International Journal of Mathematics and Mathematical Sciences

Volume 2009 (2009), Article ID 369482, 20 pages

http://dx.doi.org/10.1155/2009/369482

## Heisenberg Uncertainty Relation in Quantum Liouville Equation

Gruppo di Fisica Interdisciplinare, Dipartimento di Fisica e Tecnologie Relative, Università di Palermo and INFM-CNR, Unità di Palermo, Viale delle Scienze, Ed. 18, I-90128 Palermo, Italy

Received 21 July 2009; Accepted 26 September 2009

Academic Editor: Marianna Shubov

Copyright © 2009 Davide Valenti. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider the quantum Liouville equation and give a characterization of the solutions which satisfy the Heisenberg uncertainty relation. We analyze three cases. Initially we consider a particular solution of the quantum Liouville equation: the Wigner transform (**x**,**v**,) of a generic solution (**x**;) of the Schrödinger equation. We give a representation of (**x**, ) by the Hermite functions. We show that the values of the variances of **x** and **v** calculated by using the Wigner function (**x**,**v**,) coincide, respectively, with the variances of position operator and conjugate momentum operator obtained using the wave function (**x**,). Then we consider the
Fourier transform of the density matrix (**z**,**y**,) = (**z**,)(**y**,*t*). We find again that the variances of **x** and **v** obtained by using (**z**, **y**,) are respectively equal to the variances of and calculated in (**x**,). Finally we introduce the matrix and we show that a generic square-integrable function (**x**,**v**,) can be written as Fourier transform of a density matrix, provided that the matrix is diagonalizable.

#### 1. Introduction

In nonrelativistic quantum mechanics the state of a system formed by *N* particles is described by a state vector whose *x*-representation is given by the wave function ( is the generic vector of a 3*N*-dimensional space). The square modulus of represents the probability density that the particle is found in at the time . The time evolution of the state vector (or, more precisely, the evolution of the corresponding wave function) is given by the Schrödinger equation [1]

In classical mechanics the dynamics of a system is described by the Newton’s equations of motion which represent trajectory equations. Alternatively Lagrange or Hamilton formulations emphasize other concepts, for example, the law of energy conservation, but essentially nothing different is introduced [2].

A system formed by *N* particles, for example, a gas, is usually studied by the tools of the statistical mechanics and its state may be described instantaneously by a probability function which depends on both positions and velocities of the particles [3, 4]. The time evolution of a statistical system can be obtained by several different equations. In particular a system formed by *N* particles in a potential can be described by the classic Liouville equation [5]

The function is a probability density and it instantaneously describes the system state inside the 6 *N*-dimensional phase space . The use of the Liouville equation instead of Newton’s equations shifts the emphasis from the concept of trajectory to that of probability.

When the time evolution of a many-particle system is considered, it is useful to obtain single-particle approximation of the classic Liouville equation. The Vlasov limit [6], where the field is scaled as for particles' number approaching infinite, and the use of a one-dimensional model allow to obtain an equation which is formally identical to (1.2), the classic Liouville equation in a two-dimensional phase space

In these conditions the probability function describes the density of single particle in an unitary segment. In general, the *Vlasov equation* [7, 8] provides the probability density of finding a single particle at the position with speed at the time ( and are vectors belonging to the ordinary three-dimensional space). The use of the statistical mechanics allows to connect the mechanical properties (mycroscopical domain) of the constituting particles with the thermodynamic behaviour (macroscopical domain) of the system. Moreover the importance of a formulation of classical mechanics based on the Liouville equation is that quantum mechanics may be introduced from classical mechanics amended by suitable postulates and principles [9].

