Universal Forms for One-Dimensional Quantum Hamiltonians: A Comparison of the SUSY and the De La Peña Factorization Approaches

Abstract

We show that by linking two factorization techniques often employed to solve Schroedinger's equation one can give any one-dimensional hamiltonian the same form in terms of quantities typical of these approaches. These are the supersymmetric technique (SUSY) and the one of De La Peña's. It is shown that the linkage between them exhibits interesting peculiarities, that are illustrated in the case of a very important family of quantum potentials, namely, reflection-less ones.

1. Introduction

Due to its great importance from the conceptual, practical, and educational viewpoints, the one-dimensional Schroedinger equation (ODSE) has been the subject of continuous attention by researchers since the advent of quantum mechanics in the 1920s. ODSE-interest has been further stimulated because of its relevance in connection with interesting problems in other areas of theoretical physics. One can mention the analysis of exact multisoliton solutions to certain Hamiltonians dynamical systems governed by equations such as the Korteweg-de Vries and sine-Gordon ones [1–5]. The factorization approach [6] is an elegant method to tackle Schroedinger's equation that has been in use for more than half a century (see the excellent review of [7] and references therein). In the last 30 years two important factorization approaches have become popular. One of them is the supersymmetric one (SUSY) [1–5], fully equivalent to the Infeld-Hull-methodology, which adds the notion of shape-invariance. The other is due to De La Peña's one [8]. The SUSY approach, in particular, has lead to remarkable progress towards the completion of the program of clasifying all the exactly solvable one-dimensional potentials. The SUSY procedure has also been useful for developing powerful new approximation techniques for solving the Schroedinger equation. We wish to affect here a SUSY-de la Peña comparison and ascertain to what an extent they are equivalent and are able to shed light on each other. The task allows us to give a special universal form to any one-dimensional Hamiltonian. Reference to a specific example helps in such an endeavor, and we will use the one-dimensional well that includes a cosh⁻¹(𝑥)-squared term, a particularly interesting case of reflection-less potential [9], since it does not reflect waves at any energy.

The paper is organized as follows. We begin our considerations by briefly reviewing the supersymmetric formalism (SUSY) [1–5] in Section 2. Section 3 is a recapitulation of De La Peña's treatment. Our original materials are contained in Section 4, where we link De La Peña's methodology to SUSY. Finally, some conclusions are drawn in Section 5.

2. The Supersymmetrical Formalism

The quantum mechanical supersymmetric formalism (SUSY) [1–5] revolves around specific relations between (i) eigenenergies, (ii) eigenfunctions, and (iii) phase shifts with regards to the Hamiltonians 𝐻^1,2ℏ=−[²𝑑2𝑚][²𝑑𝑥²]+𝑉^1,2, (2.1) associated with two supersymmetric partner potentials 𝑉¹ and 𝑉². The ground state energy of 𝐻¹ is assumed to be zero. The Hamiltonian operators 𝐻^1,2 are factored as 𝐻¹=𝑄^†𝑄, and 𝐻²=𝑄𝑄^†. The operators 𝑄=(ℏ/√2𝑚)(𝑑/𝑑𝑥)+𝑊 and 𝑄^†=−(ℏ/√2𝑚)(𝑑/𝑑𝑥)+𝑊, are given in terms of the so-called quantum superpotential 𝑊(𝑥), that, in turn, is related to the two functions 𝑉^1,2 as specified below. The superpotential 𝑊 can be obtained from the ground state wave function 𝜓₀ of 𝐻¹ in the fashion ℏ𝑊(𝑥)=−(√12𝑚)[𝜓₀⁽¹⁾][𝑑𝜓₀⁽¹⁾𝑑𝑥]. (2.2) The partner Hamiltonians 𝐻¹ and 𝐻² possess exactly the same energy spectra, except for the fact that 𝐻² has one bound state less than 𝐻¹ [1–5]. The eigen-energies and eigenstates of the partner Hamiltonians 𝐻^1,2 are related in the following manner: 𝐸₀⁽¹⁾=0,𝐸_𝑛⁽²⁾=𝐸⁽¹⁾_𝑛+1(𝑛=0,1,…),

