Heisenberg Uncertainty Relation in Quantum Liouville Equation

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 𝑓(x,v,𝑡) of a generic solution 𝜓(x;𝑡) of the Schrödinger equation. We give a representation of 𝜓(x, 𝑡) by the Hermite functions. We show that the values of the variances of x and v calculated by using the Wigner function 𝑓(x,v,𝑡) coincide, respectively, with the variances of position operator 𝑋 and conjugate momentum operator 𝑃 obtained using the wave function 𝜓(x,𝑡). Then we consider the Fourier transform of the density matrix 𝜌(z,y,𝑡) = 𝜓^∗(z,𝑡)𝜓(y,t). We find again that the variances of x and v obtained by using 𝜌(z, y,𝑡) are respectively equal to the variances of 𝑋 and 𝑃 calculated in 𝜓(x,𝑡). Finally we introduce the matrix ‖𝐴_𝑛𝑛′(𝑡)‖ and we show that a generic square-integrable function 𝑔(x,v,𝑡) can be written as Fourier transform of a density matrix, provided that the matrix ‖𝐴_𝑛𝑛′(𝑡)‖ 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 𝜓(𝐱,𝑡) (𝐱 is the generic vector of a 3N-dimensional space). The square modulus of 𝜓(𝐱,𝑡) represents the probability density that the particle is found in 𝐱 at the time 𝑡. The time evolution of the state vector (or, more precisely, the evolution of the corresponding wave function) is given by the Schrödinger equation [1]

𝚤ℏ𝜕𝜓(𝐱,𝑡)ℏ𝜕𝑡=−²Δ2𝑚^𝐱𝜓(𝐱,𝑡)+𝑉(𝐱,𝑡)𝜓(𝐱,𝑡). (1.1)

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 𝑉(𝐱,𝑡) can be described by the classic Liouville equation [5]

𝜕𝑓𝜕𝑡+𝐯⋅∇_𝐱𝑓−∇_𝐱𝑉⋅∇_𝐯𝑓=0. (1.2)

The function 𝑓(𝐱,𝐯,𝑡) is a probability density and it instantaneously describes the system state inside the 6 N-dimensional phase space {𝐱,𝐯}. 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 1/𝑁 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

𝜕𝑓(𝑥,𝑣,𝑡)𝜕𝑡+𝑣𝜕𝑓(𝑥,𝑣,𝑡)−𝜕𝑥𝜕𝑉(𝑥,𝑡)𝜕𝑥𝜕𝑓(𝑥,𝑣,𝑡)𝜕𝑣=0. (1.3)

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 𝐱 with speed 𝐯 at the time 𝑡 (𝐱 and 𝐯 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 𝑁=1). A one-dimensional model will be studied without loss of generality. Considerations and results can be successively extended to the multiparticle case. For 𝑁=1 the one-dimensional Schrödinger equation is

𝚤ℏ𝜕𝜓(𝑥,𝑡)ℏ𝜕𝑡=−²𝜕2𝑚²𝜓(𝑥,𝑡)𝜕𝑥²+𝑉(𝑥,𝑡)𝜓(𝑥,𝑡). (1.4) Defining the density matrix

𝜌(𝑧,𝑦,𝑡)=𝜓^∗(𝑧,𝑡)𝜓(𝑦,𝑡), (1.5) where 𝜓^∗ represents the conjugate complex of 𝜓, from Schrödinger equation (1.4) one obtains

𝚤ℏ𝜕𝜌ℏ𝜕𝑡=−²𝜕2𝑚²𝜌𝜕𝑦²+ℏ²𝜕2𝑚²𝜌𝜕𝑧²−𝑉(𝑧)𝜌+𝑉(𝑦)𝜌. (1.6) Setting

ℏ𝑧=𝑥+ℏ2𝑚𝑠,𝑦=𝑥−ℏ2𝑚𝑠,𝑢(𝑥,𝑠,𝑡)=𝜌(𝑥+ℏ2𝑚𝑠,𝑥−2𝑚𝑠,𝑡), (1.7)(1.8)  (1.6) becomes

𝜕𝑢𝜕𝜕𝑡+𝚤(𝜕𝑠𝜕𝑢𝜕𝑥)−𝚤𝑉(𝑥+(ℏ/2𝑚)𝑠,𝑡)−𝑉(𝑥−(ℏ/2𝑚)𝑠,𝑡)ℏ𝑢=0. (1.9) Now the Wigner distribution function may be introduced [26]

1𝑓=2𝜋_𝑅𝑠𝑢(𝑥,𝑠,𝑡)𝑒^𝚤𝑠𝑣𝑑𝑠. (1.10) From (1.9) the one-dimensional quantum Liouville equation [5] is obtained

𝜕𝑓(𝑥,𝑣,𝑡)𝜕𝑡+𝑣𝜕𝑓(𝑥,𝑣,𝑡)𝜕𝑥+𝑊𝑓=0 (1.11) with

1𝑊𝑓=−𝚤2𝜋_{𝑅𝑠𝑅𝑣′}𝑉(𝑥+(ℏ/2𝑚)𝑠,𝑡)−𝑉(𝑥−(ℏ/2𝑚)𝑠,𝑡)ℏ⋅𝑓(𝑥,𝑣^′,𝑡)𝑒^{𝚤𝑠(𝑣−𝑣′)}𝑑𝑣^′𝑑𝑠. (1.12)

