Supersymmetric Partners of the One-Dimensional Infinite Square Well Hamiltonian

The study of one dimensional models in quantum mechanics is useful in order to gain a better understanding of the properties of quantum systems. In particular, the construction of supersymmetric (SUSY) partners of given potentials allow for an analysis of one dimensional Hamiltonians that often keep similarities with the original ones. Many studies have been done in this field and a brief account of references [1,2,3,4,5,6,7,8,9,10,11,12,13,14] only covers a small part of all previous work.

In the present paper, we intend to investigate the properties of the SUSY partners of the self-adjoint determinations of the operator $-d^2/d{x}^2$ on $L^2[-a,a]$, $a>0$ and finite, with appropriate boundary conditions at the points $-a$ and a. Note that this problem is closely related to the problem of the definition of the “free” Hamiltonian on the one dimensional infinite square well potential.

From our point of view, SUSY quantum mechanics is a method that pursues the identification of the class of Hamiltonians for which their spectral problem can be algebraically solved. Traditionally, people have investigated SUSY partners of well studied exactly solvable Hamiltonians that give rise to other Hamiltonians for which the spectrum coincides with the spectrum of the original Hamiltonian except for one eigenvalue. In addition, there are several examples in which one original Hamiltonian produces an infinite chain of Hamiltonians, the first element of the chain being its SUSY partner and each of the others is a partner of the previous and the next one. Here, we explore the possibility of obtaining the whole chain of partners corresponding to self-adjoint extensions of a symmetric one dimensional Hamiltonian with equal deficiency indices. Since in our case, the variety of self-adjoint extensions is quite wide, depending on four real parameters, we have expected to find interesting new results in the field as it happened to be.

The analysis of these self-adjoint extension has been done in [15]. The task of computing the SUSY partners of all the self-adjoint determinations (also called extensions) of $-d^2/d{x}^2$ on $L^2[-a,a]$, their spectra and their wave functions is not trivial, although can be carried out systematically.

Although the idea of self-adjoint extensions of symmetric (or Hermitian) operators on (infinite dimensional) Hilbert spaces is not yet very popular among physicists, it is, however, possible to find recent papers on the topic [16,17,18,19,20,21,22,23,24]. Standard quantum mechanics textbooks refer to the one dimensional infinite square well potential or the harmonic oscillator as if they were described by a unique self-adjoint Hamiltonian, which produces a neatly calculable spectrum. The mathematical reality is much more complex and may give many more possibilities for the study of quantum mechanics systems. Let us briefly address to this problem, for which a more thorough presentation can be found in mathematical textbooks [25] as well as papers addressed to the Physics community [15].

Concerning terminology, an operator, A, on a infinitely dimensional separable Hilbert space (the Hilbert space must be infinite dimensional, since otherwise all operators are continuous and defined on the whole space. In such a case, this argumentation does not make sense. A separable Hilbert space is one with a countable orthonormal basis, which is always the case in ordinary quantum mechanics) $\mathcal{H}$ is symmetric, or equivalently Hermitian if for any pair of vectors $\phi ,\psi \in \mathcal{D}\left(A\right)$, where $\mathcal{D}\left(A\right)$ is the domain of A, which must be densely defined, one has that $\langle A\phi |\psi \rangle =\langle \phi |A\psi \rangle$, where $\langle -|-\rangle$ denotes the scalar product on $\mathcal{H}$. This means that the adjoint, $A^{†}$, of A extends A, $A\prec A^{†}$ (i.e., $\mathcal{D}\left(A\right)\subset \mathcal{D}\left(A^{†}\right)$ and $A\psi =A^{†}\psi$, for all $\psi \in \mathcal{D}\left(A\right)$). The deficiency indices are $n_{\pm }:=dim\mathrm{Ran}(A^{†}\pm iI)$, where $\mathrm{Ran}\left(B\right)$ is the range (image space) of the operator B and I is the identity operator. A symmetric (or Hermitian) operator has self-adjoint determinations (or extensions) if and only if $n_{+}=n_{-}$ [25]. If $n_{+}=n_{-}=0$, this extension is unique. On the other hand, if $n_{+}=n_{-}\ne 0$, the number of extensions is infinite and, in the case of Hilbert spaces of functions, they usually can be determined by some matching or boundary conditions that the functions in the domain of the extensions should fulfill at some points [15,25,26,27].

