# Reduction of Time-Dependent Schrödinger Equations with Effective Mass to Stationary Schrödinger Equations

Authors: Axel Schulze-Halberg
Journal: Research Letters in Physics
 (2008)

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## 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 preservesL2-normalizability.


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## Body

## 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 L2-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 L2-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(1)12MΦ′′+(E−U)Φ=0, where the mass M is a positive
constant, U=U(x) denotes the
potential with constant energy E, and Φ=Φ(x) is the
solution. Next, consider the effective mass TDSE(2)iΨt+12mΨxx−mx2m2Ψx−VΨ=0,  where m=m(x,t) is the
real-valued and positive mass, V=V(x,t) stands for the
potential, and Ψ=Ψ(x,t) is the
solution. Suppose that the potential V in the
effective mass TDSE can be expressed in the form(3)V=UI32+2ABM3/2I1−2(AMI1)2−B2MI2−AMI1I2−18I22−7mx232m3+mxx8m2+∫((AmM+mt4m)I2+m2(I2)t+(2A′mM+AmtMm)I1+B′mM+Bmt2mM)dx+i(c′−AM)−B22M,  where the
following abbreviations have been used:(4)U=U((1M)I3I1+1M∫I3Bdt),(5)I1=∫mdx,I2=∫mtmdx,I3=exp(2M∫Adt).  Then the
solutions of the Schrödinger  (1) and
(2) are related to each other
via the following point canonical transformation:(6)Ψ(x,t)=exp(f(x,t)−iEv(t))Φ(u(x,t)),  where the
functions f, v, and u read as
follows:(7)f=∫(mx4m−i(m2(∫mtmdx)+2AmM(∫mdx)+BmM))dx+c,(8)u=1Mexp(2M∫Adt)(∫mdx)+1M∫exp(2M∫tAdt′)Bdt,(9)v=∫exp(4M∫Adt)dt,  with arbitrary,
real-valued A=A(t),B=B(t), and c=c(t). 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 m to be a
constant. Thus, suppose that m=M is constant and
note that mx=mt=0. From (5), we then infer that I1=Mx and I2=0. Now, it is easy to see that the potential (3) for
constant m=M takes the form(10)V=UI32+(A′−2A2M)x2+(B′−2ABM)x−B22M+i(c′−AM),  which coincides
with the potential given in [14].


### 2.3. Preservation ofL2-Normalizability

In this
paragraph, we show that the mapping (6)
preserves L2-normalizability.
To this end, we assume that the solution Ψ of the
effective mass TDSE (2) has domain D⊂ℝ, and infer from (6) and (7) that(11)∥Ψ∥2=∫D|exp(f−iEv)Φ|2dx=∫Dexp(∫xmx′2mdx′)|Φ|2dx.  Suppose now
that the ratio mx/m is not constant
or purely time-dependent. We change the integration variable from x to u, where the latter is given in (8):(12)∫Dexp(∫xmx′2mdx′)|Φ|2dx=∫u(D)m|Φ|2MI3mdu=∫u(D)m|Φ|21u′(x)du(13)=MI3∥Φ∥2. Thus, if Φ is L2-normalizable
on u(D), then Ψ is L2-normalizable
on D. Now, if the ratio mx/m is constant or
purely time-dependent, the integral evaluation in (12) is invalid. Note that a
constant ratio mx/m implies that
the mass has the form of an exponential, that is,(14)m=aexp(bx), where a and b are purely
time-dependent or constants. Insertion of the latter mass into (12) gives(15)∫Dexp(∫xmx′2mdx′)|Φ|2dx=∫u(D)exp(b2x)|Φ|21u′(x)du=∫u(D)exp(b2x)|Φ|2MI3mdu=∫u(D)exp(b2x)|Φ|2MI3aexp((b/2)x)du=MI3a∥Φ∥2.  As in (13), if Φ is L2-normalizable
on u(D), then Ψ is L2-normalizable
on D.