The term −(𝜕𝑉(𝑥,𝑡)/𝜕𝑥)(𝜕𝑓(𝑥,𝑣,𝑡)/𝜕𝑣) in the classic Liouville equation (1.3) is replaced in the quantum extension (1.11) by the term 𝑊𝑓. The linear operator 𝑊 is named pseudodifferential operator and it is applied by a product operation acting on the Fourier transform 𝑓. The multiplicator 𝑊, that is, the symbol of the pseudodifferential operator, is obtained comparing (1.9) and (1.11)

𝑊=−𝚤𝑉(𝑥+(ℏ/2𝑚)𝑠,𝑡)−𝑉(𝑥−(ℏ/2𝑚)𝑠,𝑡)ℏ. (1.13)

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 𝜓_𝑗(𝑗=1,2,…): a state is obtained as a mixture, with probabilities 𝜆_𝑗(0≤𝜆_𝑗≤1;∑_𝑗𝜆_𝑗=1), of the orthonormal solutions. The corresponding density matrix is given by

𝜌(𝑧,𝑦,𝑡)=_𝑗𝜆_𝑗𝜓^∗_𝑗(𝑧,𝑡)𝜓_𝑗(𝑦,𝑡)=_𝑗𝜆_𝑗𝜌_𝑗(𝑧,𝑦,𝑡), (1.14) where 𝜌_𝑗(𝑧,𝑦,𝑡) is the density matrix of the quantum state represented by the wave function 𝜓_𝑗 (pure case). Since 𝜌(𝑧,𝑦,𝑡) is a linear combination of solutions of (1.6), it is a solution of the same equation. Defining 𝑢 and 𝑓 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 𝑓(𝑥,𝑣,𝑡) 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 ℏ→0 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 Δ𝑥 and Δ𝑣 calculated by using a solution of the QLE coincide identically with the variances Δ𝑋 and Δ𝑃 of position operator and conjugate momentum operator obtained by using the corresponding Schrödinger solution 𝜓(𝑥,𝑡).

2. Square-Integrable Functions 𝜓(𝑋,𝑇): The Space 𝐿²

Any arbitrary solution 𝜓(𝑥) of the Schrödinger equation, due to its expandibility in plane waves,

1𝜓(𝑥)=√2𝜋^+∞−∞𝑑𝑘𝜓(𝑘)𝑒^𝚤𝑘𝑥, (2.1) verifies the inequality[1]

Δℏ𝑋Δ𝑃≥2, (2.2) where 𝑋 and 𝑃 are observables which satisfy the commutation rule

[𝑋,𝑃]=𝚤ℏ. (2.3)

The probabilistic interpretation [33, 34] of the wave function 𝜓(𝑥,𝑡) of a particle implies that |𝜓(𝑥,𝑡)|²𝑑𝑥 is the probability of measuring, at the time 𝑡, the particle inside a segment with amplitude 𝑑𝑥 and centered in 𝑥 (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

^+∞_−∞𝑑𝑥|𝜓(𝑥,𝑡)|²=1. (2.4) With each pair of square-integrable functions 𝜙(𝑥,𝑡), 𝜓(𝑥,𝑡) a complex number is associated by the definition of scalar product

(𝜙,𝜓)=^+∞_−∞𝑑𝑥𝜙^∗(𝑥)𝜓(𝑥). (2.5)

The set of the square-integrable functions conjunctly with the correspondence defined in (2.5) is called 𝐿² and it has the structure of a Hilbert space [35–37]. 𝐿² 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 𝐿² 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 𝑥₀ 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 𝐿² the set 𝐹 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 𝜌(𝑥+(ℏ/2𝑚)𝑠,𝑥−(ℏ/2𝑚)𝑠,𝑡) as a function of 𝑥 and 𝑠 to the Wigner function 𝑓(𝑥,𝑣,𝑡) has been performed. It is not obvious that the variances Δ𝑥 and Δ𝑣, obtained using the Wigner functions 𝑓(𝑥,𝑣,𝑡), 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 𝑓(𝑥,𝑣,𝑡) of an arbitrary Schrödinger function 𝜓(𝑥,𝑡) (pure state). Expanding 𝜓(𝑥,𝑡) by Hermite functions, it is shown that the variances of 𝑥 and 𝑣 obtained using the Wigner function 𝑓(𝑥,𝑣,𝑡) coincide respectively with the variances of the operators position and conjugate momentum calculated using the wave function 𝜓(𝑥,𝑡).

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 𝑆² of the square-integrable functions. We show that the Heisenberg inequality is verified by an arbitrary function 𝑓(𝑥,𝑣,𝑡)∈𝑆² provided that, given any arbitrary basis {𝑇_𝑛𝑛′(𝑥,𝑣)} in 𝑆², the matrix ‖𝐴_𝑛𝑛′(𝑡)‖ of the coefficients of the expansion

𝑓(𝑥,𝑣,𝑡)=_𝑛𝑛′𝐴_𝑛𝑛′(𝑡)𝑇_𝑛𝑛′(𝑥,𝑣) (2.6) 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

Δℏ𝑋Δ𝑃≥2, (3.1) where 𝑋 and 𝑃 are observables which satisfy the commutation rule [1]

[𝑋,𝑃]=𝚤ℏ. (3.2) In 𝑥 representation 𝑋 and 𝑃 coincide respectively with the operator which multiplies by 𝑥 and with the differential operator (ℏ/𝑖)(𝜕/𝜕𝑥).

