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Geometric Methods in Physics. XXX Workshop 2011
Trends in Mathematics, 229–237
c 2013 Springer Basel
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Factorization Method and the
Position-dependent Mass Problem
Sara Cruz y Cruz
To Professor Bogdan Mielnik, for all his contributions in Physics
Abstract. The dynamics of position-dependent mass systems is considered
from both, classical and quantum mechanical points of view, by means of the
factorization method. Some examples are presented, with particular choices of
the mass function, for the harmonic oscillator in order to illustrate our results.
In the quantum regime, new isospectral position-dependent mass potentials
are also constructed by the intertwining technique.
Mathematics Subject Classification (2010). 81Q60; 34L10.
Keywords. Position-dependent mass, factorization method, isospectral potentials.
1. Introduction
The problem of describing the motion of systems endowed with position-dependent
mass (PDM) has attracted interest since they appear in many physical problems.
These include, e.g., the study of the electronic properties of semiconductors [1–3],
quantum dots [4], the description of the dynamics of non linear oscillators [5, 6]
as well as classical systems in curved spaces [7], just to mention few ones. The
very concept of a PDM system is a fundamental problem which is far from being completely understood. Many contributions have been developed over the last
years in different approaches [8–19]. In the quantum mechanical regime, it is well
known that an ambiguity in ordering of the mass and the momentum operators
appears and the goal is to choose the proper Hamiltonian. Some arguments have
been given to this respect, e.g., the Galilean invariance [8] and the correspondence
between classical and quantum PDM potentials [16]. In some other cases the ordering is fixed by the boundary conditions imposed on a particular system [19].
The generation of exactly solvable PDM problems has also been considered. The
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factorization method [20–22] has been explored in [10–17]. In this work we present
the factorization method applied to the solution of the PDM problem in the classical as well as in the quantum mechanical frames. The paper is organized as follows.
In Section 2 the classical case is considered and some examples are presented for
the harmonic oscillator algebra. In Section 3 the quantum mechanical problem is
discussed and some new PDM potentials isospectral to the harmonic oscillator are
constructed. We end this contribution with some general remarks.
2. Classical position-dependent mass systems
Consider the classical position-dependent mass system described by the standard
Hamiltonian
𝑝2
+ 𝒱(𝑥)
(1)
ℋ=
2𝑚(𝑥)
where 𝑥 and 𝑝 are the canonical variables of position and linear momentum. The
mass 𝑚(𝑥) > 0 and the potential 𝒱(𝑥) are position-dependent functions setting
the domain of definition 𝒟(ℋ) of the Hamiltonian. The problem can be addressed
from two points of view: in the first one 𝒱 and 𝑚 are known and the phase space
motion is determined by reducing the PDM problem to an equivalent CM one;
in the second case, it is assumed that there is an algebraic structure fixing the
potential and the phase space trajectories in terms of 𝑚(𝑥) [23] (see also [16]). In
this work, the second approach is considered: the explicit form of the potential as
well as the dynamics are determined from the algebraic properties of the system,
by the factorization method.
Suppose that the Hamiltonian ℋ can be factorized in terms of two complex
functions [23]
𝑝
+ 𝒲(𝑥)𝜙(ℋ)
(2)
𝒜± = ∓𝑖𝑓 (𝑥) √
2𝑚(𝑥)
in the form
ℋ = 𝒜− 𝒜+ + 𝜖 = 𝒜+ 𝒜− + 𝜖,
(3)
with 𝜖 the factorization constant, 𝑓 , 𝒲 functions of the position and 𝜙 a function
of the energy of the system.
Suppose, additionally, that 𝒜± , ℋ close the following algebra in terms of
Poisson brackets
{ ± }
{ − +}
(4)
= 𝑖𝛾𝜙(ℋ),
𝒜 , ℋ = ±𝑖𝛾𝜙(ℋ)𝒜± ,
𝒜 ,𝒜
where 𝛾 is a constant. Observe that two complex-conjugate, non autonomous integrals of motion can be constructed in the form
𝒬± = 𝒜± 𝑒∓𝑖𝛾𝜙(ℋ)𝑡 ,
± 2
(5)
whose values 𝑞 ± fulfill√𝑞 − 𝑞 + = ∣𝑞 ∣ = ℰ −𝜖, ℰ being the total energy of the system.
