0 Quantum Mechanics and Statistical Description of Results of Measurement

Quantum mechanics and its meaning have been discussed in a large number of publications from many different points of view (see e.g. books (Auletta, 2001; Wheeler & Zurek, 1981)). It shows that quantum mechanics is, despite its successful applications, difficult to understand. In this chapter, we discuss quantum mechanics from the point of view of mathematical statistics and show that the most important parts of the mathematical formalism of quantum mechanics can be derived from the statistical description of results of measurement. Various aspects of this approach can be found for example in (Frieden, 1998; 2004; Frieden & Soffer, 1995; Kapsa & Skála, 2009; 2011; Kapsa et al., 2010; Reginatto, 1998; 1999; Skála & Kapsa, 2005a;b; 2007a;b; 2011; Skála, Čížek & Kapsa, 2011). One of the main differences between classical and quantum mechanics is consistent statistical description of results of measurement in quantum mechanics. In contrast to classical mechanics according to which physical measurement can be made in principle arbitrarily exact, quantum mechanics takes into consideration physical reality confirmed by experiments and describes physical measurement statistically. The most important points of the statistical description of measurement of the space coordinate x are summarized in Section 2. An important quantity appearing in this approach is the probability density ρ(x, t) of obtaining the value x in measurement made at time t. For the sake of simplicity, only one spatial coordinate x is taken here. Due to the normalization condition for the probability density corresponding to the fact that the measured system must be somewhere in space the probability density ρ must obey the continuity equation analogous to that known from classical continuum mechanics. Therefore, except for ρ, we have to take into account also the corresponding probability density current j(x, t) appearing in the continuity equation. We note that the density current j is also necessary for describing the motion in space. To describe the statistical state of the system, both quantities ρ and j are necessary. It is shown in Section 3 that instead of two real quantities ρ and j, we can use also two real functions s1(x, t) and s2(x, t) given by equations ρ = exp(−2s2/h̄) and j = ρ v = ρp/m = ρ(∂s1/∂x)/m, where s1 corresponds to the Hamilton action S in the expression p = ∂S/∂x known from the Hamilton–Jacobi theory of classical mechanics. More compact way of describing the statistical state of the system is to use the complex wave function ψ = exp[(is1 − s2)/h̄] as it is done in quantum mechanics. We note that the expression for the probability density current j = ρ(∂s1/∂x)/m is equivalent to the expression for the probability density current known from quantum mechanics. 10


Introduction
Quantum mechanics and its meaning have been discussed in a large number of publications from many different points of view (see e.g.books (Auletta, 2001;Wheeler & Zurek, 1981)).It shows that quantum mechanics is, despite its successful applications, difficult to understand.In this chapter, we discuss quantum mechanics from the point of view of mathematical statistics and show that the most important parts of the mathematical formalism of quantum mechanics can be derived from the statistical description of results of measurement.Various aspects of this approach can be found for example in (Frieden, 1998;2004;Frieden & Soffer, 1995;Kapsa & Skála, 2009;2011;Kapsa et al., 2010;Reginatto, 1998;1999;Skála & Kapsa, 2005a;b;2007a;b;2011;Skála, Čížek & Kapsa, 2011).One of the main differences between classical and quantum mechanics is consistent statistical description of results of measurement in quantum mechanics.In contrast to classical mechanics according to which physical measurement can be made in principle arbitrarily exact, quantum mechanics takes into consideration physical reality confirmed by experiments and describes physical measurement statistically.The most important points of the statistical description of measurement of the space coordinate x are summarized in Section 2. An important quantity appearing in this approach is the probability density ρ(x, t) of obtaining the value x in measurement made at time t.For the sake of simplicity, only one spatial coordinate x is taken here.Due to the normalization condition for the probability density corresponding to the fact that the measured system must be somewhere in space the probability density ρ must obey the continuity equation analogous to that known from classical continuum mechanics.Therefore, except for ρ, we have to take into account also the corresponding probability density current j(x, t) appearing in the continuity equation.We note that the density current j is also necessary for describing the motion in space.To describe the statistical state of the system, both quantities ρ and j are necessary.It is shown in Section 3 that instead of two real quantities ρ and j, we can use also two real functions s 1 (x, t) and s 2 (x, t) given by equations ρ = exp(−2s 2 /h) and j = ρ v = ρp/m = ρ(∂s 1 /∂x)/m, where s 1 corresponds to the Hamilton action S in the expression p = ∂S/∂x known from the Hamilton-Jacobi theory of classical mechanics.More compact way of describing the statistical state of the system is to use the complex wave function ψ = exp[(is 1 − s 2 )/h] as it is done in quantum mechanics.We note that the expression for the probability density current j = ρ(∂s 1 /∂x)/m is equivalent to the expression for the probability density current known from quantum mechanics.