Self-adjoint determinations of the operator $-d^2/d{x}^2$ defined on functions supporting whatever interval, $\mathcal{K}$, in the real line $\mathbb{R}$ are used to define the so call contact potentials [26,28,29,30,31]. These are perturbations of the “free operator” $H_0=-d^2/d{x}^2$, which are supported on a single point $x_0\in \mathcal{K}$. Typical examples of contact potentials are the Dirac delta $\delta (x-x_0)$ or its derivative ${\delta }^{\prime }(x-x_0)$, which define Hamiltonians of the type $H_0+\delta (x-x_0)$ or $H_0+{\delta }^{\prime }(x-x_0)$ as well defined self-adjoint operators on the Hilbert space $L^2\left(\mathcal{K}\right)$ [27]. These types of perturbations may serve as a good and tractable approximation for a very localized spatial perturbation and are defined via matching conditions that must satisfy the functions on the domain of the operator at $x_0$. Concerning the operator $-d^2/d{x}^2$ on $L^2[-a,a]$, some relations have been found among the boundary conditions at the borders $-a$ and a and matching conditions defining a $\delta$ or ${\delta }^{\prime }$ perturbation at the origin [32,33].

A comment is of relevance here. Let us consider the subspace, ${\mathcal{D}}_0$, of all twice differentiable square integrable functions, $\phi \left(x\right)$, in the interval $[-a,a]$, with second derivative in $L^2[-a,a]$, verifying the boundary conditions $\phi (-a)=\phi \left(a\right)={\phi }^{\prime }(-a)={\phi }^{\prime }\left(a\right)=0$, and a differential operator of the form where $p_1\left(x\right)$ and $p_2\left(x\right)$ are continuous real functions (with $p_1\left(x\right)$ differentiable) on $[-a,a]$. Then D is Hermitian on ${\mathcal{D}}_0$ with deficiency indices (2, 2). It has been proven in [34] (vol. 2, p. 90) that all self-adjoint extensions of D have a purely discrete spectrum. This is precisely the case of all the self-adjoint determinations of $-d^2/d{x}^2$ under our study [15]. These self-adjoint extensions are characterized by a set of four real parameters, so that one particular choice of these parameters gives a unique self-adjoint determination of $-d^2/d{x}^2$ on $L^2[-a,a]$ and vice-versa. Although this is much less known, a similar situation emerges in the study of the one dimensional harmonic oscillator [35].

The present article intends in the first place, to complete as far as possible, the classification of the self-adjoint extensions of $-d^2/d{x}^2$ on $L^2[-a,a]$ given by [15]. Once this task has been done, we intend to obtain the whole chain of SUSY partners of each of the self-adjoint extensions using standard methods already developed in the theory [1]. This kind of supersymmetry intends to construct a series of potentials (in our case one-dimensional), with an energy spectrum closely related and that can be obtained from the spectrum of the original potential. Thus, being given one of our original self-adjoint extensions and being known the solution of the spectral problem, we should be able to obtain an infinite sequence of Hamiltonians such that their spectra coincides with the spectra of the previous one except for one eigenvalue, and hence from the original one except for a finite number of energy levels. We must add that all self-adjoint extensions of $-d^2/d{x}^2$ on $L^2[-a,a]$ have a purely discreet spectrum with an infinite number of energy levels.

The ground state for each of these extensions either has a strictly positive, zero or negative energy. Obviously, in the latter case, this fact comes from extensions which are not definitely positive. This is somehow paradoxical, due to the form of the original operator, which is $-d^2/d{x}^2$. This paradox is solved in [15]. For those extensions with a ground state with strictly positive energy, we have constructed the whole sequence of its SUSY partners and have given the eigenvalues and eigenfunctions for these partners. As mentioned earlier, the set of eigenvalues for each partner comes from the set of eigenvalues of the extension from which we construct the sequence of partners.