In order to characterize the solutions of the quantum Liouville equation which preserves the Heisenberg relation, we calculate the variances Δ𝑥 and Δ𝑣 using the Wigner transform of a generic solution 𝜓(𝑥,𝑡) 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

𝑆_𝑛1=(𝜋)^1/4𝑒^−𝑥2/21√2^𝑛𝐻𝑛!^𝑛(𝑥), (3.3) where 𝐻_𝑛(𝑥) is the Hermite polynomial of degree 𝑛 given by [38, 39]

𝐻_𝑛(𝑥)=(−1)^𝑛𝑒^𝑥2𝑑^𝑛𝑑𝑥^𝑛𝑒^−𝑥2, (3.4) constitute a basis in 𝐹.

We may expand the generic wave function 𝜓(𝑥,𝑡) on this basis obtaining

1𝜓(𝑥,𝑡)=(𝜋)^1/4𝑒^−𝑥2/2_𝑛𝐴_𝑛(𝑡)√2^𝑛𝐻𝑛!^𝑛(𝑥) (3.5) with 𝐴_𝑛(𝑡) given by

𝐴_𝑛(𝑡)=_𝑅𝑥𝑑𝑥𝜓^∗(𝑥,𝑡)𝑆_𝑛(𝑥). (3.6) For the mean values of 𝑋 and 𝑋² it results

⟨𝑋⟩=_𝑅𝑥𝑑𝑥𝜓^∗(𝑥,𝑡)𝑥𝜓(𝑥,𝑡),⟨𝑋²⟩=_𝑅𝑥𝑑𝑥𝜓^∗(𝑥,𝑡)𝑥²𝜓(𝑥,𝑡). (3.7)(3.8)

The integrals present in (3.7) and (3.8) may be worked out by means of the expansion of 𝜓(𝑥,𝑡) in Hermite functions. By using (3.5) and (3.7) the expression for ⟨𝑋⟩ becomes

1⟨𝑋⟩=(𝜋)12_𝑛𝑛′𝐴^∗_𝑛(𝑡)𝐴_𝑛′(𝑡)2^𝑛′𝑛^′!2^𝑛𝑛!^+∞−∞𝑑𝑥𝑥𝑒^−𝑥2𝐻_𝑛(𝑥)𝐻_𝑛′(𝑥). (3.9) Analogously, by using (3.5) in (3.8), one obtains

⟨𝑋²1⟩=(𝜋)^1/2_𝑛𝑛′𝐴^∗_𝑛(𝑡)𝐴_𝑛′(𝑡)2^𝑛′𝑛^′!2^𝑛𝑛!^+∞−∞𝑑𝑥𝑥²𝑒^−𝑥2𝐻_𝑛(𝑥)𝐻_𝑛′(𝑥). (3.10) For the mean value of 𝑃 it results

⟨ℏ𝑃⟩=𝑖(1𝜋)^1/2_𝑛𝑛′𝐴^∗_𝑛(𝑡)𝐴_𝑛′(𝑡)2^𝑛′𝑛^′!2^𝑛𝑛!⋅[𝑛^′+∞_−∞𝑑𝑥𝑒^−𝑥2𝐻_𝑛(𝑥)𝐻_𝑛′−1(𝑥)−𝑛^+∞_−∞𝑑𝑥𝑒^−𝑥2𝐻_𝑛′(𝑥)𝐻_𝑛−1(𝑥)]. (3.11) Finally, by using (3.5), for the mean value of 𝑃² we get

⟨𝑃²⟩=−ℏ²(1𝜋)^1/2_𝑛𝑛′𝐴^∗_𝑛(𝑡)𝐴_𝑛′(𝑡)2^𝑛′𝑛^′!2^𝑛⋅1𝑛!⋅[−2^+∞_−∞𝑑𝑥𝑒^−𝑥2𝐻_𝑛(𝑥)𝐻^′_𝑛(𝑥)−2𝑛𝑛^′+∞_−∞𝑑𝑥𝑒^−𝑥2𝐻_𝑛−1(𝑥)𝐻_𝑛′−1(𝑥)+𝑛(𝑛−1)^+∞_−∞𝑑𝑥𝑒^−𝑥2𝐻_𝑛−2(𝑥)𝐻^′_𝑛(𝑥)+𝑛^′(𝑛^′−1)^+∞_−∞𝑑𝑥𝑒^−𝑥2𝐻_𝑛′−2(𝑥)𝐻_𝑛(𝑥)]. (3.12)

In order to show that the variances Δ𝑥=⟨𝑥²⟩−⟨𝑥⟩² and Δ𝑣=⟨𝑣²⟩−⟨𝑣⟩² verify the Heisenberg uncertainty relation, we may compare them with the variances Δ𝑋=⟨𝑋²⟩−⟨𝑋⟩² and Δ𝑃=⟨𝑃²⟩−⟨𝑃⟩² of the operators 𝑋 and 𝑃 calculated by using the wave function 𝜓(𝑥,𝑡). It is then necessary to calculate the mean values ⟨𝑥⟩, ⟨𝑥²⟩, ⟨𝑣⟩, ⟨𝑣²⟩ using the formalism of Wigner transform. By using the expansion given in (3.5), the Wigner transform (1.10) of the generic wave function 𝜓(𝑥,𝑡) becomes (setting 𝑚=1)

1𝑓=2𝜋_𝑛𝑛′(1𝜋)^1/2𝐴^∗_𝑛(𝑡)𝐴_𝑛′(𝑡)2^𝑛𝑛!2^𝑛′𝑛^′!⋅^+∞_−∞𝑑𝑠𝑒^𝚤𝑠𝑣𝑒^{−(𝑥+(ℏ/2)𝑠)2/2}𝑒^{−(𝑥−(ℏ/2)𝑠)2/2}𝐻_𝑛ℏ(𝑥+2𝑠)𝐻_𝑛′ℏ(𝑥−2𝑠). (3.13)