In quantum physics a system can be also described by using the tools of the statistical mechanics. The use of a stochastic formulation to describe the exciton transport in polar media [10] and the existence of relation connecting physical observables with the temperature [11] emphasize the importance of introducing statistical tools to resolve typical problems of the quantum physics. It is useful to remember that ultimately in solid state physics the study of the dressing processes of excitons and conduction electrons in polar media [12–16] has been accompained by the realization of experimental techniques allowing to create excitons within times of the order of hundred femtoseconds [17, 18]. When one deals with dressing processes in the matter, the use of the statistical mechanics allows to treat a many-particle system as an ensemble. This leads to a thermodynamic-like description of quantum phenomena [19–21]. In example, the dressing dynamics of excitons and polarons may be connected with some mean properties of the matter and the theoretical results may be compared with the experimental observations [22–25].

The statistical treatment of a quantum system may be led by introducing an equation describing the global behaviour of the system and simultaneously considering that it undergoes the laws of the quantum physics. In particular, it is possible to obtain an equation which represents the extension of the Liouville equation to the quantum mechanics. For sake of semplicity the particle system will be described referring to the Vlasov equation (i.e., the Liouville equation for ). A one-dimensional model will be studied without loss of generality. Considerations and results can be successively extended to the multiparticle case. For the one-dimensional Schrödinger equation is

Defining the density matrix

where represents the conjugate complex of , from Schrödinger equation (1.4) one obtains

Setting

(1.6) becomes

Now the Wigner distribution function may be introduced [26]

From (1.9) the one-dimensional *quantum Liouville equation* [5] is obtained

with

The term in the classic Liouville equation (1.3) is replaced in the quantum extension (1.11) by the term . The linear operator is named pseudodifferential operator and it is applied by a product operation acting on the Fourier transform . The multiplicator , that is, the symbol of the pseudodifferential operator, is obtained comparing (1.9) and (1.11)

Using the Liouville equation in the place of Schrödinger equation allows to deal with statistical mixtures of states which cannot be represented by a wave function. These states may be defined by considering a complete set of orthonormal solutions for the Schrödinger equation a state is obtained as a mixture, with probabilities , of the orthonormal solutions. The corresponding density matrix is given by

where is the density matrix of the quantum state represented by the wave function (pure case). Since is a linear combination of solutions of (1.6), it is a solution of the same equation. Defining and as in (1.8) and (1.10) we observe that generally the quantum Liouville equation allows to describe statistical mixture of states (nonpure case).

Moreover the use of tools of statistical mechanics in treating quantum problems allows to consider the effects due to the continuous reduction of the spatial domains where the solid state physics works. The dimensions of the modern semiconductor devices become comparable with the free mean path of the electrons which can cross the active zone without undergoing scattering processes. These problems can be overcrossed by considering the distribution function of the electrons and by introducing equations typical of the statistical mechanics whose solutions provide a more precise and correct description of the dynamics of these systems [27, 28]. Yet, the increasing reduction of the dimensions where these devices work does not allow to neglect the quantum effects. It is necessary to use equations including these contributions by extending the classical models to the quantum physics [29]. Then, knowing the mathematical characteristics of these equations becomes very important in order to study the physical properties of the systems considered. Particularly, it was shown that the quantum hydrodynamic equation directly obtained from the Schrödinger equation have solutions which generally do not converge to the corresponding classical solutions, when the Planck constant tends to zero [30]. This problem can be overcrossed by introducing the Wigner transform and the quantum Liouville equation given respectively by (1.10) and (1.11). In fact, it was shown that for the solutions of the quantum Liouville equation converge to those of the corresponding classical equation [31, 32]. Yet, the use of the quantum Liouville equation to describe a quantum system suggests the necessity of verifying that this equation satisfies the Heisenberg uncertainty relation. In this paper we give a characterization of the solutions of the quantum Liouville equation (QLE) which satisfy the Heisenberg uncertainty relation. At this aim we show that for suitable conditions the variances and calculated by using a solution of the QLE coincide identically with the variances and of position operator and conjugate momentum operator obtained by using the corresponding Schrödinger solution .