Thus, making 𝑞 ± = ℰ − 𝜖𝑒±𝑖𝜑0 , the phase space trajectories can be written in
Factorization Method and the Position-dependent Mass Problem
terms of two parameters (ℰ, 𝜑0 ) as
(√
)
ℰ −𝜖
−1
𝑥(𝑡) = 𝒲
cos (𝛾𝜙(ℋ)𝑡 + 𝜑0 ) ,
𝜙(ℋ)
1 √
2 (ℰ − 𝜖) 𝑚(𝑥) sin (𝛾𝜙(ℋ)𝑡 + 𝜑0 ) .
𝑝(𝑡) = −
𝑓 (𝑥)
231
(6)
(7)
As an example, let us consider the harmonic oscillator of frequency 𝜔. One can
find that for this simple system 𝑓 (𝑥) = 1, 𝜙(ℋ) = 1 and 𝛾 = 𝜔, leading to
√
(∫
)2
∫
𝑚0 𝜔 2
𝑚0 𝜔 2
𝒲(𝑥) =
𝐽(𝑥)𝑑𝑥,
𝒱(𝑥) =
𝐽(𝑥)𝑑𝑥 + 𝜖
(8)
2
2
√
with 𝐽(𝑥) = 𝑚(𝑥)/𝑚0 and 𝑚0 a constant with dimensions of mass. Hence, under
the transformation
∫
𝒫(𝑥, 𝑝) = 𝑝/𝐽(𝑥),
𝒳 (𝑥) = 𝐽(𝑥)𝑑𝑥
(9)
the Hamiltonian takes the form of a CM harmonic oscillator of position 𝒳 and
momentum 𝒫. Note, however, that for some choices of 𝑚(𝑥) the transformation
(9) may not map 𝒟(ℋ) onto the whole real line as required if 𝒳 should represent
the position of the CM oscillator [16], meaning that there are important differences
between PDM and CM problems for those cases. Below, we will consider two mass
functions in order to illustrate this approach.
In the first place consider the regular mass 𝑚1 leading to the potential 𝒱1
𝑚1 (𝑥) =
𝑚0
,
1 + (𝑘𝑥)2
𝒱1 (𝑥) =
𝑚0 𝜔 2
arcsinh2 𝑘𝑥
2𝑘 2
(10)
with 𝑘 a constant in inverse position units (observe that the case of constant mass
is recovered in the limit 𝑘 → 0). In this case we have
⎡√
⎤
2(ℰ − 𝜖) 𝑘
1
cos (𝜔𝑡 + 𝜑0 )⎦
(11)
𝑥1 (𝑡) = sinh ⎣
𝑘
𝑚0 𝜔
√
2𝑚0 (ℰ − 𝜖)
𝑝1 (𝑡)− =
sin (𝜔𝑡 + 𝜑0 ) .
(12)
1 + (𝑘𝑥(𝑡))2
Figure 1 shows the potential 𝒱1 and some phase trajectories for different values of the total energy of the system. One can note that they are soft deformations
of that of the CM oscillator, with the position and momentum taking, in principle,
arbitrary values.
Next, we consider the singular mass 𝑚2 with potential 𝒱2
𝑚2 (𝑥) =
𝑚0
,
(𝑘𝑥)2
𝒱2 (𝑥) =
𝑚0 𝜔 2 2
ln 𝑘𝑥
2𝑘 2
(13)
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S. Cruz y Cruz
Figure 1. The potential and phase space trajectories for 𝑚1 with ℰ =
0.1, 0.3, 0.5 and 𝑚2 for ℰ = 0.1, 0.2, 0.3. In these graphics 𝑚0 = 2, 𝑘 = 2,
𝜔 = 0.8, 𝜖 = 0.5 and 𝜙 = 𝜋. Inner curves correspond to lower energies.
for which
⎡√
⎤
2(ℰ − 𝜖) 𝑘
1
exp ⎣
cos (𝜔𝑡 + 𝜑0 )⎦
𝑘
𝑚0 𝜔
√
2𝑚0 (ℰ − 𝜖)
sin (𝜔𝑡 + 𝜙0 ) .