(𝐸⁽¹⁾_𝑛+1)^−1/2𝑄𝜓⁽¹⁾_𝑛+1=𝜓_𝑛⁽²⁾,(𝐸_𝑛⁽²⁾)^−1/2𝑄^†𝜓_𝑛⁽²⁾=𝜓⁽¹⁾_𝑛+1(𝑛=0,1,…). (2.3) The above factorization formalism generalizes that of the harmonic oscillator (HO) Hamiltonian, that introduces the creation-annihilation operators 𝑎 and 𝑎^† so as to express it as a product of them. From here on we set ℏ=𝑚=1, which redefines 𝑊: 1𝑊(𝑥)=−(√21)[𝜓₀⁽¹⁾][𝑑𝜓₀⁽¹⁾𝑑𝑥]. (2.4) The two Hamiltonians' spectra can be derived one from the other, which is handy when one of them is exactly solvable. Then, 𝑄^± reads 𝑄^±=1√2𝑑[∓+√𝑑𝑥2𝑊(𝑥)], (2.5) and one has 𝑉¹1=(−√2𝑊^′+𝑊²),𝑉²1=(√2𝑊^′+𝑊²), (2.6) that is, 𝑉¹+𝑉²=2𝑊²,𝑉²−𝑉¹=√2𝑊^′. (2.7) Also, integration of (2.4) yields 𝜓¹₀=𝑁exp(−𝑑𝑥𝑊(𝑥)). (2.8)𝑁 above is a normalization constant. Equation (2.8) can be inverted if a suitable function 𝑊 is given, which, in turn, would led us to a specific Hamiltonian using the prescription described in the preceding subsection.