Now we may obtain the mean value of 𝑥, 𝑥², 𝑣, and 𝑣² using the distribution function given in (3.13). For ⟨𝑥⟩ it results

1⟨𝑥⟩=(𝜋)^1/2_𝑛𝑛′𝐴^∗_𝑛(𝑡)𝐴_𝑛′(𝑡)2^𝑛𝑛!2^𝑛′𝑛^′!^+∞−∞𝑑𝑥𝑥𝑒^−𝑥2𝐻_𝑛(𝑥)𝐻_𝑛′(𝑥). (3.14) By following an analogous procedure we obtain

⟨𝑥²1⟩=(𝜋)^1/2_𝑛𝑛′𝐴^∗_𝑛(𝑡)𝐴_𝑛′(𝑡)2^𝑛𝑛!2^𝑛′𝑛^′!^+∞−∞𝑑𝑥𝑥²𝑒^−𝑥2𝐻_𝑛(𝑥)𝐻_𝑛′(𝑥). (3.15) In a similar way one obtains the expression giving the mean value of 𝑣

⟨𝑣⟩=𝑖ℏ_𝑛𝑛′(1𝜋)^1/2𝐴^∗_𝑛(𝑡)𝐴_𝑛′(𝑡)2^𝑛𝑛!2^𝑛′𝑛^′!⋅^+∞_−∞𝑑𝑥𝑒^−𝑥2[𝑛𝐻_𝑛−1(𝑥)𝐻_𝑛′(𝑥)−𝑛^′𝐻_𝑛′−1(𝑥)𝐻_𝑛(𝑥)]. (3.16) Finally the expression for the mean value of 𝑣² is obtained

⟨𝑣²⟩=−ℏ²(1𝜋)^1/2_𝑛𝑛′𝐴^∗_𝑛(𝑡)𝐴_𝑛′(𝑡)2^𝑛𝑛!2^𝑛′𝑛^′!^+∞−∞𝑑𝑥𝑒^−𝑥21⋅[−2𝐻_𝑛(𝑥)𝐻_𝑛′(𝑥)+𝑛(𝑛−1)𝐻_𝑛−2(𝑥)𝐻_𝑛′(𝑥)+𝑛^′(𝑛^′−1)𝐻_𝑛(𝑥)𝐻_𝑛′−2(𝑥)−2𝑛𝑛^′𝐻_𝑛−1(𝑥)𝐻_𝑛′−1(𝑥)]. (3.17)

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 Δ𝑋 and Δ𝑃 of the quantum operators 𝑋 and 𝑃 calculated by the use of a generic wave function 𝜓(𝑥,𝑡) coincide respectively with the variances Δ𝑥 and Δ𝑣 of the variables 𝑥 and 𝑣 obtained by using the Wigner transform 𝑓(𝑥,𝑣,𝑡) 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 𝜌(𝑥+(ℏ/2𝑚)𝑠,𝑥−(ℏ/2𝑚)𝑠,𝑡). 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 {𝑟_𝑗} is a basis of the vectorial space F, an arbitrary density matrix is given by

ℏ𝜌(𝑥+ℏ2𝑚𝑠,𝑥−2𝑚𝑠,𝑡)=_𝑗𝜆_𝑗𝑟^∗_𝑗ℏ(𝑥+2𝑚𝑠)𝑟_𝑗ℏ(𝑥−=2𝑚𝑠)_𝑗𝜆_𝑗𝜌_𝑗ℏ(𝑥+ℏ2𝑚𝑠,𝑥−2𝑚𝑠) (4.1) with

𝜌_𝑗ℏ(𝑥+ℏ2𝑚𝑠,𝑥−2𝑚𝑠)=𝑟^∗_𝑗ℏ(𝑥+2𝑚𝑠)𝑟_𝑗ℏ(𝑥−2𝑚𝑠). (4.2) The coefficients 𝜆_𝑗, which determine the time evolution of the system, satisfy the relation

_𝑗𝜆_𝑗=1. (4.3)

Due to the presence of the single contributions 𝜌_𝑗, (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 𝜌(𝑥+(ℏ/2𝑚)𝑠,𝑥−(ℏ/2𝑚)𝑠,𝑡) and it can be expressed by the linear combination of the mean values calculated in each pure state

⟨𝑥⟩=_𝑗𝜆_𝑗⟨𝑥⟩_𝑗,⟨𝑥²⟩=_𝑗𝜆_𝑗⟨𝑥²⟩_𝑗,⟨𝑣⟩=_𝑗𝜆_𝑗⟨𝑣⟩_𝑗,⟨𝑣²⟩=_𝑗𝜆_𝑗⟨𝑣²⟩_𝑗, (4.4) where ⟨⟩_𝑗 represents the mean value obtained using the Wigner function

𝑓_𝑗1(𝑥,𝑣,𝑡)=2𝜋^+∞_−∞𝑑𝑠𝑒^𝚤𝑠𝑣𝜌_𝑗ℏ(𝑥+ℏ2𝑚𝑠,𝑥−2𝑚𝑠). (4.5) We must evaluate if the product Δ𝑥Δ𝑣 satisfies the Heisenberg relation for every set of 𝜆_𝑗. We then consider a statistical mixture of N quantum states which is described, at the time 𝑡, by the density matrix given in (4.1). The corresponding Wigner function is given by