### 2.4. Application

We now apply
the scheme developed above as follows. First, we choose a potentialU, such that the associated stationary Schrödinger equation
(1) is exactly solvable, and calculate its solution. In the second
step, we choose an effective mass m and fix the
free parameters A,B,c that appear in
(3). By means of this data, we compute the transformation components f,u,v 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 U, which we choose to be a harmonic oscillator:(16)U=x2.  The associated
stationary Schrödinger equation (1) has the following solutions that are L2-normalizable
on any subset of ℝ:(17)Φn=exp(−M2x2)Hn((2M)1/4x).  Here, Hn denotes the
Hermite polynomial of order n, where n is a
natural number or zero [17]. The corresponding energies read(18)En=12M(2n+1).  Now let α=α(t) be an
arbitrary, real, and positive function and set(19)m=αexp(x),A=−Mα′4α,B=0,c=0.On substituting
the latter settings into (7)–(9), we get(20)f=14x,u=2Mexp(12x),v=∫1αdt.  We now insert (16)
and (19) into
the time-dependent potential (3), which simplifies to(21)V=4Mαexp(x)−332αexp(−x).  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):(22)Ψn=exp(−8Mexp(x)+14x−i(2n+1)2M∫1αdt)×Hn((32M)1/4exp(12x)).  Let us now
exemplarily verify that  the
solution as given in  (22) is
L2-normalizable.
For the sake of simplicity, we restrict ourselves to the case
n=0 and M=1/2. It is well known that the functions Φn as given in (17)
are L2-normalizable
on the whole real line. In this application, we will restrict the domain of Φn to (0,∞), which preserves normalizability of Φn. 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(23)Φn(0){>=<}0ifn={4k2k+14k+2},k∈ℕ∪{0}.  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 Φn are L2-normalizable
on (0,∞) and we find for Φ0 that(24)∥Φ0∥2=∫0∞exp(−x2)dx=12π.  Before we can
compute the norm of the solution Ψ as given in (22),
according to (11), we need to find the domain D. Since (0,∞)=u(D), by means of u in (20), we
must have D=ℝ. We find(25)∥Ψ0∥2=∫−∞∞exp(−8exp(x)+12x)dx=π8.  On comparing (24)
and (25), we get(26)∥Ψ0∥2=12∥Φ0∥2.  We compare this
with (15).
According to (5),
for M=1/2, we have I3=1/α, and according to (14), we have a=α, which gives(27)MI3a=12,coinciding with
the factor in (26).


## 2.1. The Point Canonical Transformation

Let us consider
the stationary Schrödinger equation(1)12MΦ′′+(E−U)Φ=0, where the mass M is a positive
constant, U=U(x) denotes the
potential with constant energy E, and Φ=Φ(x) is the
solution. Next, consider the effective mass TDSE(2)iΨt+12mΨxx−mx2m2Ψx−VΨ=0,  where m=m(x,t) is the
real-valued and positive mass, V=V(x,t) stands for the
potential, and Ψ=Ψ(x,t) is the
solution. Suppose that the potential V in the
effective mass TDSE can be expressed in the form(3)V=UI32+2ABM3/2I1−2(AMI1)2−B2MI2−AMI1I2−18I22−7mx232m3+mxx8m2+∫((AmM+mt4m)I2+m2(I2)t+(2A′mM+AmtMm)I1+B′mM+Bmt2mM)dx+i(c′−AM)−B22M,  where the
following abbreviations have been used:(4)U=U((1M)I3I1+1M∫I3Bdt),(5)I1=∫mdx,I2=∫mtmdx,I3=exp(2M∫Adt).  Then the
solutions of the Schrödinger  (1) and
(2) are related to each other
via the following point canonical transformation:(6)Ψ(x,t)=exp(f(x,t)−iEv(t))Φ(u(x,t)),  where the
functions f, v, and u read as
follows:(7)f=∫(mx4m−i(m2(∫mtmdx)+2AmM(∫mdx)+BmM))dx+c,(8)u=1Mexp(2M∫Adt)(∫mdx)+1M∫exp(2M∫tAdt′)Bdt,(9)v=∫exp(4M∫Adt)dt,  with arbitrary,
real-valued A=A(t),B=B(t), and c=c(t). 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 m to be a
constant. Thus, suppose that m=M is constant and
note that mx=mt=0. From (5), we then infer that I1=Mx and I2=0. Now, it is easy to see that the potential (3) for
constant m=M takes the form(10)V=UI32+(A′−2A2M)x2+(B′−2ABM)x−B22M+i(c′−AM),  which coincides
with the potential given in [14].


