Deformed quantum mechanics and Hermitian operators
Abstract
Starting on the basis of the noncommutative differential calculus, we introduce a generalized deformed Schrödinger equation. It can be viewed as the quantum stochastic counterpart of a generalized classical kinetic equation, which reproduces at the equilibrium the wellknown deformed exponential stationary distribution. In this framework, deformed adjoint of an operator and hermitian operator properties occur in a natural way in order to satisfy the basic quantum mechanics assumptions.
1 Introduction
In the recent past, there has been a great deal of interest in the study of quantum algebra and quantum groups in connection between several physical fields [1]. From the seminal work of Biedenharn [2] and Macfarlane [3], it was clear that the calculus, originally introduced in the study of the basic hypergeometric series [4, 5, 6], plays a central role in the representation of the quantum groups with a deep physical meaning and not merely a mathematical exercise. Many physical applications have been investigated on the basis of the deformation of the Heisenberg algebra [7, 8, 9, 10, 11]. In Ref.[12, 13] it was shown that a natural realization of quantum thermostatistics of deformed bosons and fermions can be built on the formalism of calculus. In Ref.[14], a deformed Poisson bracket, invariant under the action of the symplectic group, has been derived and a classical deformed thermostatistics has been proposed in Ref.[15]. Furthermore, it is remarkable to observe that calculus is very well suited for to describe fractal and multifractal systems. As soon as the system exhibits a discretescale invariance, the natural tool is provided by Jackson derivative and integral, which constitute the natural generalization of the regular derivative and integral for discretely selfsimilar systems [16].
In the past, the study of generalized linear and nonlinear Schrödinger equations has attracted a lot of interest because of many collective effects in quantum manybody models can be described by means of effective theories with generalized oneparticle Schrödinger equation [17, 18, 19, 20]. On the other hand, it is relevant to mention that in the last years many investigations in literature has been devoted to nonHermitian and pseudoHermitian quantum mechanics [21, 22, 23, 24, 25, 27].
In the framework of the Heisenberg algebra, a deformed Schrödinger equations have been proposed [28, 29]. Although the proposed quantum dynamics is based on the noncummutative differential structure on configuration space, we believe that a fully consistent deformed formalism of the quantum dynamics, based on the properties of the calculus, has been still lacking.
In this paper, starting on a generalized classical kinetic equation reproducing as stationary distribution of the wellknow exponential function, we study a generalization of the quantum dynamics consistently with the prescriptions of the differential calculus. At this scope, we introduce a deformed Schrödinger equation with a deformed Hamiltonian which is a nonHermitian operator with respect to the standard (undeformed) operators properties but its dynamics satisfies the basic assumptions of the quantum mechanics under generalized operators properties, such as the definition of adjoint and hermitian operator.
2 Noncommutative differential calculus
We shall briefly review the main features of the noncommutative differential calculus for real numbers. It is based on the following commutative relation among the operators and ,
(1) 
with a real and positive parameter.
A realization of the above
algebra in terms of ordinary real numbers can be accomplished by the
replacement [14, 30]
(2)  
(3) 
where is the Jackson derivative [4] defined as
(4) 
where
(5) 
is the dilatation operator. Its action on an arbitrary real function is given by
(6) 
The Jackson derivative satisfies some simple proprieties which will be useful in the following. For instance, its action on a monomial is given by
(7) 
and
(8) 
where and
(9) 
are the socalled basicnumbers. Moreover, we can easily verify the following version of the Leibnitz rule
(10)  
A relevant role in the algebra, as developed by Jackson, is given by the basicbinomial series defined by
(11)  
where
(12) 
is known as the binomial coefficient which reduces to the ordinary binomial coefficient in the limit [6]. We should remark that Eq.(12) holds for , while it is assumed to vanish otherwise and we have defined . Remarkably, a analogue of the Taylor expansion has been introduced in Ref. [4] by means of a basicbinomial (11) as
(13) 
where and so on.
Consistently with the calculus, we also introduce the basicintegration
(14) 
where and for whilst and for [5, 6, 15, 16]. Clearly, Eq.(14) is reminiscent of the Riemann quadrature formula performed now in a nonuniform hierarchical lattice with a variable step . It is trivial to verify that
(15) 
for any .
Let us now introduce the following deformed exponential function defined by the series
(16) 
which will play an important role in the framework we are
introducing. The function (16) defines the basicexponential, well known in the literature since a long time
ago, originally introduced in the study of basic hypergeometric
series [5, 6]. In this context, let us observe that
definition (16) is fully consistent with
its Taylor expansion, as given by Eq.(13).
The basicexponential is a monotonically increasing function,
, convex, , with and reducing to the ordinary exponential in the
limit: . An important property satisfied
by the exponential can be written formally as [6]
(17) 
where the left hand side of Eq.(17) must be considered by means of its series expansion in terms of basicbinomials:
(18) 
By observing that for any , since , from Eq.(17) we can see that [15]
(19) 
The above property will be crucial in the following introduction to a consistent deformed quantum mechanics.
3 Classical deformed kinetic equation
Starting from the realization of the algebra, defined in Eq.s(2)(3), we can write for homogeneous system the following deformed FokkerPlanck equation [31]
(22) 
where and are the drift and diffusion coefficients, respectively.
The above equation has stationary solution that can be written as
(23) 
where is a normalization constant, is the deformed exponential function defined in Eq.(16) and we have defined ^{1}^{1}1In the following, for simplicity, we limit ourselves to consider the drift coefficient as a monomial function of .