1𝑓(𝑥,𝑣,𝑡)=2𝜋^+∞_−∞𝑑𝑠𝑒^𝚤𝑠𝑣ℏ𝜌(𝑥+ℏ2𝑚𝑠,𝑥−=12𝑚𝑠,𝑡)2𝜋_𝑗𝜆_𝑗^+∞_−∞𝑑𝑠𝑒^𝚤𝑠𝑣𝜌_𝑗ℏ(𝑥+ℏ2𝑚𝑠,𝑥−2𝑚𝑠). (4.6) The mean square deviations of 𝑥 and 𝑣 calculated in 𝑓(𝑥,𝑣,𝑡) are given by

(Δ𝑥)²=_𝑗𝜆_𝑗⟨𝑥²⟩_𝑗−_𝑗𝜆²_𝑗⟨𝑥⟩²_𝑗−_𝑖≠𝑗𝜆_𝑖𝜆_𝑗⟨𝑥⟩_𝑖⟨𝑥⟩_𝑗,(Δ𝑣)²=_𝑗𝜆_𝑗⟨𝑣²⟩_𝑗−_𝑗𝜆²_𝑗⟨𝑣⟩²_𝑗−_𝑖≠𝑗𝜆_𝑖𝜆_𝑗⟨𝑣⟩_𝑖⟨𝑣⟩_𝑗. (4.7)(4.8) Note that (4.7) can be written

(Δ𝑥)²=_𝑗𝜆_𝑗(⟨𝑥²⟩_𝑗−𝜆_𝑗⟨𝑥⟩²_𝑗)−_𝑖≠𝑗𝜆_𝑖𝜆_𝑗⟨𝑥⟩_𝑖⟨𝑥⟩_𝑗=_𝑗𝜆_𝑗[(Δ𝑥_𝑗)²+(1−𝜆_𝑗)⟨𝑥⟩²_𝑗]−_𝑖≠𝑗𝜆_𝑖𝜆_𝑗⟨𝑥⟩_𝑖⟨𝑥⟩_𝑗=_𝑗𝜆_𝑗(Δ𝑥_𝑗)²+_𝑖≠𝑗𝜆_𝑖𝜆_𝑗⟨𝑥⟩²_𝑗−_𝑖≠𝑗𝜆_𝑖𝜆_𝑗⟨𝑥⟩_𝑖⟨𝑥⟩_𝑗=_𝑗𝜆_𝑗(Δ𝑥_𝑗)²−_𝑖<𝑗𝜆_𝑖𝜆_𝑗(⟨𝑥⟩_𝑗−⟨𝑥⟩_𝑖)². (4.9) By an analogous procedure, from (4.8) one gets

(Δ𝑣)²=_𝑗𝜆_𝑗(Δ𝑣_𝑗)²−_𝑖<𝑗𝜆_𝑖𝜆_𝑗(⟨𝑣⟩_𝑗−⟨𝑣⟩_𝑖)². (4.10) Finally using (4.9) and (4.10) one gets

(Δ𝑥)²(Δ𝑣)²=_𝑗,𝑗′𝜆_𝑗(Δ𝑥)²_𝑗𝜆_𝑗′(Δ𝑣)²_𝑗′+_{𝑖<𝑗,𝑗′}𝜆_𝑖𝜆_𝑗(⟨𝑥⟩_𝑗−⟨𝑥⟩_𝑖)²𝜆_𝑗′(Δ𝑣)²_𝑗′+_{𝑗,𝑖′<𝑗′}𝜆_𝑗(Δ𝑥)²_𝑗𝜆_𝑖′𝜆_𝑗′(⟨𝑣⟩_𝑗−⟨𝑣⟩_𝑖)²+_{𝑖<𝑗,𝑖′<𝑗′}𝜆_𝑖𝜆_𝑗(⟨𝑥⟩_𝑖−⟨𝑥⟩_𝑗)²𝜆_𝑖′𝜆_𝑗′(⟨𝑣⟩_𝑖′−⟨𝑣⟩_𝑗′)². (4.11) Setting

Θ=_{𝑖<𝑗,𝑗′}𝜆_𝑖𝜆_𝑗(⟨𝑥⟩_𝑗−⟨𝑥⟩_𝑖)²𝜆_𝑗′(Δ𝑣)²_𝑗′+_{𝑗,𝑖′<𝑗′}𝜆_𝑗(Δ𝑥)²_𝑗𝜆_𝑖′𝜆_𝑗′(⟨𝑣⟩_𝑗−⟨𝑣⟩_𝑖)²+_{𝑖<𝑗,𝑖′<𝑗′}𝜆_𝑖𝜆_𝑗(⟨𝑥⟩_𝑖−⟨𝑥⟩_𝑗)²𝜆_𝑖′𝜆_𝑗′(⟨𝑣⟩_𝑖′−⟨𝑣⟩_𝑗′)², (4.12)  (4.11) becomes

(Δ𝑥)²(Δ𝑣)²=_𝑗,𝑗′𝜆_𝑗(Δ𝑥)²_𝑗𝜆_𝑗′(Δ𝑣)²_𝑗′+Θ. (4.13) From (4.13) it results

(Δ𝑥)²(Δ𝑣)²=_𝑗′<𝑗𝜆_𝑗𝜆_𝑗′[(Δ𝑥)²_𝑗(Δ𝑣)²_𝑗′+(Δ𝑥)²_𝑗′(Δ𝑣)²_𝑗]+_𝑗𝜆²_𝑗(Δ𝑥)²_𝑗(Δ𝑣)²_𝑗+Θ. (4.14)