#### 2. Square-Integrable Functions : The Space

Any arbitrary solution of the Schrödinger equation, due to its expandibility in plane waves,

verifies the inequality[1]

where and are observables which satisfy the commutation rule

The probabilistic interpretation [33, 34] of the wave function of a particle implies that is the probability of measuring, at the time , the particle inside a segment with amplitude and centered in (one-dimensional model). The total probability of detecting the particle somewhere in space has to be obviously equal to 1 (probability conservation), so we must have

With each pair of square-integrable functions , a complex number is associated by the definition of *scalar product *

The set of the square-integrable functions conjunctly with the correspondence defined in (2.5) is called and it has the structure of a Hilbert space [35–37]. satisfies all the criteria of a vector space where the scalar product is given by the operation defined in (2.5). From a physical point of view the set is too wide. It is then useful to reduce the set of square-integrable functions to the ones which possess certain properties of regularity and describe real physical systems. The functions have to be everywhere defined, (a probability value must be associated to the particle at every point in the space); the functions must be everywhere continuous (the probability amplitude in an arbitrary point of the space cannot depend on the size of the volume which contains the particle); the functions have to be infinitely differentiable (this also assures the possibility of approximating them by a Taylor's expansion). No complete list of the necessary properties is given; the prescription is only considering in the set constituted by functions which are “enough” regular, that is functions which are suitable for describing the behavior of real physical systems.

As we said in the previous sections, the dynamics of a quantum system can be described applying the formalism of the statistic mechanics. This treatment can be performed by using the *quantum Liouville equation* introduced in the first section. In order to obtain the QLE, the passage from the density matrix as a function of and to the Wigner function has been performed. It is not obvious that the variances and , obtained using the Wigner functions , satisfy the Heisenberg uncertainty relation. In the following, a study of the solutions of the quantum Liouville equation is performed. A solution of the quantum Liouville equation is obtained considering the Wigner transform of an arbitrary Schrödinger function (pure state). Expanding by Hermite functions, it is shown that the variances of and obtained using the Wigner function coincide respectively with the variances of the operators position and conjugate momentum calculated using the wave function .

This comparison is repeated exploiting a more general solution, that is the Fourier transform of an arbitrary density matrix (Wigner function for a statistical mixture of states). The results show that any solution of the quantum Liouville equation, defined as Fourier transform of any density matrix, also verifies the Heisenberg uncertainty relation.

Finally a larger characterization is presented for functions which contemporaneously satisfy both the quantum Liouville equation and the Heisenberg relation. This characterization is obtained by defining the space of the square-integrable functions. We show that the Heisenberg inequality is verified by an arbitrary function provided that, given any arbitrary basis in , the matrix of the coefficients of the expansion

is diagonalizable.

#### 3. Heisenberg Relation and Wigner Distribution Function: Pure State

The “enough” regular solutions of the Schrödinger equation which belong to the space *F*, verify instantaneously the Heisenberg uncertainty relation

where and are observables which satisfy the commutation rule [1]

In representation and coincide respectively with the operator which multiplies by and with the differential operator .

In order to characterize the solutions of the quantum Liouville equation which preserves the Heisenberg relation, we calculate the variances and using the Wigner transform of a generic solution of the one-dimensional Schrödinger equation (1.4). For this purpose we introduce the basis composed of the Hermite functions. Consider an arbitrary solution of (1.4) (Schrödinger equation) and its Wigner transform given by (1.10) (pure state case). It is known that the Hermite functions, which are defined by

where is the Hermite polynomial of degree given by [38, 39]

constitute a basis in .

We may expand the generic wave function on this basis obtaining

with given by

For the mean values of and it results

The integrals present in (3.7) and (3.8) may be worked out by means of the expansion of in Hermite functions. By using (3.5) and (3.7) the expression for becomes

Analogously, by using (3.5) in (3.8), one obtains

For the mean value of it results

Finally, by using (3.5), for the mean value of we get

In order to show that the variances and verify the Heisenberg uncertainty relation, we may compare them with the variances and of the operators and calculated by using the wave function . It is then necessary to calculate the mean values , , , using the formalism of Wigner transform. By using the expansion given in (3.5), the Wigner transform (1.10) of the generic wave function becomes (setting )

Now we may obtain the mean value of , , , and using the distribution function given in (3.13). For it results

By following an analogous procedure we obtain

In a similar way one obtains the expression giving the mean value of

Finally the expression for the mean value of is obtained

The comparison between the set constituted by (3.9), (3.10), (3.11), (3.12) and that formed by (3.14), (3.15), (3.16), (3.17) indicates that the variances and of the quantum operators and calculated by the use of a generic wave function coincide respectively with the variances and of the variables and obtained by using the Wigner transform defined in (1.10).

