factorization method and new potentials with the oscillator spectrum

8/13/2019 Factorization Method and New Potentials With the Oscillator Spectrum

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Factorization method and new potentials with the oscillator spectrumBogdan MielnikCitation: J. Math. Phys. 25, 3387 (1984); doi: 10.1063/1.526108View online: http://dx.doi.org/10.1063/1.526108View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v25/i12Published by theAmerican Institute of Physics.Related ArticlesCentral configurations with a quasihomogeneous potential functionJ. Math. Phys. 49, 052901 (2008)GazeauKlauder coherent states for trigonometric RosenMorse potentialJ. Math. Phys. 49, 022104 (2008)The support of the limit distribution of optimal Riesz energy points on sets of revolution in 3J. Math. Phys. 48, 122901 (2007)Scattering in one dimension: The coupled Schrdinger equation, threshold behaviour and Levinsons theoremJ. Math. Phys. 37, 6033 (1996)

Addition formulas for qBessel functionsJ. Math. Phys. 33, 2984 (1992)Additional information on J Math PhysJournal Homepage: http://jmp.aip.org/Journal Information: http://jmp.aip.org/about/about_the_journalTop downloads: http://jmp.aip.org/features/most_downloadedInformation for Authors: http://jmp.aip.org/authors

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Factorization method and new potentials with the oscillator spectrumBogdan Mielnika)Departamento de Flsica, Centro de Investigaci6n y de Estudios Avanzados del l P. N., 07000 Mexico D. F.,Mexico(Received 2 March 1984; accepted for publication 15 June 1984)A one-parameter family of potentials in one dimension is constructed with the energy spectrumcoinciding with that of the harmonic oscillator. This is a new derivation of a class of potentialspreviously obtained by Abraham and Moses with the help of the Gelfand-Levitan formalism.PACS numbers: 02.30.Em, 03.65. - w

I. INTRODUCTIONContrasted with general relativity, where new solutionsare found every now and then, the class of exactly solubleproblems of quantum mechanics (QM) has not greatly expanded. The two most commonly used exact methods todetermine the spectra in QM are the method of the orthogonal polynomials and the algebraic method of factorization. The potentials for which exact solutions exist form arather narrow family, including the elastic and Coulomb po

tentials (modified by the 1/r 2 terms), the Morse potential,the square potential wells, and a few others, and the commonopinion is that this is everything exactly soluble in Schrodinger's quantum mechanics. Hence, it might be of interestto notice that in some occasions the factorization methodseems not yet completely explored. n particular, it allowsthe construction of a class of potentials in one dimension,which have the oscillator spectrum, but which are differentfrom the potential of the harmonic oscillator. This class hasbeen previously derived using the Gelfand-Levitan formalism.II CLASSICAL FACTORIZATION METHOD

The factorization method in its most classical form wasfirst used to determine the spectrum of the Hamiltonian ofthe harmonic oscillator in one dimension:1 d 2 1H= - + 22 dx 2 2 (2.1)

The method consisted of introducing the operators ofcreation and annihilation_ 1 (d + ) _ 1 _ x /2 d ,'/2a - - - x - e -----e,J2 dx J2 dx (2.2)* 1 ( d ) 1 x /2 d _ x /2a = J2 - dx x = - J2 e dx e ,

with the propertiesa*a = H* _ H 1 =>[a,a*] = 1.aa - +2

Hence,Ha* = a*(H + 1),Ha=a(H-l ) .

(2.3)

(2.4)

(2.5)(2.6)

al On leave of absence from Department of Physics, Warsaw University,Warsaw, Poland.

These relations allow the construction of the eigenvectors and eigenvalues of H f 1/1 is an eigenvector of H(H1/I = ,1,1/1 the functions a*1/I and a1/l [provided that they arenonzero and belong to L 2 R )] are new eigenvectors corresponding to the eigenvalues A + 1 and A - 1, respectively:

H(a*1/I) = a*(H + 1 1/1 = A + l)a*1/I,H(a1/l) = a(H - 1 1/1 = A - l)a1/l.

(2.7)(2.8)

Since the operator H is positively definite, one immediately finds the lowest energy eigenstate 1/10 as the one forwhich

.1. ----' d x /2.1. ----'.1. ) C - x /2a f/O = V-r -e f /o=V-r f /OX = oe ,dx (2.9)and one checks that the corresponding eigenvalue is ,1,0 =Using now the operator a*, one subsequently constructs theladder of other eigenvectors 1 In corresponding to the nexteigenvalues An = n + :

1 In = Cn a*r1/lo = Cn( - l)n[ :x: e - X'/2]e - x'/22.10)

where Hn x) are the Hermite polynomials given by the Rodriguez formula(2.11)

The nonexistence of any other spectrum points and eigenstates follows from the completeness of the Hermite polynomials. The above method was first employed by Dirac. 1 Itsextension for the hydrogen atom was found by Infeld andHull.2 A generalized presentation is due to Plebafiski. 3 Thegroup theoretical meaning is owed to Moshinsky,4 Wolf,5and other authors. Yet, there is still one aspect of he methodrelatively unexplored. t can be used not only to find theinterdependence between different spectral subspaces of thesame operator but also to transform one Hamiltonian intoanother.III MODIFIED HAMILTONIAN