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

(Δ𝑥)²_𝑗(Δ𝑣)²_𝑗≥ℏ²4⇒(Δ𝑣)²_𝑗≥ℏ²41(Δ𝑥)²_𝑗 (4.15) which implies

(Δ𝑥)²_𝑗′(Δ𝑣)²_𝑗≥ℏ²4(Δ𝑥)²_𝑗′(Δ𝑥)²_𝑗. (4.16) Using (4.15) and (4.16) in (4.14) the following inequality is obtained:

(Δ𝑥)²(Δ𝑣)²≥ℏ²4_𝑗𝜆²_𝑗+ℏ²4_𝑗′<𝑗𝜆_𝑗𝜆_𝑗′[(Δ𝑥)²_𝑗(Δ𝑥)²_𝑗′+(Δ𝑥)²_𝑗′(Δ𝑥)²_𝑗=ℏ]+Θ₂4_𝑗𝜆²_𝑗+ℏ²4_𝑗′<𝑗𝜆_𝑗𝜆_𝑗′[𝑦_𝑗𝑗′+1𝑦_𝑗𝑗′]+Θ, (4.17) where the nonnegative variable

𝑦_𝑗𝑗′=(Δ𝑥)²_𝑗(Δ𝑥)²_𝑗′ (4.18) has been defined.

Noting

𝑦_𝑗𝑗′+1𝑦_𝑗𝑗′≥2,∀𝑦_𝑗𝑗′≥0, (4.19) (4.17) allows to write

(Δ𝑥)²(Δ𝑣)²≥ℏ²4[_𝑗𝜆²_𝑗+2_𝑗′<𝑗𝜆_𝑗𝜆_𝑗′ℏ]+Θ=²4[_𝑗𝜆_𝑗]²+Θ. (4.20) Finally, remembering that for the coefficients 𝜆_𝑗 the normalization relation holds

_𝑗𝜆_𝑗=1 (4.21) and that Θ is defined nonnegative, from (4.20) the following inequality is obtained:

ℏΔ𝑥Δ𝑣≥2 (4.22) which represents the Heisenberg uncertainty relation within the two-dimensional phase space {𝑥,𝑣}.

5. The Set of the Square-Integrable Functions: The Space 𝑆²

Consider the set {𝑓(𝑥,𝑣,𝑡)} constituted by the functions which obey the relation:

^+∞_−∞𝑑𝑥^+∞_−∞𝑑𝑣𝑓(𝑥,𝑣,𝑡)=1. (5.1)

Inside the set {𝑓(𝑥,𝑣,𝑡)} we characterize the subset formed by the square-integrable functions, that is the functions which satisfy the following relation

^+∞_−∞𝑑𝑥^+∞_−∞𝑑𝑣|𝑓(𝑥,𝑣,𝑡)|²<∞. (5.2)

This set, named 𝑆², possesses the structure of a vectorial space. In fact, if we consider two arbitrary functions 𝑓(𝑥,𝑣,𝑡), 𝑔(𝑥,𝑣,𝑡)∈𝑆² and two arbitrary complex numbers 𝜆₁, 𝜆₂, for the function 𝜆₁𝑓(𝑥,𝑣,𝑡)+𝜆₂𝑔(𝑥,𝑣,𝑡) the following relation holds:

^+∞_−∞𝑑𝑥^+∞_−∞𝑑𝑣|𝜆₁𝑓(𝑥,𝑣,𝑡)+𝜆₂𝑔(𝑥,𝑣,𝑡)|²≤∞. (5.3) 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 𝑆² satisfies all the properties of a vectorial space.

Moreover a scalar product (𝑓,𝑔) can be defined

(𝑓,𝑔)=^+∞_−∞𝑑𝑥^+∞_−∞𝑑𝑣𝑓^∗(𝑥,𝑣)𝑔(𝑥,𝑣), (5.4) which associates each pair of elements of 𝑆² with a complex number.

The set of the functions belonging to 𝑆² with the scalar product defined in (5.4) constitutes a Hilbert space.

6. A Basis of 𝑆²

In order to construct a basis of the vectorial space 𝑆² 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 𝑆_𝑛(𝑥) it is possible to introduce the set {𝑓_𝑛𝑛′(𝑥,𝑣)} with

𝑓_𝑛𝑛′(𝑥,𝑣)=^+∞_−∞𝑑𝑠𝑒^𝚤𝑠𝑣𝑒^{−(𝑥+(ℏ/2)𝑠)2/2}𝑒^{−(𝑥−(ℏ/2)𝑠)2/2}𝐻_𝑛(𝑥+(ℏ/2)𝑠)𝜋^1/22^𝑛𝐻𝑛!^𝑛′(𝑥−(ℏ/2)𝑠)𝜋^1/22^𝑛′𝑛^′!. (6.1) We propose to show that the set {𝑓_𝑛𝑛′(𝑥,𝑣)} constitutes a basis of 𝑆². This may be obtained by proving that {𝑓_𝑛𝑛′(𝑥,𝑣)} is an orthonormal and complete set of this space.

