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

In nonrelativistic quantum mechanics the state of a system formed by

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 [

A system formed by

The function

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 [

In these conditions the probability function describes the density of single particle in an unitary segment. In general, the

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 [

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

The term

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

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

Any arbitrary solution

The probabilistic interpretation [

The set of the square-integrable functions conjunctly with the correspondence defined in (

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

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

The “enough” regular solutions of the Schrödinger equation which belong to the space

In order to characterize the solutions of the quantum Liouville equation which preserves the Heisenberg relation, we calculate the variances

We may expand the generic wave function

The integrals present in (

In order to show that the variances

Now we may obtain the mean value of

The comparison between the set constituted by (

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

Due to the presence of the single contributions

The term

Noting

Consider the set

Inside the set

This set, named

Moreover a

The set of the functions belonging to

In order to construct a basis of the vectorial space

In order to prove the orthonormality of the set

To prove that the set

Consider an arbitrary function

It is useful noting that (

By following a procedure similar to that used for the set

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

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

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.

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.