#### 4. Heisenberg Relation and Wigner Distribution Function: Statistical Mixture of States

Within the usual one-dimensional model we consider a quantum particle whose dynamics is given by the Liouville equation. The state of the particle is represented by the Wigner distribution function obtained as Fourier transform of the density matrix . In the most general case, as we said, the density matrix (and then the corresponding Wigner function) describes a statistical mixture of states which is not associated with a wave function. Using a basis of the space *F,* the density matrix of the system can be expressed in terms of density matrices of pure states. For example, if is a basis of the vectorial space *F*, an arbitrary density matrix is given by

with

The coefficients , which determine the time evolution of the system, satisfy the relation

Due to the presence of the single contributions , (4.1) allows to introduce again the idea of wave function. Each of these contributions represents the density matrix of a pure state. The mean value of an arbitrary operator is then obtained using the Wigner distribution function calculated as Fourier transform of and it can be expressed by the linear combination of the mean values calculated in each pure state

where represents the mean value obtained using the Wigner function

We must evaluate if the product satisfies the Heisenberg relation for every set of . We then consider a statistical mixture of *N* quantum states which is described, at the time , by the density matrix given in (4.1). The corresponding Wigner function is given by

The mean square deviations of and calculated in are given by

Note that (4.7) can be written

By an analogous procedure, from (4.8) one gets

Finally using (4.9) and (4.10) one gets

Setting

(4.11) becomes

From (4.13) it results

The term is nonnegative. Moreover, as we showed in the previous section, the variances and obtained using Wigner transform of pure states coincide with the variances and calculated by the corresponding wave functions and they satisfy the Heisenberg uncertainty relation

which implies

Using (4.15) and (4.16) in (4.14) the following inequality is obtained:

where the nonnegative variable

has been defined.

Noting

(4.17) allows to write

Finally, remembering that for the coefficients the normalization relation holds

and that is defined nonnegative, from (4.20) the following inequality is obtained:

which represents the Heisenberg uncertainty relation within the two-dimensional phase space .

#### 5. The Set of the Square-Integrable Functions: The Space

Consider the set constituted by the functions which obey the relation:

Inside the set we characterize the subset formed by the square-integrable functions, that is the functions which satisfy the following relation

This set, named , possesses the structure of a vectorial space. In fact, if we consider two arbitrary functions , and two arbitrary complex numbers , , for the function the following relation holds:

Equation (5.3) indicates that the sum of two square-integrable functions is itself a square-integrable function. The set is then closed respect to the sum. It is easy to verify the set satisfies all the properties of a vectorial space.

Moreover a *scalar product * can be defined

which associates each pair of elements of with a complex number.

The set of the functions belonging to with the scalar product defined in (5.4) constitutes a Hilbert space.

#### 6. A Basis of

In order to construct a basis of the vectorial space we consider the Hermite functions defined in (3.3). They constitute a basis of the space formed by the solutions of the Schrödinger equation. Using the generic Hermite function it is possible to introduce the set with

We propose to show that the set constitutes a basis of . This may be obtained by proving that is an orthonormal and complete set of this space.

##### 6.1. Orthonormality

In order to prove the orthonormality of the set , we calculate the following integral:

where , are two arbitrary functions belonging to . Performing the variable change

and using the orthonormality of the Hermite functions, from (6.2) one gets

which represents the orthonormality relation for the functions .