6.1. Orthonormality

In order to prove the orthonormality of the set {𝑓_𝑛𝑛′(𝑥,𝑣)}, we calculate the following integral:

^+∞_−∞𝑑𝑥^+∞_−∞𝑑𝑣𝑓_𝑛𝑛′(𝑥,𝑣)𝑓^∗_𝑚𝑚′=(𝑥,𝑣)_+∞−∞𝑑𝑥^+∞_−∞𝑑𝑣^+∞_−∞𝑑𝑠𝑒^𝚤𝑠𝑣𝑒^{−(𝑥+(ℏ/2)𝑠)2/2}𝑒^{−(𝑥−(ℏ/2)𝑠)2/2}×𝐻_𝑛(𝑥+(ℏ/2)𝑠)𝜋^1/22^𝑛𝐻𝑛!^𝑛′(𝑥−(ℏ/2)𝑠)𝜋^1/22^𝑛′𝑛^′!⋅^+∞_−∞𝑑𝑠^′𝑒^{−𝚤𝑠′𝑣}𝑒^{−(𝑥+(ℏ/2)𝑠′)2/2}𝑒^{−(𝑥−(ℏ/2)𝑠′)2/2}𝐻_𝑚(𝑥+(ℏ/2)𝑠^′)𝜋^1/22^𝑚𝐻𝑚!^𝑚′(𝑥−(ℏ/2)𝑠^′)𝜋^1/22^𝑚′𝑚^′!, (6.2) where 𝑓_𝑛𝑛′(𝑥,𝑣), 𝑓_𝑚𝑚′(𝑥,𝑣) are two arbitrary functions belonging to 𝑆². Performing the variable change

ℏ𝑦=𝑥+2ℏ𝑠,𝑧=𝑥−2𝑠, (6.3) and using the orthonormality of the Hermite functions, from (6.2) one gets

^+∞_−∞𝑑𝑥^+∞_−∞𝑑𝑣𝑓_𝑛𝑛′(𝑥,𝑣)𝑓^∗_𝑚𝑚′(𝑥,𝑣)=2𝜋𝛿_𝑛𝑚𝛿_{𝑛′𝑚′}, (6.4) which represents the orthonormality relation for the functions 𝑓_𝑛𝑛′(𝑥,𝑣).

6.2. Closure

To prove that the set {𝑓_𝑛𝑛′(𝑥,𝑣)} satisfies closure relation, we write

_𝑛𝑛′𝑓_𝑛𝑛′(𝑥,𝑣)𝑓^∗_𝑛𝑛′(𝑥^′,𝑣^′)=_𝑛𝑛′^+∞_−∞𝑑𝑠𝑒^𝚤𝑠𝑣𝑒^{−(𝑥+(ℏ/2)𝑠)2/2}𝑒^{−(𝑥−(ℏ/2)𝑠)2/2}×𝐻_𝑛(𝑥+(ℏ/2)𝑠)𝜋^1/22^𝑛𝐻𝑛!^𝑛′(𝑥−(ℏ/2)𝑠)𝜋^1/22^𝑛′𝑛^′!⋅^+∞_−∞𝑑𝑠^′𝑒^{−𝚤𝑠′𝑣′}𝑒^{−(𝑥′+(ℏ/2)𝑠′)2/2}𝑒^{−(𝑥′−(ℏ/2)𝑠′)2/2}𝐻_𝑛(𝑥^′+(ℏ/2)𝑠^′)𝜋^1/22^𝑛𝐻𝑛!^𝑛′(𝑥^′−(ℏ/2)𝑠^′)𝜋^1/22^𝑛′𝑛^′!. (6.5) Applying the closure relation of the Hermite functions, from (6.5), one gets

_𝑛𝑛′𝑓_𝑛𝑛′(𝑥,𝑣)𝑓^∗_𝑛𝑛′(𝑥^′,𝑣^′)=2𝜋𝛿(𝑥−𝑥^′)𝛿(𝑣−𝑣^′), (6.6) which expresses the closure relation. The set {𝑓_𝑛𝑛′(𝑥,𝑣)}, which is orthonormal and complete, constitutes a basis of the space 𝑆².

7. Functions of the Space 𝑆² and Solutions of the Quantum Liouville Equation

Consider an arbitrary function 𝑓(𝑥,𝑣,𝑡)∈𝑆². Its expansion in terms of basis vectors 𝑓_𝑛𝑛′(𝑥,𝑣) is given by

𝑓(𝑥,𝑣,𝑡)=_𝑛𝑛′𝐴_𝑛𝑛′(𝑡)𝑓_𝑛𝑛′(𝑥,𝑣)=_𝑛𝑛′𝐴_𝑛𝑛′(𝑡)^+∞_−∞𝑑𝑠𝑒^𝚤𝑠𝑣𝑆_𝑛ℏ(𝑥+2𝑠)𝑆_𝑛′ℏ(𝑥−2𝑠), (7.1) where the coefficients 𝐴_𝑛𝑛′(𝑡) are complex numbers.

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

^+∞_−∞𝑑𝑥^+∞_−∞𝑑𝑣_𝑛𝑛′𝐴_𝑛𝑛′(𝑡)^+∞_−∞𝑑𝑠𝑒^𝚤𝑠𝑣𝑆_𝑛ℏ(𝑥+2𝑠)𝑆_𝑛′ℏ(𝑥−2𝑠)=1, (7.2) which determines

_𝑛𝑛′𝐴_𝑛𝑛′(𝑡)^+∞_−∞𝑑𝑠^+∞_−∞𝑑𝑣𝑒^{𝚤𝑠𝑣+∞}_−∞𝑑𝑥𝑆_𝑛ℏ(𝑥+2𝑠)𝑆_𝑛′ℏ(𝑥−2𝑠)=2𝜋_𝑛𝑛′𝐴_𝑛𝑛′(𝑡)^+∞_−∞𝑑𝑠𝛿(𝑠)^+∞_−∞𝑑𝑥𝑆_𝑛ℏ(𝑥+2𝑠)𝑆_𝑛′ℏ(𝑥−2𝑠)=2𝜋_𝑛𝑛′𝐴_𝑛𝑛′(𝑡)^+∞_−∞𝑑𝑥𝑆_𝑛(𝑥)𝑆_𝑛′(𝑥)=2𝜋_𝑛𝑛′𝐴_𝑛𝑛′(𝑡)𝛿_𝑛𝑛′=2𝜋_𝑛𝐴_𝑛𝑛(𝑡)=1. (7.3) Analogously, using (5.2) and (7.1), we get