𝑝2 (𝑥) = −
𝑘𝑥(𝑡)
𝑥2 (𝑡) =
(14)
(15)
Figure 1 shows the potential and phase trajectories for 𝑚2 . In contrast to the
previous case, it is evident the presence of a singularity, confining the motion of
the system to a region given by the domain of definition of 𝑚(𝑥). It is worthwhile
to mention that, even the unusual form of the mass, the behavior of the phase
space variables is quite regular. The presence of a divergence in the mass function
appears as a potential barrier suggesting that one can define oscillators in bounded
domains by introducing masses with singularities.
3. Quantum position-dependent mass systems
In the quantum mechanical regime, it is well known that the canonical variables
𝑥, 𝑝 do not commute and an ambiguity ordering appears in expressions containing
products of these variables. A general hermitian Hamiltonian in this case can be
defined as
1
1
𝐻𝑎 = 𝑚𝑎 𝑝 𝑚2𝑏 𝑝 𝑚𝑎 + 𝑉𝑎 (𝑥),
(16)
𝑎+𝑏=− ,
2
2
with 𝑎 the ordering parameter (𝑏 = −𝑎 − 1/2). As mentioned before, the choice
of this parameter has been addressed in several ways [8, 16, 19]. In this work it is
kept arbitrary, with no more assumptions on a particular ordering of 𝑝 and 𝑚.
Similar to the classical case, the form of the potential is found from the algebraic
structure underlying the system. Therefore, the eigenvalue equation
𝐻𝑎 𝜓(𝑥) = 𝐸𝜓(𝑥)
(17)
for which the spectrum is well known, can be studied by means of the factorization
method.
Factorization Method and the Position-dependent Mass Problem
233
Suppose then that 𝐻𝑎 can be factorized in terms of two linear operators
𝑖 𝑎
𝑖 𝑏
𝑏
𝑎
𝐴+
𝐴−
𝑎 = − √ 𝑚 𝑝 𝑚 + 𝑊𝑎 (𝑥),
𝑎 = √ 𝑚 𝑝 𝑚 + 𝑊𝑎 (𝑥)
2
2
in the form
−
𝐻𝑎 = 𝐴+
𝑎 𝐴𝑎 + 𝜖.
In the position representation 𝑝 = −𝑖ℏ𝑑/𝑑𝑥; hence, defining the differential
ator
𝑑
1
,
D= √
𝑚(𝑥) 𝑑𝑥
one may write
(
)
√
ℏ
1
√
𝐴+
D
+
=
−
2ℏ
𝑎
+
D (ln 𝐽(𝑥)) + 𝑊𝑎 (𝑥)
𝑎
2
2
√
ℏ
𝐴−
2ℏ𝑎D (ln 𝐽(𝑥)) + 𝑊𝑎 (𝑥).
𝑎 = √ D+
2
(18)
(19)
oper(20)
(21)
(22)
It is not difficult to show that the function 𝑊𝑎 (𝑥) must satisfy the Riccati equation
(
)
√
1
ℏ
(23)
− √ D𝑊𝑎 + 2 2ℏ 𝑎 +
(D ln 𝐽) 𝑊𝑎 + 𝑊𝑎2 = 𝑉𝑎 − 𝜖
4
2
while
(
)
1
= 2ℏD𝑊𝑎 + 2ℏ 𝑎 +
(24)
D2 ln 𝐽.
4
For the case in which the factorizing operators close the harmonic oscillator alge+
bra, i.e., [𝐴−
𝑎 , 𝐴𝑎 ] = ℏ𝜔, we have
√
(
)
∫
√
𝑚0 𝜔 2
1
𝑊𝑎 (𝑥) =
𝐽(𝑥)𝑑𝑥 − 2ℏ 𝑎 +
D ln 𝐽(𝑥),
(25)
2
4
[
+
𝐴−
𝑎 , 𝐴𝑎
]
fixing 𝑉𝑎 (𝑥) as
√
2
(
)2
)
1
2
𝐽(𝑥)𝑑𝑥 + ℏ 𝑎 +
D2 ln 𝐽(𝑥)
4
(
)2
1
− 2ℏ2 𝑎 +
(D ln 𝐽(𝑥))2 ,
(26)
4
) }
{
(
which is isospectral to the CM harmonic oscillator: 𝑆𝑝(𝐻𝑎 ) = 𝐸𝑛 = 𝑛 + 12 ℏ𝜔 ,
and lead to wave functions 𝜓𝑛 (𝑥) given by
1 ( + )𝑛
𝐴𝑎 𝜓0 (𝑥)
(27)
𝜓𝑛 (𝑥) = √
𝑛!