2.1. SUSY and Reflection-Less Potentials

Reflection-less potentials are those associated with a zero reflection coefficient. Resonant tunnelling phenomena are of great interest. We recapitulate here some well-known results [10–17]. The penetrability coefficient of the pertinent barrier during the resonant tunnelling becomes large to a maximum extent. The attention devoted to reflection-less potentials is due to the fact that they exhibit a penetrability coefficient of almost unity in a whole region of the energy spectrum, whereas resonant tunnelling exists only at selected energy levels. The number of papers devoted to study of properties of the reflection-less quantum systems has been lately increasing. Here we mention just two excellent reviews [18, 19] in which one finds both (i) methods for a detailed analysis of the properties of one- and multichannel reflection-less quantum systems, and also (ii) several simple approaches for their qualitative understanding. All these methods have found application in scattering theory (direct and inverse). As an interesting illustration consider potentials of the form 𝑉¹=12(𝑛²−𝑛(𝑛+1)cosh²𝑥);𝑉²=12(𝑛²−𝑛(𝑛−1)cosh²𝑥), (2.9) which are supersymmetric partners. In the case 𝑛=1 (related to the sine-Gordon theory [20, 21]) we have 𝑉²=12(freeparticle);𝑉¹=122(1−cosh²𝑥). (2.10) We remark that 𝑉¹ is a potential that plays a critical role with regards to soliton-solutions in de Korteweg-de Vries (KdV) hierarchy. With the 𝑛=1 solutions on can also get the eigenfunctions and eigenvalues for 𝑛=2 (related to the 𝜙⁴-theory [22, 23]), and so on. The 𝑉¹-partner is the free-particle Hamiltonian and 𝜓¹₀=1√2cosh⁻¹𝑥 (2.11) of energy 𝐸¹₀=0. In turn, for other potentials of the type (2.9) with 𝑛>1, that is, for 𝜓²_𝑛 and 𝐸²_𝑛, one has 𝜓²_𝑛=𝑒^𝑖𝑛𝑥,𝐸²_𝑛=12(1+𝑛²), (2.12) while 𝜓¹_𝑛+1=1𝐸²_𝑛𝑄⁺𝜓²_𝑛=(tanh𝑥−𝑖𝑛)√1+𝑛²𝑒^𝑖𝑛𝑥, (2.13) and we see that the states 𝜓¹_𝑛+1 do not contain reflected waves. The energy eigenvalues are 𝐄²_𝑛. Successive reiteration of the approach yields the eigenstates of the potentials 𝑛(𝑛−1)/2cosh²𝑥 starting from the free-particle instance. The number of bound states is 𝑛. For the gs one gets, from (𝑑/𝑑𝑥+𝑛tanh𝑥)𝜓¹₀=0, 𝜓¹₀1(𝑥)∝cosh²𝑥. (2.14) The gs-energy equals 0, and the remaining eigenvalues are identical to those for 𝑉². Application of 𝑄⁺_𝑛=(1/√2)(−𝑑/𝑑𝑥+𝑛tanh𝑥) gives the remaining eigenstates. SUSY allows one to relate transmission and reflection coefficients for potentials associated to continuous spectra. Define now 𝑊(𝑥→±∞)≡𝑊_±, (2.15) and thus, 𝑉_1,2→𝑊²_±for𝑥→±∞. (2.16) Consider an incoming (from 𝑥→−∞) plane wave 𝑒^𝑖𝑘𝑥 of energy 𝐸 [11–17]. After 𝑉_1,2-scattering, we get transmitted waves 𝑇_1,2(𝑘)𝑒^{𝑖𝑘′𝑥} and reflected ones 𝑅_1,2(𝑘)𝑒^{−𝑖𝑘𝑥}, so that [11–17] 𝜓^(1,2)(𝑘,𝑥→−∞)→𝑒^𝑖𝑘𝑥+𝑅_1,2𝑒^{−𝑖𝑘𝑥},𝜓^(1,2)(𝑘,𝑥→+∞)→𝑇_1,2𝑒^{𝑖𝑘′𝑥}, (2.17) with 𝑘 and 𝑘^′ being of the form [11–17] 𝑘=(𝐸−𝑊²₋)^1/2,𝑘^′=(𝐸−𝑊²₊)^1/2. (2.18) SUSY connects continuous same-energy w.f.s of, respectively, 𝐻₁, 𝐻₂ as in the discrete instance. One has [11–17] 𝜓_𝑛⁽²⁾=[𝐸⁽¹⁾_𝑛+1]^−1/2𝑄𝜓⁽¹⁾_𝑛+1,𝜓⁽¹⁾_𝑛+1=[𝐸_𝑛^(𝑛)]^−1/2𝑄^†𝜓_𝑛⁽²⁾,𝑒^𝑖𝑘𝑥+𝑅₁𝑒^{−𝑖𝑘𝑥}=𝑁[(−𝑖𝑘+𝑊₋)𝑒^𝑖𝑘𝑥+(𝑖𝑘+𝑊₋)𝑒^{−𝑖𝑘𝑥}𝑅₂𝑇],₁𝑒^{𝑖𝑘′𝑥}=𝑁[(−𝑖𝑘^′+𝑊₊)𝑒^{𝑖𝑘′𝑥}𝑇₂], (2.19) with 𝑁 being a normalization constant. Equating same-exponent-terms we find, after elimination of 𝑁 [11–17] 𝑅₁𝑊(𝑘)=(⁻+𝑖𝑘𝑊₋−𝑖𝑘)𝑅₂𝑇(𝑘),₁𝑊(𝑘)=(⁺−𝑖𝑘^′𝑊₋−𝑖𝑘)𝑇₂(𝑘). (2.20) Note that (i) |𝑅₁|²=|𝑅₂|² and also |𝑇₁|²=|𝑇₂|². (ii) 𝑅₁(𝑇₁) and 𝑅₂(𝑇₂) share poles in the complex plane, saving for the fact that 𝑅₁(𝑇₁) has an extra-pole at 𝑘=−𝑖𝑊₋. The pole lies on the positive imaginary axis only if 𝑊₋<0, corresponding to a bound state of null energy. (iii) For 𝑊₊=𝑊₋ one has 𝑇₁(𝑘)=𝑇₂(𝑘). (iv) For 𝑊₋=0,  𝑅₁(𝑘)=−𝑅₂(𝑘). If one of the partner-potentials is a constant (free particle), the the other partner is reflectionless. Consequently, potentials of the type 𝑉(𝑥)=𝐴secℎ²𝛼𝑥 play a critical role in soliton-studies.