## 2.3. Preservation ofL2-Normalizability

In this
paragraph, we show that the mapping (6)
preserves L2-normalizability.
To this end, we assume that the solution Ψ of the
effective mass TDSE (2) has domain D⊂ℝ, and infer from (6) and (7) that(11)∥Ψ∥2=∫D|exp(f−iEv)Φ|2dx=∫Dexp(∫xmx′2mdx′)|Φ|2dx.  Suppose now
that the ratio mx/m is not constant
or purely time-dependent. We change the integration variable from x to u, where the latter is given in (8):(12)∫Dexp(∫xmx′2mdx′)|Φ|2dx=∫u(D)m|Φ|2MI3mdu=∫u(D)m|Φ|21u′(x)du(13)=MI3∥Φ∥2. Thus, if Φ is L2-normalizable
on u(D), then Ψ is L2-normalizable
on D. Now, if the ratio mx/m is constant or
purely time-dependent, the integral evaluation in (12) is invalid. Note that a
constant ratio mx/m implies that
the mass has the form of an exponential, that is,(14)m=aexp(bx), where a and b are purely
time-dependent or constants. Insertion of the latter mass into (12) gives(15)∫Dexp(∫xmx′2mdx′)|Φ|2dx=∫u(D)exp(b2x)|Φ|21u′(x)du=∫u(D)exp(b2x)|Φ|2MI3mdu=∫u(D)exp(b2x)|Φ|2MI3aexp((b/2)x)du=MI3a∥Φ∥2.  As in (13), if Φ is L2-normalizable
on u(D), then Ψ is L2-normalizable
on D.


## 2.4. Application

We now apply
the scheme developed above as follows. First, we choose a potentialU, such that the associated stationary Schrödinger equation
(1) is exactly solvable, and calculate its solution. In the second
step, we choose an effective mass m and fix the
free parameters A,B,c that appear in
(3). By means of this data, we compute the transformation components f,u,v 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 U, which we choose to be a harmonic oscillator:(16)U=x2.  The associated
stationary Schrödinger equation (1) has the following solutions that are L2-normalizable
on any subset of ℝ:(17)Φn=exp(−M2x2)Hn((2M)1/4x).  Here, Hn denotes the
Hermite polynomial of order n, where n is a
natural number or zero [17]. The corresponding energies read(18)En=12M(2n+1).  Now let α=α(t) be an
arbitrary, real, and positive function and set(19)m=αexp(x),A=−Mα′4α,B=0,c=0.On substituting
the latter settings into (7)–(9), we get(20)f=14x,u=2Mexp(12x),v=∫1αdt.  We now insert (16)
and (19) into
the time-dependent potential (3), which simplifies to(21)V=4Mαexp(x)−332αexp(−x).  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):(22)Ψn=exp(−8Mexp(x)+14x−i(2n+1)2M∫1αdt)×Hn((32M)1/4exp(12x)).  Let us now
exemplarily verify that  the
solution as given in  (22) is
L2-normalizable.
For the sake of simplicity, we restrict ourselves to the case
n=0 and M=1/2. It is well known that the functions Φn as given in (17)
are L2-normalizable
on the whole real line. In this application, we will restrict the domain of Φn to (0,∞), which preserves normalizability of Φn. 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(23)Φn(0){>=<}0ifn={4k2k+14k+2},k∈ℕ∪{0}.  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 Φn are L2-normalizable
on (0,∞) and we find for Φ0 that(24)∥Φ0∥2=∫0∞exp(−x2)dx=12π.  Before we can
compute the norm of the solution Ψ as given in (22),
according to (11), we need to find the domain D. Since (0,∞)=u(D), by means of u in (20), we
must have D=ℝ. We find(25)∥Ψ0∥2=∫−∞∞exp(−8exp(x)+12x)dx=π8.  On comparing (24)
and (25), we get(26)∥Ψ0∥2=12∥Φ0∥2.  We compare this
with (15).
According to (5),
for M=1/2, we have I3=1/α, and according to (14), we have a=α, which gives(27)MI3a=12,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.



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