(24) 
If we postulate a generalized Brownian motion in a deformed classical dynamics by mean the following definition of the drift and diffusion coefficients
(25) 
where is the friction constant, is a constant depending on the system and is the dilatation operator (5), the stationary solution of the above FokkerPlanck equation can be obtained as solution of the following stationary differential equation
(26) 
It easy to show that the solution of the above equation can be written as
(27) 
4 deformed Schrödinger equation
We are now able to derive a deformed Schrödinger equation by means of a stochastic quantization method [32].
Starting from the following transformation of the probability density
(28) 
where is the function defined in Eq.(24), the deformed FokkerPlanck equation (22) can be written as
(29) 
where
(30) 
The above equation has the same structure of the time dependent Schrödinger equation. In fact, the stochastic quantization of the Eq.(22) can be realized with the transformations
(31)  
(32) 
getting the generalized Schrödinger equation
(33) 
where
(34) 
is the deformed Hamiltonian. Let us note that the Hamiltonian (34) is a notHermitian operator with respect to the standard definition based on the ordinary (undeformed) scalar product of squareintegrable functions [9, 14]. In the following section, we will see as this aspect can be overridden by means the introduction of a deformed scalar product and generalized properties of operators inspired to calculus.
The above equation admits factorized solution , where satisfies to the equation
(35) 
with the standard (undeformed) solution
(36) 
while is the solution of timeindependent Schrödinger equation
(37) 
In one dimensional case, for a free particle () described by the wave function , Eq.(37) becomes
(38) 
where . The solution of the previous equation can be written as
(39) 
The above equation generalizes the plane wave function in the framework of the calculus.
5 deformed products and Hermitian operators
In order to develop a consistent deformed quantum dynamics, we have to generalize the products between functions and properties of the operators in the framework of the calculus.
Let us start on the basis of Eq.(19), which implies
(40)  
(41) 
and in terms of the plane wave (39)
(42) 
Inspired to the above equation, it appears natural to introduce the complex conjugation of a function as
(43) 
and, consequently, the probability density of a single particle in a finite space as
(44) 
Thus, the wave functions must be squareintegrable functions of configuration space, that is to say the functions such that the integral
(45) 
converges.
The function space define above it is a linear space. If and are two squareintegrable functions, any linear combinations , where and are arbitrarily chosen complex numbers, are also squareintegrable functions.
Following this line, it is possible to define a scalar product of the function by the function as
(46) 
This is linear with respect to , the norm of a function is a real, nonnegative number: and
(47) 
Analogously to the undeformed case, it is easy to see that from the above properties of the scalar product follows the Schwarz inequality
(48) 
Consistently with the above definitions, the adjoint of an operator is defined by means of the relation
(49) 
and, by definition, a linear operator is Hermitian if it is its own adjoint. More explicitly, an operator is Hermitian if for any two states and we have
(50) 
First of all, the above properties are crucial to have a consistent conservation in time of the probability densities, defined in Eq.(44). In fact, by taking the complex conjugation of Eq.(33), summing and integrating term by term the two equations, we get
(51) 
where the last equivalence follows from the fact that the operator Hamiltonian is Hermitian. In this context, it is relevant to observe that it is possible to verify the above property by using the timespatial factorization solution of the Schrödinger equation. In fact, we have
(52) 
From the stationary Schrödinger equation (37) and its complex conjugation we have directly
(53) 
and the terms in the square bracket of Eq.(52) goes to zero.
6 Observables in deformed quantum mechanics
On the basis of the above properties, we have the recipe to generalize the definition of observables in the framework of deformed theory by postulating that:

with the dynamical variable is associate the linear operator ;

the mean value of this dynamical variable, when the system is in the dynamical (normalized) state , is
(54)
Observables are real quantities, hence the expectation value (54) must be real for any state :
(55) 
therefore, on the basis of Eq.(50), observables must be represented by Hermitian operators.
If we require there is a state for which the result of measuring the observable is unique, in other words that the fluctuations
(56) 
must vanish, we obtain the following eigenvalue equation of a Hermitian operator with eigenvalue
(57) 
As a consequence, the eigenvalues of a Hermitian operator are real because is real for any state; in particular for an eigenstate with the eigenvalue for which .
Furthermore, as in the undeformed case, two eigenfunctions and of the Hermitian operator , corresponding to different eigenvalues and , are orthogonal. We can always normalize the eigenfunction, therefore we can chose all the eigenvalues of a Hermitian operator orthonormal, i.e.
(58) 
Consequently, two eigenfunctions and belonging to different eigenvalues are linearly independent.
It easy to see that, adapting step by step the undeformed case to the introduced deformed framework, the totality of the linearly independent eigenfunctions of Hermitian operator form a complete (orthonormal) set in the space of the previously introduced squareintegrable functions. In other words, if is any state of a system, then it can be expanded in terms of the eigenfunctions (with a discrete spectrum) of the corresponding Hermitian operator associate to the observable:
(59) 
where
(60) 
The above expansion allows us, as usual, to write the expectation value of in the normed state as
(61) 
where are the set of eigenvalues (assumed, for simplicity, discrete and nondegenerate) and the normalization condition of the wave function can be written in the form
(62) 
7 Conclusions
On the basis of the stochastic quantization procedure and on the differential calculus, we have obtained a generalized linear Schrödinger equation which involves a deformed Hamiltonian that is nonHermitian with respect to the standard (undeformed) definition. However, under an appropriate generalization of the operators properties and the introduction of a deformed scalar product in the space of squareintegral wave functions, such equation of motion satisfies the basic quantum mechanics assumptions.
Although a complete physical and mathematical description of the introduced quantum dynamical equations lies out the scope of this paper, we think that the results derived here appear to provide a deeper insight into a full consistent deformed quantum mechanics in the framework of the calculus and may be a relevant starting point for future investigations.
Acknowledgment
It is a pleasure to thank A M Scarfone for useful discussions.
References
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