3. De La Peña's Treatment

We pass now to consider the alternative approach of [8]. This is a factorization method that provides an easily implementable algorithm for the purpose, based in the commutator and anticommutator of the creation and destruction operators that affect the factorization. These two operators play here the role of the SUSY-partner potentials. From a practical viewpoint the two techniques are equivalent. Notice that De La Peña's one was advanced one year before Witten's celebrated paper [1–5], so that one cannot ask for it to be superior in any sense to SUSY. As a matter of fact, De La Peña introduced his ideas in a journal that devotes much space to teaching techniques so as to provide beginners with an intuitive grasp of the factorization method (we will follow below the notation of [8]), as applied to any Hermitian operator 𝑃. His technique is then of utility to anyone without great experience in Quantum Mechanics. He denotes with {|𝑛⟩} its eigenstate-set, a complete basis for Hilbert's space, that for simplicity's sake we assume to be characterized by the single quantum number 𝑛. Let {𝑝_𝑛} be the 𝑃-spectrum. Then 𝑃|𝑛⟩=𝑝_𝑛|𝑛⟩. (3.1) This operator 𝑃 is to be later identified with the Hamiltonian of the system under consideration. Creation and destruction operators associated to 𝑃 are to be denoted with 𝜂^†-𝜂: 𝜂^†=_𝑛𝐶_𝑛|𝑛+1⟩⟨𝑛|;𝜂=_𝑛𝐶_𝑛−1|𝑛−1⟩⟨𝑛|, (3.2) for as yet unspecified coefficients 𝐶_𝑛 that, when applied to |𝑛⟩≡|𝑘⟩, yield 𝜂^†|𝑘⟩=𝐶_𝑘|𝑘+1⟩;𝜂|𝑘⟩=𝐶_𝑘−1|𝑘−1⟩. (3.3) In determining the coefficients 𝐶_𝑛 De La Peña leaves two of them undetermined, so as to get useful degrees of freedom. The 𝐶_𝑛 are related to the eigenvalues of the bilinear operators 𝜂𝜂^† and 𝜂^†𝜂, as seen from (3.3): 𝜂𝜂^†|𝑘⟩=|𝐶_𝑘|²|𝑘⟩;𝜂^†𝜂|𝑘⟩=|𝐶_𝑘−1|²|𝑘⟩. (3.4) Since |𝑛=0⟩ should reasonably be the “vacuum,’’ here 𝜂|0⟩=0;entailing𝐶₋₁=0. (3.5) Next one considers whether an 𝑛-upper bound exists or not. If the affirmative is the case, 𝑛=𝑁, then 𝜂^†|𝑁⟩=0;entailing𝐶_𝑁=0. (3.6) Otherwise, such condition is missing. The degrees of choice mentioned above can be put to good use by requiring 𝑃 to be of a linear nature 𝑃=𝑎₀₀+𝑎₁₀𝜂𝜂^†+𝑎₀₁𝜂^†𝜂. (3.7) With this form for 𝑃 and (3.5) we evaluate its eigenvalues 𝑝_𝑚 according to 𝑝_𝑚=𝑎₀₀+𝑎₁₀|𝐶_𝑚|²+𝑎₀₁|𝐶_𝑚−1|². (3.8) At this stage we are left with (𝑁+3) unknowns, namely, 𝑎₀₀,  𝑎₁₀,  𝑎₀₁, and {|𝐶_𝑛|,𝑛=0,1,2,…,𝑁−1}. To determine them we have at our disposal the 𝑁+1 conditions arising from (3.8), with 𝑛=0,1,2,…,𝑁, plus normalization, that is, scale-fixing, and its “origin’’-choice, which suffices for reproducing the 𝑃-spectrum. We realize now that for representing an arbitrary operator it is easier to work with the product of the operators 𝜂 and 𝜂^† than with these operators themselves. This forces us to introduce yet two new operators as linear combinations of the just mentioned product: 𝐴=[𝜂,𝜂^†]=𝜂𝜂^†−𝜂^†𝜂;𝑆={𝜂,𝜂^†}=𝜂𝜂^†+𝜂^†𝜂, (3.9) so that one casts 𝑃 in a fashion that constitutes the main advantage of the approach 𝑃=𝑞₀+𝑞_𝑎𝐴+𝑞_𝑠𝑆, (3.10) where the constants in the equation are fitted by comparison with the known form of 𝑃. The eigenvalues of the just introduced operators 𝐴and𝑆 are called, respectively, 𝑎_𝑘 and 𝑠_𝑘 and are obtained via (3.3) and (3.9): 𝐴|𝑘⟩=𝑎_𝑘|𝑘⟩=(|𝐶_𝑘|²−|𝐶_𝑘−1|²)|𝑘⟩;𝑆|𝑘⟩=𝑠_𝑘|𝑘⟩=(|𝐶_𝑘|²+|𝐶_𝑘−1|²)|𝑘⟩, (3.11) entailing 𝑎_𝑘=|𝐶_𝑘|²−|𝐶_𝑘−1|²;𝑠_𝑘=|𝐶_𝑘|²+|𝐶_𝑘−1|², (3.12) which in turn are manipulated to yield, after summation and subtraction of them, |𝐶_𝑘|²=12(𝑠_𝑘+𝑎_𝑘1)=2(𝑠_𝑘+1−𝑎_𝑘+1). (3.13) With these results one ascertains that the spectra of 𝐴-𝑆 fulfill a set of consistency relations. We get, (i) from (3.12) and (3.13), 𝑠_𝑘≥0;𝑠_𝑘+𝑎_𝑘≥0;𝑠_𝑘−𝑎_𝑘≥0, (3.14) further (ii) from (3.13), 𝑠_𝑘+1−𝑠_𝑘=𝑎_𝑘+1+𝑎_𝑘, (3.15) and (iii), from (3.12) and (3.6), 𝑠₀=𝑎₀;𝑠_𝑁=−𝑎_𝑁. (3.16) These relations can be interpreted as follows. Equations (3.14) show that the spectrum of 𝑆 is nonnegative and with lower bound 𝑎₀. The basic condition (3.15) fixes the 𝑃-spectrum's structure. Lower and (possible) upper spectral bounds are determined by (3.16). If application of (3.16) leads to a contradiction, this entails that no upper bound exists. The consistency relations (3.14), (3.15), and (3.16) contain all possible information concerning 𝑃-spectrum in a most compact and accessible fashion, given that one knows the operators 𝐴 and 𝑆 associated to any 𝑃.