Consider once more the factorized expressionH + = aa*. (3.1)

Are the operators a and a* here unique? Define the newoperatorsb = _ ~ + P X ) ) ,J2 dx (3.2)

3387 J. Math. Phys. 25 (12), December 1984 0022-2488/84/123387-03 02.50 1984 American Institute o PhysiCS 3387



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b * = _ + {3 (X)), 3.3)..fi dxand demand that H + be written alternatively as

H + = bb . (3.4)This leads to

1 d 2 1 2 1 1 ( d 2 {3' 2)- T dx 2 + T X + T = T - dx 2 + +{3,(3.5)

and so, the condition for {3 is the Ricatti equation{3' + 32 = 1+ x 2 3.6)The occurrence of the Ricatti equation in the factoriza

tion problems is a typical phenomenon. 2,3 In general, theexplicit solution of his type of equat ion is not known. This isnot the case in (3.6), where one has one particular solution{3 = x. Hence, the general solution can be obtained putting{3 = x + /J x). This yieldst/J' + 2tjJx +t/J 2 = ~ / t / J 2 + 2x(1/t/J) + 1 = 0. (3.7)

Introducing now a new function y = 1/ fJ, one ends upwith a first-order linear inhomogeneous equation

- y' + 2xy + 1 = 0,whose general solu tion is

(3.8)

y= (r +f - x d X ) ~ , rER.Hence,

e - ~ e - ~ifJ(x) = ., ~ { 3 x ) = x ++ e - x dx' r + e - . o ' n _ I) =(for n = 1, 2, ... ~ b i f J o = 0, (3.17)

and so, the "missing vector" is found from the first-orderdifferential equation

bifJo = _ [ ~ + {3 (X)]ifJo = ..fi dx~ o = coe - x'/2exp(lX fJ (x')dx). (3.18)By the very definition (3.18), ifJo is another eigenvector ofH

corresponding to the eigenvalue Ao = 1:H't/Jo = b *b + )ifJo = ifJo' 3.19)

As the system of vectors ifJO,ifJl' is complete in L 2(R ), theoperator (3.12) is a new Hamiltonian, whose spectrum is thatof the harmonic oscillator, although the potential is not.Since our initial Hamiltonian is parity invariant, the finalconclusion should remain valid when x_ - x. Indeed,though each one of the potentials (3.13) is not parity invar-iant, the whole class is: V( - x,r) = V(x, - r) (Irl > [ii).The reader can verify that what we have obtained here is thesame class of potentials that Abraham and Moses obtainedby using the Gelfand-Levitan formalism (Ref. 6, Sec. III,starting on the top of p. 1336; see also papers by Nieto andGutschick7 and NietoS).Remark: Differently than for the oscillator, the eigenvectors ifJ admit no first-order differential "rising operator." The ifJ,,'s are constructed not by a rising operation, butdue to their relation to the " 's, which can be schematicallyrepresented as in Fig. 1. This means, however, that for theifJ 's there is a differential "rising operator," but it is of thethird order: A * = b *a*b. As far as we know, the use of thehigher-order rising and lowering operators in spectral problems has not yet been explored.

H' I HA- i l ~ ~.,I,J'0 00/0

A= babH'A=A- (H' + IH A=A(H -I I

FIG. I. The relation between the t/Jn s and the Pn 'soBogdan Mielnik 3388



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CKNOWLEDGMENTSThe author is grateful to Augusto Garcia for his kindhospitality at the Departamento de Flsica del Centro de In-

vestigaci6n y de Estudios A vanzados in Mexico. Thanks aredue to Jerzy Plebafiski and other colleagues at the Departa-mento de Flsica for their interest in this work and helpfuldiscussions. The discussion with Marcos Moshinsky in Oax-tepec in 1982 is remembered.

3389 J Math. Phys. Vol. 25 No. 12 December 1984

Ip. A. M. Dirac, The Principles o Quantum Mechanics (Oxford U. P., Ox-ford, 1958).2L. Infe1d and T. E. Hull, Rev. Mod. Phys. 23, 21 1951).3J. Plebaiiski, Notes from Lectures o Elementary Quantum Mechanics(CINVESTAV, Mexico, 1966).4M. Moshinsky, The Harmonic Oscillator in Modern Physics, in Fromtoms to Quarks (Gordon and Breach, New York, 1969).'K. B Wolf, J. Math. Phys. 24,478 1983).6p. B. Abraham and H. E. Moses, Phys. Rev. A 22,1333 1980).7M. M. Nieto and V. P. Gutschick, Phys. Rev. D 23, 922 1981).8M. M. Nieto, Phys. Rev. D 24,1030 1981).

Bogdan Mielnik 3389

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factorization method and new potentials with the oscillator spectrum

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