners consists just of the harmonic oscillator shifted in energy by the amount of level spacings: energies aud states depend on only one pa­ rameter n, the number of nodes of the , instead of N,l:

n = N-l-l,

E(n, A) = E(N, £, A) = 2n +1 + X(Vn\VM- (3.12) The changes of the level spacings

6n = 6NS = E(n, A) - E(n - 1, A) - 2 (3.13) obey

2nSn = \(

2nSn = A(v>n-il(V72 - *V)|^,), (3.15) 10 the rewritten form of (3.3) and (3.4). By subtracting (3.14) from (3.15) for n +1 we get

(« + l)*»+i - nSn = (A/2)(v>n|Hv»>, (3.16) and ^n = (V2)E(^IHv.), (3.17) a very simple formula, implyinp that all the level spacings of a harmonic oscillator increase under the perturbation of a potential with positive second derivative, or decrease if V" < 0. Starting with (3.16) we can iterate the procedure and do with (v>ir|V"|yv) what we have done with (wl^lv«/)» arriving at

2n{«n + l)Sn+l - nSn) - (nSn - (n - l)*n_,)] = (A/2) £V|VHv>*>, (3.18) i/=0 which gives *he recursion relation

n(n + l)(*n+1 - 6n) = (n - l)n(6n - Sn.t) + (A/4) £fa,|V-V). (3-19) and finally

n(n + l)(*n+1 - *B) = (A/4) J> - v)(. (3.20)

So the sign of the fourth derivative of the perturbing potential tells us about the increase or decrease of the level spacings with increasing n. This procedure can be iterated again and again, yielding formulas for k-th order differences of the energies related to the 2&-th differential of the potential.

4 Exact Inequalities Beyond Perturbation Theory 4.1 The Main Theorem When we intend to compare the eigenvalues of two Hamiltonians H, H\ defined on the same set in R, with the same kinetic energy operator, but with different potentials, we use some supersymmetry relations and also a set of comparison potentials. So the intuitive argument of comparing the effective atomic potentials with Coulomb comparison potentials is turned into a strict proof. In order to find comparison potentials for the general case, we have to define a function G(x), such that the difference of the potentials, which is the difference of the Hamiltoni­ ans, V, - V = Hi - H = 2G'{x) (4.1) 11 holds. For technical reasons we need a strict inequality

G'(x) > 0. (4.2)

Then we can compare the energies En(V) with En-i(V + 2G'), where n counts the eigenvalues, it denotes the number of nodes of the corresponding wave functions.

Now we know of a family of comparison potentials Ve, where

En(Ve) = En-1(Ve + 2G') (4.3) because of supersymmetry:

Vc = Gl-G'e + r,, (4.4)

Gc(x) = G{x) + c/2, ij arbitrary. (4.5)

If, for example, we wish to compare En{V + ———•) to En-\\y + ^ -), we

2 have 2G' = 2{t + l)/x , Gc(x) = -{t + l)/r + c/2, so we have the Coulomb potentials with Z = (£ + l)c as comparison potentials. As in the intuitive argument we need the convexity or concavity property of V, that

V lies above or below the comparison potential Ve. Ve is that comparison potential which has the properties of touching V at xo, with the same slope:

Ve(x0) = V(x0), Vc'(x0) = V'(xo) (4.6)

at a fixed point x0. For the intuitive argument we had to choose for x0 some mean value.

For the strict proof of inequalities we need comparison for each point x0. The convexity or concavity property which is needed, is that V(x) lies above or below all the comparison potentials tangent to V. If the comparison potentials were linear, this would be the usual convexity or concavity property. Here we may use the fact that

Ve - V0 = cG + s linear in G. Then we may consider V — VQ as a function l of G and observe that we have to look for the convexity or concavity of (V — VQ) o G~. When stated in derivatives in x: (v-W&vM < H<"-<«

Then we may also construct a supersymmetric partner to H = --j-r + V(x). For this dx* we need $(x) = ~u'(x)/u(x), (4.8) with u(x) a positive solution of

The hard work to be done is the proof of the following 12

Proposition: If u is the positive square integrable wave function of the ground state of --£r + V(x), and if (V - V ) c G~x is ( amWK \ and not affine, then dx2 0 \ concave J

?'(*){ <}tf(x). (4.9)

For the inequalities between g' and G' to hold strictly everywhere, it sufficies that the convexity inequalities (4.7) hold strictly ac least at one single point x. With the use of this proposition one can prove

Theorem 3: If (v-j - -™j-){V -Vo)\ < [ 0» not everywhere = 0, then

En(V)l^\En.i{V + 2G'). . (4.10)

Proof:

Propoaition+Mini—Max-Principle

BJY) «ET Bn.,{V + V) { < } *-i(V + 2G').