##### 6.2. Closure

To prove that the set satisfies closure relation, we write

Applying the closure relation of the Hermite functions, from (6.5), one gets

which expresses the closure relation. The set , which is orthonormal and complete, constitutes a basis of the space .

#### 7. Functions of the Space and Solutions of the Quantum Liouville Equation

Consider an arbitrary function . Its expansion in terms of basis vectors is given by

where the coefficients are complex numbers.

It is useful noting that (5.1) and (5.2) determine some properties of the coefficients . In fact, using condition (5.1) in (7.1), one gets

which determines

Analogously, using (5.2) and (7.1), we get

From (7.3) and (7.4) it results that the following conditions hold:

Consider again the expansion (7.1) which holds for an arbitrary function . Now we introduce another hypothesis for the coefficients , that is, we suppose that the matrix can be put in diagonal form in ( is the space which contains the square-integrable Schrödinger functions). If is the matrix which diagonalizes , it results

with

This transformation induces in the basis change

where are the vectors belonging to the new basis. The matrix elements of represent in the space the coefficients of the expansion (7.1) on the basis vectors . The diagonalization defined by (7.6) induces within the space the transformation

with

By following a procedure similar to that used for the set it is easy to verify that the functions satisfy relations both of ortonormality and closure. Therefore they constitute a new basis of . Equation (7.1) then becomes

where we set

Now we define the density matrix

Using (7.13) in (7.11) allows to write

Equation (7.14) indicates that any function , whose matrix is diagonalizable, may be expressed as Fourier transform of a density matrix and then it verifies the Heisenberg uncertainty relation given in (4.22).

#### 8. Conclusions

The quantum Liouville equation allows to deal with a quantum system using methods and tools of the statistic mechanics. This equation is derived from a typical quantum equation, that is the Schrödinger equation. In order to characterize the set of solutions of the quantum Liouville equation which satisfy the Heisenberg uncertainty principle we investigated both the Schrödinger equation and the quantum Liouville equation. So, we recalled that an arbitrary solution of the Schrödinger equation, because of its expansibility in plane waves,

verifies the inequality

where and are observables which satisfy the commutation rule

Afterwards we investigated the Heisenberg inequality with reference to the quantum Liouville equation. We studied three different cases. Initially a particular solution of the quantum Liouville equation has been considered, this solution being the Wigner transform of an arbitrary wave function . So, we found that the product of the variances and , which are defined within a two-dimensional phase space, verifies the Heisenberg uncertainty relation. Then we considered a more general case: the quantum Liouville equation has been resolved by using the Fourier transform of the density matrix of an arbitrary quantum state. This allows to deal with states which cannot be represented by a wave function. The expressions obtained for the variances and verify the Heisenberg relation and allow to extend the result obtained for the pure state case to the statistical mixture of states. Finally, we showed that an arbitrary function admits the following expansion:

and this expression can be written as Fourier transform of a density matrix provided that the matrix is diagonalizable.

In conclusion we applied an alternative procedure, based on the use of the Hermite functions, to characterize the solutions of the quantum Liouville equation which verify the Heisenberg uncertainty relation.

#### Acknowledgments

The author is grateful to Professor Carlo Cercignani who inspired this work and Professor Bernardo Spagnolo for useful discussions and suggestions. The author also acknowledges financial support by MIUR.