𝑚0 𝜔 2
𝑉𝑎 (𝑥) =
2
(∫
where 𝜓0 (𝑥) is the ground state defined by 𝐴−
𝑎 𝜓0 (𝑥) = 0.
At this point, it is important to stress that the subscript 𝑎 in 𝑉𝑎 distinguishes
different potentials for different orderings of the kinetic term. However, the Hamiltonian 𝐻𝑎 is the same for any value of 𝑎, and the subscript only labels different
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S. Cruz y Cruz
orderings of 𝑝 and 𝑚. Therefore, neither the spectrum, nor the eigenfunctions of
𝐻𝑎 should depend on 𝑎. Indeed, the substitution of (25) into (18) gives
√
∫
𝑚0 𝜔 2
ℏ
ℏ
±
±
𝐽𝑑𝑥
(28)
𝐴𝑎 = 𝐴 = ∓ √ D ± √ (D ln 𝐽) +
2
2
2 2
which are actually independent of the ordering parameter (see [17]). Note also that
)
(
√
∫
𝑚0 𝜔 2
ℏ
± 1/2
1/2
𝐴 𝐽
𝐽𝑑𝑥 = 𝐽 1/2 a± ,
∓√ D +
=𝐽
(29)
2
2
of the CM harmonic oscillator
where we can identify to a± as the
∫ ladder operators
√
𝑑
by making the correspondence 𝐽𝑑𝑥 → 𝑦(𝑥), 𝑚0 D → 𝑑𝑦
.
√
In this way, if 𝜓0 (𝑥) = 𝐽(𝑥)𝜙0 (𝑦(𝑥)), then 𝜙0 (𝑦) must satisfy
(
)
√
ℏ
𝑑
𝑚0 𝜔 2
√
+
𝑦 𝜙0 (𝑦) = 0
(30)
2
2𝑚0 𝑑𝑦
which is nothing but the equation defining the ground state of the CM harmonic
oscillator. The whole set of wave functions 𝜓𝑛 (𝑥) are hence constructed as
(∫
)
1/2
𝜓𝑛 (𝑥) = 𝐽 (𝑥)𝜙𝑛 𝐽(𝑥)𝑑𝑥 ,
(31)
with 𝜙𝑛 (𝑦) the wave functions of the constant mass harmonic oscillator, consistently with the point canonical transformation [9]. Some plots of potential and
corresponding wave functions are presented in Figure 2.
Figure 2. Position-dependent mass potentials and its corresponding
first 4 wave functions for masses 𝑚1 (left) and 𝑚2 (right). Observe
that the potentials depend on the ordering parameter 𝑎, upper curves
correspond to smaller values of 𝑎. Note though, that the wave functions
are the same for any value of 𝑎. Here we have used 𝑚0 = 2, 𝑘 = 2,
𝜔 = 0.8 and 𝑎 = 0, 0.25, 0.35, 0.5.
Observe that the PDM harmonic oscillator Hamiltonian 𝐻𝑎 can be also factorized as
ℏ𝜔
.