4. Relation between SUSY and De La Peña

As recounted above, De La Peña introduces his two operators 𝜂, 𝜂^† [8] so as to be in a position to cast his all-important operator in the bilinear fashion (3.7) and then employs two auxiliary operators 𝐴, 𝑆 (3.9) so as to cast 𝑃 in the final form (3.10). It seems rather natural to identify with each other the creation-annihilation operators of the two factorization approaches, that is, SUSY's 𝑄⁻, 𝑄⁺ with, respectively, 𝜂 and 𝜂^†. Such a procedure then tells us that 𝐴𝜓(𝑥)=𝑄⁻𝑄⁺𝜓−𝑄⁺𝑄⁻𝜓=12(𝑑+√𝑑𝑥𝑑2𝑊)(−+√𝑑𝑥12𝑊)𝜓−2𝑑(−+√𝑑𝑥𝑑2𝑊)(+√𝑑𝑥2𝑊)𝜓, (4.1) which, after suitable manipulations, leads to 𝐴𝜓(𝑥)=√2𝑊^′𝜓(𝑥), (4.2) entailing 𝐴=√2𝑊^′𝟏, (4.3) involving the unity operator.

This is an important relation, our first new result, linking the De La Peña operator 𝐴 to the SUSY superpotential. It generalizes the HO-commutation rule, in which [𝑎,𝑎^†]=1. In other words, De La Peña's 𝐴 carries information regarding the way the SUSY's 𝑄^± differ from their HO-counterparts. The 𝑄^± commutators become now nontrivial operators instead of the customary, HO-related constants. Analogously (cf. (2.2)), 𝑆𝜓(𝑥)=𝑄⁻𝑄⁺𝜓+𝑄⁺𝑄⁻𝑑𝜓=−²𝜓𝑑𝑥²+2𝑊²𝑑𝜓⇒𝑆=−²𝑑𝑥²+2𝑊²1=2[−(2)𝑑²𝑑𝑥²+𝑊²]. (4.4) Notice that 𝑆 is proportional to a Hamiltonian form in which the square of 𝑊 acts as the potential. More specifically, De La Peña's 𝑆 is the Hamiltonian of a particle of mass 4𝑚 (cf. (2.1)) moving in a 2𝑊²-well.