The general formalism can also be applied to obtain a sequence of Hamiltonians when the ground state of the original self-adjoint extension of $-d^2/d{x}^2$ on $L^2[-a,a]$ has zero or negative energy. In this case, partner Hamiltonians may be very different from the original one in the sense that they may have a finite number of eigenvalues or simply no eigenvalues. This is due to the presence of nodes in the wave function of the ground state. Nevertheless, these partners may be obtained and classified, although this discussion is left for a future publication.

This paper is organized as follows—in Section 2 we reformulate the classification given by [15] of the self-adjoint extensions of $-d^2/d{x}^2$ on $L^2[-a,a]$. In Section 3, we classify these extensions in terms of some other sets of parameters, not considered in [15]. In Section 4, we construct the first SUSY partners for those extensions with positive ground level energy and give the precise form of its eigenfunctions. In Section 5, we give the complete sequence of SUSY partners for each of these extensions. We close this article with a Conclusions Section and an Appendix in which we show what the correct form for the wave functions for the energy levels should be.

Let us go back to the differential operator $H_0:=-d^2/d{x}^2$ defined on $L^2[-a,a]$, $a>0$ and with domain ${\mathcal{D}}_0$ as above, just before (1). On ${\mathcal{D}}_0$, $H_0$ is symmetric (Hermitian) with deficiency indices (2, 2) [15]. According to the von-Neumann theorem [25], $H_0$ admits an infinite number of self-adjoint extensions labeled by four real parameters.

The adjoint operator $H_0^{†}$ acts as $-d^2/d{x}^2$ on the functions of its domain (see [36,37] for a definition of the domain of the adjoint of a given densely defined operator and its properties). If $\varphi$ is a function of such domain, we get integrating by parts: where $\langle -,-\rangle$ denotes the scalar product on $L^2[-a,a]$ and the prime being the derivative with respect to the variable x and the asterisk meaning complex conjugate. The self-adjoint extensions of $H_0$ are equal to $-d^2/d{x}^2$ as an operator acting on the subdomains of the domain of $H_0^{†}$ of functions with $B(\varphi ,\varphi )=0$. This happens if and only if there exists a $2\times 2$ unitary matrix U such that (see [15] and references quoted therein):

The set of self-adjoint extensions of $H_0$ is in one to one correspondence with the set of $2\times 2$ unitary operators U. Thus, each of these extensions will be labeled by its corresponding operator as $H_{\alpha }$. Since there is a set of four real independent parameters that characterize the set of operators U, then, the set of self-adjoint extensions of $H_{\alpha }$ is also characterized by the same parameters [15]. Each of the operators U has the following form [15]:

Here, $\psi$ and $m_i$, $i=0,1,2,3$ are real parameters so that $\psi \in [0,\pi ]$ and $m_0^2+m_1^2+m_2^2+m_3^2=1$, which means that only four parameters are independent [15]. The latter relation is a consequence of unitarity: the modulus of the determinant of U must be a number of modulus one.

There are some of these extensions with a clear physical interest, which does not mean that the others are irrelevant from the physics point of view. In [15], the authors distinguish three categories of extensions:

Apart from these three categories, there are some other extensions. The reason why the authors of [15] single out those extensions that preserve positivity is due to the existence of extensions with negative energies. In fact, as proven in [34] (Theorem 16, vol 2, page 44), $H_{\alpha }$ may have one (which may be doubly degenerate) or two (with no degeneration) negative energy states. All other extensions have non-negative eigenvalues and are called positivity preserving. Only three of these positivity preserving extensions with special simplicity are discussed in [15]. We want to determine the energy levels in this situation.