(32)
𝐻𝑎 = 𝐴− 𝐴+ −
2
Factorization Method and the Position-dependent Mass Problem
235
It is well known, for the CM case, that the operators 𝐴± are not unique [20]. It is
not difficult to prove this fact also for the PDM potentials, indeed, 𝑊𝑎 (𝑥) fulfills
the Riccati equation
(
(
)
)
√
ℏ
1
ℏ𝜔
1
2
2
√ D𝑊𝑎 + 2 2ℏ 𝑎 +
(33)
(Dln𝐽)𝑊𝑎 + 𝑊𝑎 = 𝑉𝑎 − 2ℏ 𝑎 +
D2 ln𝐽 +
4
4
2
2
with the general solution
√
(
)
∫
√
𝑚0 𝜔 2
1
𝑊𝑎 (𝑥, Γ) =
𝐽(𝑥)𝑑𝑥 − 2ℏ 𝑎 +
D ln 𝐽(𝑥)
2
4
]
[
√
∫ ∫𝐽𝑑𝑥
𝑚 𝜔
𝑚0 𝜔
ℏ2
− ℏ0 𝑡2
𝑒
𝑑𝑡 ,
(34)
+ √ D ln Γ +
ℏ
2
0
leading to new (𝑎-independent) operators
[
]
√
∫ ∫𝐽𝑑𝑥
2
𝑚0 𝜔 2
𝑚
𝜔
ℏ
0
𝐵 ± = 𝐴± + √ D ln Γ +
𝑒− ℏ 𝑡 𝑑𝑡
ℏ
2
0
(35)
such that 𝐻𝑎 = 𝐵 − 𝐵 + − ℏ𝜔/2. It is clear that these operators do not close the
˜ 𝑎 by apHeisenberg algebra, meaning that we can construct new Hamiltonians 𝐻
plying a Darboux transformation [20]
with
˜ 𝑎 (Γ) = 𝐵 + 𝐵 − + ℏ𝜔 = 1 𝑚𝑎 𝑝𝑚2𝑏 𝑝𝑚𝑎 + 𝑉˜𝑎 (𝑥, Γ)
𝐻
2
2
[
𝑉˜𝑎 (𝑥, Γ) = 𝑉𝑎 (𝑥) − ℏ2 D2 ln Γ +
which is non singular whenever ∣Γ∣ >
shown below.
√
𝜋
2 .
√
𝑚0 𝜔
ℏ
∫
0
∫
𝐽𝑑𝑥
𝑒
(36)
]
𝑚 𝜔
− ℏ0 𝑡2
𝑑𝑡 ,
(37)
Some plots for the new potentials are
Figure 3. Some new PDM potentials isospectral to the harmonic oscillator for different choices of the new parameter Γ. Plots on (a) correspond to 𝑚1 while those in (b) to 𝑚2 . In this graphics 𝑚0 = 2,
𝑘 = 2,𝜔 = 0.8, 𝑎 = 0, Γ = 0.75, 0.8, 1 and Γ → ∞.
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S. Cruz y Cruz
Additionally, both Hamiltonians show the intertwining relations 𝐻𝑎 𝐵 − =
˜
˜ 𝑎 𝐵 + , and the wave functions 𝜃𝑛 (𝑥) of 𝐻
˜ 𝑎 can be easily con𝐵 𝐻𝑎 , 𝐵 + 𝐻𝑎 = 𝐻
±
structed by the application of 𝐵 on the wave functions of 𝐻𝑎 :
−
𝜃𝑛 (𝑥) = 𝐵 + 𝜓𝑛−1 (𝑥),
𝑛 = 1, 2, 3, . . .
(38)
corresponding to the spectral values 𝐸𝑛 . There is, though, an isolated eigenvector
˜ 𝑎 , orthonormal to the whole set {𝜃𝑛 (𝑥), 𝑛 = 1, 2, . . .}, but not connected
𝜃0 (𝑥) of 𝐻
to {𝜓𝑛 (𝑥), 𝑛 = 0, 1, 2, . . .} by 𝐵 ± defined as
𝐵 − 𝜃0 (𝑥) = 0,
(39)
and corresponding to the eigenvalue 𝐸0 [20].
4. Concluding remarks
We have considered the PDM harmonic oscillator from classical and quantum mechanical points of view. In both cases the problem was addressed by means of the
factorization method. The technique is consistent with the point canonical transformation. Some examples were presented in order to show the effect of a regular
and singular variable mass in the dynamics of the system. In the quantum case,
the solution was given for a generalized ordering between 𝑚 and 𝑝. New potentials,
isospectral to the CM harmonic oscillators, were obtained from the intertwining
relations. The factorization method can be also generalized for different underlying algebraic structure of both, classical and quantum PDM problems [23]. In the
quantum case, new PDM supersymmetric partners can be also defined [22,24], and
different families of PDM coherent states can be constructed [25]. Results of these
generalizations can be found elsewhere [26].
Acknowledgment
This work was completed with the support of Projects
SIP20113705 and SIP20111061
of IPN-Mexico. The author thanks the Organizers of the XXX Workshop on Geometric Methods in Physics for the invitation to participate in the Conference, and
for the kind hospitality at the Bia̷lowieża Forest.
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Sara Cruz y Cruz
SEPI-UPIITA, Instituto Politécnico Nacional
Av. Instituto Politécnico Nacional 2580
La Laguna Ticomán, CP 07340
México D.F., Mexico
e-mail: [email protected]