We reach thus an interesting conclusion from the present linkage between two factorization methods: De La Peña's way of splitting its operator 𝑃, to be later identified with the system's Hamiltonian 𝐻¹, shows that it has a pure “Hamiltonian’’ component plus one that relates to the form in which the supersymmetric 𝑄^±differ from their HO-counterparts. Combining now (2.7), (4.3), and (4.4) leads immediately to 𝑆=𝐻¹+𝐻²;𝐴=𝑉²−𝑉¹=𝐻²−𝐻¹, (4.5) which entails expressing De La Peña's all important operator 𝑃 in SUSY terms 𝑃=𝑞₀+𝑞_𝑎𝐴+𝑞_𝑠𝑆=𝑞₀+𝑞_𝑎(𝐻²−𝐻¹)+𝑞_𝑠(𝐻²+𝐻¹)=𝑞₀+2𝑞_𝑠[𝐾+𝑊²]+√2𝑞_𝑎𝑊^′=𝑞₀+2𝑞_𝑠𝐻^𝑆+√2𝑞_𝑎𝑊^′, (4.6) where 𝐻^𝑆 is a putative Hamiltonian whose potential is the square of the superpotential. We immediately realize that De La Peña's approach seems to require some additional information vis-a-vis SUSY, namely, the three parameters 𝑞₀,𝑞_𝑠,𝑞_𝑎, which might, at first sight, be regarded as superfluous information, since SUSY (a complete treatment) does not need them. This is not so, as the following example will show.

4.1. Application of the De La Peña-SUSY Linkage

As an application we tackle the potentials discussed in Secion 1. In dealing with a potential of the form 𝑉¹=122[1−cosh²𝑥], (4.7) that is derived from the general instance 𝑉¹=12[𝑛²−𝑛(𝑛+1)cosh²𝑥], (4.8) for 𝑛=1 we are left with a Hamiltonian 𝐻¹≡𝑝𝐻=²2+122[1−cosh²𝑥𝑝]=²2−1cosh²𝑥+12. (4.9) Here we must identify De La Peña's 𝑃 with the Hamiltonian. We have 𝐻²=𝑝²2+12entailing𝑆=𝐻¹+𝐻²=𝑝²+1−sech²(𝑥),𝐴=𝐻²−𝐻¹=sech²(𝑥), (4.10) which agrees nicely with (4.3).

Thus, looking back at (3.10) 𝐻=𝑞₀+𝑞_𝑠𝑆+𝑞_𝑎𝐴=𝑞₀+𝑞_𝑠[𝑝²+1−sech²(𝑥)]+𝑞_𝑎sech²(𝑥) (4.11) we can determine the values of the constants 𝑞₀,𝑞_𝑎,𝑞_𝑠 that relate 𝑃≡𝐻 to 𝐴 and 𝑆. We find 𝑞_𝑠=1/2, 𝑞_𝑎=−1/2, and 𝑞₀=0. Consequently, we get the “correct’’ spectrum.

5. Conclusions

We have compared here two factorization methods in vogue lately, namely, SUSY and De La Peña's (the S-DLP linkage). We were able to establish a relation between them that yields some new insight into supersymmetry. Given a Hamiltonian 𝐻 that is to be factorized via the creation-destruction operators 𝑄^±, the above linkage shows that 𝐻 gets partitioned into a linear combination of two DLP-terms, involving the DLP-operators 𝑆 and 𝐴. The first one has the Hamiltonian aspect, but with the square of the superpotential as a pseudopotential, while 𝐴 refers to the way in which the 𝑄^± differ from HO's creation-destruction operators. We also see that the commutator of the supersymmetric creation-destruction operators becomes a nontrivial operator, linked to De La Peña's operator 𝐴. Summing up, we have shown that by linking two factorization techniques one can give any one-dimensional Hamiltonian the same form in terms of quantities typical of these approaches.

Acknowledgment

This work was partially supported by the MEC Grant FIS2008-00781 (Spain) and by the Projects FQM-2445 and FQM-207 of the Junta de Andalucia (Spain, EU).