In order to obtain the energy levels for a specific self-adjoint extension, $H_{\alpha }$, of $H_0=-d^2/d{x}^2$ on $L^2[-a,a]$, we have to solve the Schrödinger equation and impose on its solutions the boundary conditions that characterize the extension. These boundary conditions are given by the (4) and (6). However as stated in [15], the determination of which operators U satisfy the positivity condition as stated before involves tedious considerations. To circumvent this difficulty, let us consider the general solution of the time independent Scrödinger equation $-d^2\varphi \left(x\right)/d{x}^2=E\varphi \left(x\right)$, with $E=s^2/{\left(2a\right)}^2\ge 0$, where $2a$ is the infinite square well width (Although the energy is given, in our notation, by ${\hslash }^{2}E/2m$, we are calling “energy” the quantity represented by E.). This general solution is

Here, A and B have to be fixed with two conditions: (i) $\varphi \left(x\right)$ should be normalized in $L^2[-a,a]$ and (ii) $\varphi \left(x\right)$ should fulfill the boundary conditions (4) and (5) so that $E\ge 0$. Let us use (6) in relation (4) giving the general matching conditions, so as to obtain the following homogeneous linear system: where

The eigenvalues ${\lambda }_{\pm }\left(s\right)$ of the matrix $\mathcal{N}\left(s\right)$ are given by

The trace and the determinant of $\mathcal{N}\left(s\right)$ can be easily calculated and are, respectively: and

To begin with, let us remark that in order to have non-trivial solutions of (7) we must have

Then, the set of eigenvalues of $\mathcal{N}\left(s\right)$ is given by $Tr\left(\mathcal{N}\right(s\left)\right)$ and 0, as may be immediately seen from (9). The condition (12) gives a relation between the values of the energy, determined by the real parameter s, since $E=s^2/{\left(2a\right)}^2$, and the parameters $\psi$, $m_0$ and $m_1$, as in (5). In consequence, the energy levels depend on the values of these three parameters only. From (11) and (12), we obtain the following two transcendental equations (one with plus sign and the other with minus sign):

This form of the transcendental equations is quite interesting, since it will serve as an efficient estimation of the energy levels when these values cannot be exactly calculated. Otherwise, they permit to obtain exact solutions whenever they exists. Let us now summarize three of the results provided by [15], which we will need later on:

One of the goals of our study is to solve the eigenvalue problem for all the self-adjoint extensions, $H_{\alpha }$, of the operator $H_0=-d^2/d{x}^2$ on $L^2[-a,a]$, which from the point of view of the physicist is the infinite square well with width $2a$. As we have already seen, there are only a few of these extensions for which we may obtain an exact solution, including the textbook extension. For most of these extensions the energy levels are solutions of a transcendental equation and, therefore, no explicit solutions of the eigenvalue problem for these extensions can be given.

In this section we shall consider the first and second order supersymmetry transformation applied to some of the self-adjoint extensions $H_{\alpha }$ so far considered in here.

Let us begin this Section with a summary of the notation employed so far and its meaning:

Creation ${\left(A^{\left(i\right)}\right)}^{†}$ and annihilation $A^{\left(i\right)}$ operators will be defined later.

So far, we have obtained potentials and wave functions for the first and second SUSY partners for self-adjoint extensions of $H_0=-d^2/d{x}^2$ with ground level of positive energy. With the help of the induction method, we may find potentials as well as wave functions and energy levels for arbitrary order ℓ SUSY partners for the same class of self-adjoint extensions. We have seen already that from the SUSY partners $V^{(i,1)},V^{(i,2)}$, only the last one is really interesting and we will focus on it in the sequel.

In order to apply the inductive method, let us assume that the ground state for the ℓ-th SUSY partner, $H^{(\ell ,2)}$, of $H_{\alpha }$ is given by as in the previous cases (41) and (44). Then, the super-potential takes the following form: where ${\partial }_x^2$ denotes the second derivative with respect to the variable x. Once we have the super potential, we readily obtain the partner potentials at $\ell +1$ order, which are

Note that although the label $\alpha$ is not written explicitly on the above equations and many others, potentials and wave functions must depend on $\alpha$. This dependence is hidden in $s_0$, where we have not made it explicitly for simplicity in the notation.

The Schrödinger equation for the first $(\ell +1)$-th order partner potential, $V^{(\ell +1,1)}$, is

If we change it to the z variable, (49) takes the form:

Equation (50) is a new Legendre-type equation for which solutions are known. The respective eigenfunctions and eigenvalues are