4.2 Exchange of Coupling Constants and Eigenvalues Now also in this case can we make an exchange of coupling constants and eigenvalues inv if G has a definite sign, G{x) < 0. If (V — V0) o G is convex (concave), then for each c also (V + cG — V0) o G™ is convex (concave). So the discrete eigenvalues« 6n(c) of -dVdrc2 + v + «G + 2G' and 6„(c) of -cP/dx7 + V + cG obey the inequalities

Mc){ < Un-i(c). (4.11)

These eigenvalues are decreasing functions of c which can be inverted to c,(6) and c„(6) with the property *(6)|< ja*-»(6). (4-12) Transforming the Hamiltonians as we did for factorizable system«, we transform to the coordinate y with | = |C?|,/2. (4.13) We define 2 W0(y) = (1/4)G/G - (3/16)(G/G) - G. (4.14) 13

tav Then the convexity of (V - V0) o G^* transforms to the convexity of (W - W0) o (1/G) , also to be expressed by

(^ + (2G/G - G/G)£j (W - Wo) > 0. (4.15)

So we arrive at

ComtX Theorem 4: If {Wv - W ) o x(l/G)™ is ( ) a- i not affine, then 0 ' ' ' \ concave J

1 2 En(W) { > } En.x{W + 2G/\G\ ' ). (4.16)

This theorem applies to the broken degeneracies of the levels of the harmeni; oscillator. Again it gives the same inequalities and under the same conditions on the perturbing potential, as theorem 2, but here beyond perturbation theory.

4.3 An Outline of the Proof of the Proposition We consider the case of convexity of (V — Vo) o Glnv and indicate the procedure of the proof:

a) For fixed x0, choose c such that V&XQ) = V'(x0).

b) Assume w.l.o.g. G(x0) + c > 0 (otherwise change x -» -x). If there exists no 7, such that

2 (G(*o) + 7) - G'(x0) = V(x0) - E, (4.17) continue with f). If there exist such 7, choose the larger one c) Deduce from the convexity of V — VQ and the Mini-Max-Principle, applied to the ground state, that 7 < c.

d) Vx> x0 you have

V;(i) < Ve(x) - Vc(x0) + V(x0) -E< V(x) - E (4.18)

T/ since V;(x) < Vc'(x) < '(x). e) Let 4>i be the ground state wave function of Vy, so G + T =-#,/*,. (4.19) Due to (4.18) you have the relation

u'(xo)^(x0) - ti(io)^(x0) < 0 (4.20) 14

which implies

g(x0) = -u'/u > -#,11, = G(x0) + 7 > 0 (4.21) and

2 2 j(*o) >(G(x„) + 7) . (4.22) f) The Schrödinger equation for u is equivalent to the Riccati equation

g'(x0) = g\x0) - V(x<>) + E. (4.23)

There exists no 7 solving (4.17) iff G'(x0) < E — V(x0). Inserting this inequality in (4.23) gives immediately g'(xo) > G'(x^). If such a 7 exists, insert (4.22) into (4.23), use (4.17) and you have also finishedth e proof:

2 E G x «/'(a*) > (G(x0) + 7) - V0(*o) + = '( °)'

References

[1] E. Schrödinger, A method of determining quantum-mechanical eigenvalues and eigenfunctions, Proc. Roy. Irish Acad. A46, (1940) 9-16.

[o1 Ei. Schrödinger, Further studies on solving eigenvalue problems by factorization, ' Proc. Roy. Irish Acad. A46, (1941) 183. [3] L. Infeld, T.E. Hull, The factorization medhod, Rev. Mod. Phys. 23, (1951), 21-68. [4] E. Witten, Dynamical breaking of supersymmetry, Nucl. Püys. B188, (1981), 513- 554. [5] A.K. Ramdas, S. Rodriguez, Spectroscopy of the solid-state anologues of the hydro­ gen atom: donors and acceptors in semiconductors, Rep. Progr. Phys. 44, (1981), 1297. [6] B. Baumgartner, A. Pflug, A new approach to the understanding of level ordering in atoms and nuclei, preprint UWThPh-1989-11 to appear in Amer. J. Phys. [7] P.A. Deift, Applications of a commutation formula, Duke Math. J. 45, (1978), 267. [8] B. Baumgartner, Level Comparison Theorems, Ann. Phys. 168, (1986), 484-526.