Abstract
We introduce a class of potentials for which the time-dependent Schrödinger equation with position-dependent (effective) mass allows reduction to a stationary Schrödinger equation. This reduction is done by a particular point canonical transformation which preserves -normalizability.
1. Introduction
Schrödinger equations with effective mass occur in the context of transport phenomena in crystals (e.g., semiconductors), where the electrons are not completely free, but interact with the potential of the lattice. The quantum dynamics of such electrons can be modeled by an effective mass, the behavior of which is determined by the band curvature [1–3]. Since the effective mass Schrödinger equation takes a more complicated form than the conventional Schrödinger equation [4], the identification of solvable cases is more difficult. In the stationary case, particular potentials with effective mass have been studied mainly by means of point canonical transformations [5–7] and Darboux transformations (resp., supersymmetric factorization) [8–10]. Recently, these methods have also been elaborated for the fully time-dependent case [11–13]. However, the main problem of accessing time-dependent Schrödinger equations (TDSE) with effective mass is the lack of known solvable cases. In order to attack this problem for noneffective (constant) mass, it has been shown that for a certain class of potentials, the TDSE with constant mass can be mapped onto a stationary Schrödinger equation [14], such that each solvable stationary Schrödinger equation generates a solvable TDSE. Whereas in [14] the most general class of reducible potentials is derived, particular cases have been obtained earlier, for example, for time-dependent harmonic oscillator potentials with travelling-wave terms [15]. The method of mapping time-dependent onto stationary problems has also been used for the calculation of Green's functions for time-dependent Coulomb and other potentials [16]. The purpose of the present note is to generalize the results in [14] to the effective mass case. We identify a class of potentials for which the effective mass TDSE can be reduced to a stationary Schrödinger equation by means of a point canonical transformation. Thus, each solvable stationary Schrödinger equation gives rise to a solvable effective mass TDSE. This allows the straightforward generation of time-dependent potentials with effective masses and their corresponding solutions. Furthermore, our transformation preserves -normalizability, such that physical solutions are taken into physical solutions. In the remainder of this note, we first give the point canonical transformation that relates effective mass TDSEs and stationary Schrödinger equations to each other. We prove its correct simplification for constant mass and show that -normalizability of the solutions is preserved. In the final paragraph we present an application.
2. Reduction of the Effective Mass TDSE
2.1. The Point Canonical Transformation
Let us consider the stationary Schrödinger equation
where the mass is a positive constant, denotes the potential with constant energy , and is the solution. Next, consider the effective mass TDSE
where is the real-valued and positive mass, stands for the potential, and is the solution. Suppose that the potential in the effective mass TDSE can be expressed in the form
where the following abbreviations have been used:
Then the solutions of the Schrödinger (1) and (2) are related to each other via the following point canonical transformation:
where the functions , , and read as follows:
with arbitrary, real-valued , and . The latter statement is proved in a straightforward manner by substitution of (6) with (7)–(9) into the effective mass TDSE (2). Thus, each effective mass TDSE with a potential of the form (3) can be reduced to a stationary Schrödinger equation by means of the point canonical transformation (6). Hence, if the stationary Schrödinger (1) is solvable, so is its effective mass counterpart (2).
2.2. The Constant Mass Case
It has been shown that the TDSE with constant mass allows reduction to a stationary Schrödinger equation, if its potential takes a certain form [14]. Let us now verify that our potential simplifies correctly to this form if we assume the mass to be a constant. Thus, suppose that is constant and note that . From (5), we then infer that and . Now, it is easy to see that the potential (3) for constant takes the form
which coincides with the potential given in [14].
2.3. Preservation of -Normalizability
In this paragraph, we show that the mapping (6) preserves -normalizability. To this end, we assume that the solution of the effective mass TDSE (2) has domain , and infer from (6) and (7) that
Suppose now that the ratio is not constant or purely time-dependent. We change the integration variable from to , where the latter is given in (8):
Thus, if is -normalizable on , then is -normalizable on . Now, if the ratio is constant or purely time-dependent, the integral evaluation in (12) is invalid. Note that a constant ratio implies that the mass has the form of an exponential, that is,
where and are purely time-dependent or constants. Insertion of the latter mass into (12) gives
As in (13), if is -normalizable on , then is -normalizable on .
2.4. Application
We now apply the scheme developed above as follows. First, we choose a potential , such that the associated stationary Schrödinger equation (1) is exactly solvable, and calculate its solution. In the second step, we choose an effective mass and fix the free parameters that appear in (3). By means of this data, we compute the transformation components that are defined in (7)–(9). In the final step, we obtain the time-dependent potential (3) and its solution (6). Thus, let us start with the stationary potential , which we choose to be a harmonic oscillator:
The associated stationary Schrödinger equation (1) has the following solutions that are -normalizable on any subset of :
Here, denotes the Hermite polynomial of order , where is a natural number or zero [17]. The corresponding energies read
Now let be an arbitrary, real, and positive function and set
On substituting the latter settings into (7)–(9), we get
We now insert (16) and (19) into the time-dependent potential (3), which simplifies to
Thus, the solution of the time-dependent effective mass Schrödinger equation (2) with exponential mass given in (19) and potential (21) can be obtained by insertion of (17), (18), and (20) into (6):
Let us now exemplarily verify that the solution as given in (22) is -normalizable. For the sake of simplicity, we restrict ourselves to the case and . It is well known that the functions as given in (17) are -normalizable on the whole real line. In this application, we will restrict the domain of to , which preserves normalizability of . Note that now a possible boundary condition at zero has to be taken into account, as the energy spectrum (18) depends on it. From the definition of the Hermite polynomials [17], it follows immediately that
Thus, depending on the boundary condition imposed at zero, some values of the spectrum (18) must be omitted. In the present application, we do not require boundary conditions, since we focus on the transformation (20). Hence, the functions are -normalizable on and we find for that
Before we can compute the norm of the solution as given in (22), according to (11), we need to find the domain . Since , by means of in (20), we must have . We find
On comparing (24) and (25), we get
We compare this with (15). According to (5), for we have , and according to (14), we have , which gives
coinciding with the factor in (26).
3. Concluding Remarks
The point canonical transformation (6)–(9) introduced in this note provides a method for generating solvable time-dependent potentials for effective mass TDSEs. On one hand, such potentials are interesting in themselves, on the other hand, they can be used to find more potentials of their kind, for example, by looking for their supersymmetric partner potentials.