^+∞_−∞𝑑𝑥^+∞_−∞𝑑𝑣|𝑓(𝑥,𝑣,𝑡)|²=2𝜋_𝑛𝑛′|𝐴_𝑛𝑛′(𝑡)|²<∞. (7.4) From (7.3) and (7.4) it results that the following conditions hold:

2𝜋_𝑛𝐴_𝑛𝑛(𝑡)=12𝜋_𝑛𝑛′|𝐴_𝑛𝑛′(𝑡)|²<∞. (7.5) Consider again the expansion (7.1) which holds for an arbitrary function 𝑓(𝑥,𝑣,𝑡)∈𝑆². Now we introduce another hypothesis for the coefficients 𝐴_𝑛𝑛′(𝑡), that is, we suppose that the matrix 𝐴(𝑡)=‖𝐴_𝑛𝑛′(𝑡)‖ can be put in diagonal form in 𝐿² (𝐿² is the space which contains the square-integrable Schrödinger functions). If 𝐷 is the matrix which diagonalizes 𝐴(𝑡), it results

𝐷⁻¹𝐴(𝑡)𝐷=𝐴^𝐷(𝑡) (7.6) with

𝐴^𝐷_𝑛𝑛′𝐴(𝑡)=^𝐷𝑛𝑛(𝑡),(𝑛=𝑛^′),0,(𝑛≠𝑛^′). (7.7) This transformation induces in 𝐿² the basis change

𝑆_𝑛(𝑥)=𝐷𝑃_𝑛(𝑥) (7.8) where 𝑃_𝑛(𝑥) are the vectors belonging to the new basis. The matrix elements of ‖𝐴_𝑛𝑛′(𝑡)‖ represent in the space 𝑆² the coefficients of the expansion (7.1) on the basis vectors {𝑓_𝑛𝑛′(𝑥,𝑣)}. The diagonalization defined by (7.6) induces within the space 𝑆² the transformation

𝑓_𝑛𝑛′(𝑥,𝑣)→𝑔_𝑛𝑛′(𝑥,𝑣) (7.9) with

𝑔_𝑛𝑛′(𝑥,𝑣)=^+∞_−∞𝑑𝑠𝑒^𝚤𝑠𝑣𝑃_𝑛ℏ(𝑥+2𝑠)𝑃^∗_𝑛′ℏ(𝑥−2𝑠). (7.10)

By following a procedure similar to that used for the set {𝑓_𝑛𝑛′(𝑥,𝑣)} it is easy to verify that the functions 𝑔_𝑛𝑛′(𝑥,𝑣) satisfy relations both of ortonormality and closure. Therefore they constitute a new basis of 𝑆². Equation (7.1) then becomes

𝑓(𝑥,𝑣,𝑡)=_𝑛𝑛′𝐴^𝐷_𝑛𝑛′(𝑡)𝛿_𝑛𝑛′𝑔_𝑛𝑛′(𝑥,𝑣)=^+∞_−∞𝑑𝑠𝑒^𝚤𝑠𝑣_𝑛𝜆_𝑛(𝑡)𝜌_𝑛(𝑥,𝑠), (7.11) where we set

𝜆_𝑛(𝑡)=𝐴^𝐷_𝑛𝑛𝜌(𝑡),_𝑛(𝑥,𝑠)=𝑃_𝑛ℏ(𝑥+2𝑠)𝑃^∗_𝑛ℏ(𝑥−2𝑠). (7.12) Now we define the density matrix

𝜌(𝑥,𝑠,𝑡)=_𝑛𝜆_𝑛(𝑡)𝜌_𝑛(𝑥,𝑠). (7.13) Using (7.13) in (7.11) allows to write

𝑓(𝑥,𝑣,𝑡)=^+∞_−∞𝑑𝑠𝑒^𝚤𝑠𝑣𝜌(𝑥,𝑠,𝑡). (7.14) Equation (7.14) indicates that any function 𝑓(𝑥,𝑣,𝑡)∈𝑆², whose matrix ‖𝐴_𝑛𝑛′(𝑡)‖ 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 𝜓(𝑥) of the Schrödinger equation, because of its expansibility in plane waves,

1𝜓(𝑥)=√2𝜋^+∞−∞𝑑𝑘𝜓(𝑘)𝑒^𝚤𝑘𝑥, (8.1) verifies the inequality

Δℏ𝑋Δ𝑃≥2, (8.2) where 𝑋 and 𝑃 are observables which satisfy the commutation rule

[𝑋,𝑃]=𝚤ℏ. (8.3)

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 𝑓(𝑥,𝑣,𝑡) of an arbitrary wave function 𝜓(𝑥,𝑡). So, we found that the product of the variances Δ𝑥 and Δ𝑣, 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 Δ𝑥 and Δ𝑣 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 𝑓(𝑥,𝑣,𝑡)∈𝑆² admits the following expansion:

𝑓(𝑥,𝑣,𝑡)=_𝑛𝑛′𝐴_𝑛𝑛′(𝑡)^+∞_−∞𝑑𝑠𝑒^𝚤𝑠𝑣𝑆_𝑛ℏ(𝑥+2𝑠)𝑆_𝑛′ℏ(𝑥−2𝑠), (8.4) and this expression can be written as Fourier transform of a density matrix provided that the matrix ‖𝐴_𝑛𝑛′(𝑡)‖ 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.