#### References

- C. Cohen-Tannoudji, B. Diu, and F. Laloë,
*Quantum Mechanics*, John Wiley & Sons, New York, NY, USA, 1990. - H. Goldstein,
*Classical Mechanics*, Addison-Wesley, Reading, Mass, USA, 2nd edition, 1980. View at MathSciNet - P. M. Morse,
*Thermal Physics*, W. A. Benjamin, New York, NY, USA, 1969. View at MathSciNet - M. W. Zemansky and R. H. Dittman,
*Heat and Thermodynamics*, Mc Graw-Hill, New York, NY, USA, 1981. - P. A. Markowich, C. A. Ringhofer, and C. Schmeiser,
*Semiconductor Equations*, Springer, New York, NY, USA, 1990. View at Zentralblatt MATH · View at MathSciNet - W. Braun and K. Hepp, “The Vlasov dynamics and its fluctuations in the $1/N$ limit of interacting classical particles,”
*Communications in Mathematical Physics*, vol. 56, no. 2, pp. 101–113, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. A. Vlasov, “The vibrational properties of an electron gas,”
*Soviet Physics Uspekhi*, vol. 10, no. 6, pp. 721–733, 1968. View at Publisher · View at Google Scholar - R. L. Dobrushin, “Vlasov equations,”
*Functional Analysis and Its Applications*, vol. 13, no. 2, pp. 115–123, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - N. Klipa and S. D. Bosanac, “Quantum effects from the classical principles,”
*International Journal of Theoretical Physics, Group Theory, and Nonlinear Optics*, vol. 7, p. 15, 2000. View at Google Scholar - K. Lindenberg and B. J. West, “Exciton line shapes at finite temperatures,”
*Physical Review Letters*, vol. 51, no. 15, pp. 1370–1373, 1983. View at Publisher · View at Google Scholar · View at Scopus - B. J. West and K. Lindenberg, “Energy transfer in condensed media. I. Two-level systems,”
*The Journal of Chemical Physics*, vol. 83, no. 8, pp. 4118–4135, 1985. View at Publisher · View at Google Scholar · View at MathSciNet - B. Iadonisi and F. Bassani, “Polaronic correction to the exciton effective mass,”
*Nuovo Cimento D*, vol. 9, no. 6, pp. 703–714, 1987. View at Publisher · View at Google Scholar - T. D. Lee, F. E. Low, and D. Pines, “The motion of slow electrons in a polar crystal,”
*Physical Review A*, vol. 90, pp. 297–302, 1953. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. W. Brown, K. Lindenberg, and B. J. West, “On the dynamics of polaron formation in a deformable medium,”
*The Journal of Chemical Physics*, vol. 84, no. 3, pp. 1574–1582, 1986. View at Publisher · View at Google Scholar · View at Scopus - D. W. Brown, K. Lindenberg, B. J. West, J. A. Cina, and R. Silbey, “Polaron formation in the acoustic chain,”
*The Journal of Chemical Physics*, vol. 87, no. 11, pp. 6700–6705, 1987. View at Publisher · View at Google Scholar · View at Scopus - D. Valenti and G. Compagno, “Self-dressing dynamics of slow electrons in covalent crystals,” in
*Proceedings of the 6th Conferenza Scientifica Triennale del Comitato Regionale Ricerche Nucleari e di Struttura della Materia*, A. Messina, Ed., AIP Press, Palermo, Italy, 1999. View at Publisher · View at Google Scholar - A. P. Heberle, J. J. Baumberg, and K. Köhler, “Ultrafast coherent control and destruction of excitons in quantum wells,”
*Physical Review Letters*, vol. 75, no. 13, pp. 2598–2601, 1995. View at Publisher · View at Google Scholar · View at Scopus - A. Watson, “Laser pulses make fast work of an optical switch,”
*Science*, vol. 270, no. 5233, p. 29, 1995. View at Publisher · View at Google Scholar · View at Scopus - R. Passante, T. Petrosky, and I. Prigogine, “Virtual transitions, self-dressing and indirect spectroscopy,”
*Optics Communications*, vol. 99, no. 1-2, pp. 55–60, 1993. View at Publisher · View at Google Scholar · View at Scopus - R. Passante, T. Petrosky, and I. Prigogine, “Long-time behaviour of self-dressing and indirect spectroscopy,”
*Physica A*, vol. 218, no. 3-4, pp. 437–456, 1995. View at Publisher · View at Google Scholar · View at Scopus - E. Karpov, I. Prigogine, T. Petrosky, and G. Pronko, “Friedrichs model with virtual transitions. Exact solution and indirect spectroscopy,”
*Journal of Mathematical Physics*, vol. 41, no. 1, pp. 118–131, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Bányai, Q. T. Vu, B. Mieck, and H. Haug, “Ultrafast quantum kinetics of time-dependent RPA-screened coulomb scattering,”
*Physical Review Letters*, vol. 81, no. 4, pp. 882–885, 1998. View at Publisher · View at Google Scholar · View at Scopus - Q. T. Vu and H. Haug, “Time-dependent screening of the carrier-phonon and carrier-carrier interactions in nonequilibrium systems,”
*Physical Review B*, vol. 62, no. 11, pp. 7179–7185, 2000. View at Publisher · View at Google Scholar · View at Scopus - M. Betz, G. Göger, A. Laubereau et al., “Subthreshold carrier-LO phonon dynamics in semiconductors with intermediate polaron coupling: a purely quantum kinetic relaxation channel,”
*Physical Review Letters*, vol. 86, no. 20, pp. 4684–4687, 2001. View at Publisher · View at Google Scholar · View at Scopus - R. Huber, F. Tauser, A. Brodschelm, M. Bichler, G. Abstreiter, and A. Leitenstorfer, “How many-particle interactions develop after ultrafast excitation of an electron—hole plasma,”
*Nature*, vol. 414, no. 6861, pp. 286–289, 2001. View at Publisher · View at Google Scholar · View at Scopus - E. Wigner, “On the quantum correction for thermodynamic equilibrium,”
*Physical Review*, vol. 40, no. 5, pp. 749–759, 1932. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - C. Toeppfer and C. Cercignani, “Analytical results for the Boltzmann equation,”
*Contributions to Plasma Physics*, vol. 37, no. 2-3, pp. 279–291, 1997. View at Publisher · View at Google Scholar - Ch. Dalitz, “Half-space problem of the Boltzmann equation for charged particles,”
*Journal of Statistical Physics*, vol. 88, no. 1-2, pp. 129–144, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - N. Ben Abdallah, P. Degond, and P. A. Markowich, “On a one-dimensional Schrödinger-Poisson scattering model,”
*Zeitschrift für Angewandte Mathematik und Mechanik*, vol. 48, no. 1, pp. 135–155, 1997. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I. Gasser and P. A. Markowich, “Quantum hydrodynamics, wigner transforms and the classical limit,”
*Asymptotic Analysis*, vol. 14, no. 2, pp. 97–116, 1997. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. Gérard, P. A. Markowich, N. J. Mauser, and F. Poupaud, “Homogenization limits and wigner transforms,”
*Communications on Pure and Applied Mathematics*, vol. L, pp. 323–379, 1997. View at Google Scholar · View at Zentralblatt MATH - P. A. Markowich and C. A. Ringhofer, “An analysis of the quantum Liouville equation,”
*Zeitschrift für Angewandte Mathematik und Mechanik*, vol. 69, no. 3, pp. 121–127, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Born, “Zur Quantenmechanik der Stoßvorgänge,”
*Zeitschrift für Physik*, vol. 37, no. 12, pp. 863–867, 1926. View at Publisher · View at Google Scholar · View at Scopus - J. A. Wheeler and W. H. Zurek, Eds.,
*Quantum Theory and Measurement*, Princeton University Press, Princeton, NJ, USA, 1983. - P. R. Halmos,
*Introduction to Hilbert Space*, Chelsea Publishing Company, New York, NY, USA, 1957. - K. O. Friedrichs,
*Perturbation of Spectra in Hilbert Space*, American Mathematical Society, Providence, RI, USA, 1965. View at MathSciNet - P. R. Halmos,
*Finite-Dimensional Vector Space*, D. Van Nostrand Company, Princeton, NJ, USA, 1958. View at MathSciNet - M. Abramowitz and I. A. Stegun,
*Handbook of Mathematical Functions*, Dover, New York, NY, USA, 1970. - G. B. Arfken and H. J. Weber,
*Mathematical Methods for Physicists*, Academic Press, New York, NY, USA, 4th edition, 1995. View at MathSciNet