By analogy with the expression for the momentum p = ∂S/∂x one can make an attempt to represent the momentum by the function p = ∂s 1 /∂x.It is shown in Sections 4 and 5 that in case of the mean momentum p and the mean value xp this definition gives the same results as the quantum-mechanical representation of the momentum p = −ih(∂/∂x).However, it is not true in more complicated cases when the operator representation of the momentum has to be used.One of important quantities appearing in mathematical statistics is the Fisher information.It is shown in Section 6 that the Fisher information I x = (4/h 2 ) ∞ −∞ (∂s 2 /∂x) 2 ρ dx fulfills the inequality (x − a) 2 I x ≥ 1, where a is a real constant.This inequality is analogous to the uncertainty relations known from quantum mechanics and is general property of statistical theories similar to that used in quantum mechanics.It is shown in Section 7 that the kinetic energy in quantum mechanics can be written as a sum of two terms.The first term is statistical generalization of the kinetic energy known from classical mechanics.The second part of the kinetic energy is proportional to the Fisher information I x and does not have its counterpart in classical mechanics.Therefore, in contrast to classical mechanics, the Fisher information is an important part of the kinetic energy in quantum mechanics.Similarly to the kinetic energy, the mean value (Δp) 2 appearing in the Heisenberg uncertainty relation (Δx) 2 (Δp) 2 ≥ h2 /4 can be written as a sum of two terms (Δp) 2 = (Δp 1 ) 2 + (Δp 2 ) 2 (Section 8).Again, the first term can be understood as statistical generalization of the expression known from classical mechanics.The second term is proportional to the Fisher information I x .If the first term equals zero, the Heisenberg uncertainty relation is equivalent to the inequality for the Fisher information mentioned above.It shows that the inequality for the Fisher information is in quantum mechanics correctly respected.It is shown in Section 9 that the Heisenberg uncertainty relation can be replaced by two stronger uncertainty relations for (Δp 1 ) 2 and (Δp 2 ) 2 .The sum of these two uncertainty relations is equivalent to the Robertson-Schrödinger uncertainty relation (Section 10).By neglecting one term at the right-hand side of the Robertson-Schrödinger uncertainty relation the Heisenberg uncertainty relation is obtained.Therefore, two uncertainty relations discussed in Section 9 are stronger than the corresponding Heisenberg and Robertson-Schrödinger uncertainty relations.It is worth noting that the second uncertainty relation equivalent to the inequality for the Fisher information depends only on the function s 2 or the envelop of the wave function |ψ|.Since it does not depend on s 1 , inequality in this relation can be achieved for much larger class of the wave functions than in case of the Heisenberg and Robertson-Schrödinger uncertainty relations.It may be important in some applications as for example in the theory of the most efficient information transfer.Two examples illustrating results of Sections 8-10, namely the gaussian wave packet for a free particle and the linear harmonic oscillator are discussed in Sections 11 and 12.By using the normalization condition for ρ = |ψ| 2 it is possible to derive the equation indicating validity of the commutation relation [x, p] = ih (Section 13).This commutation relation shows that it is possible to replace the momentum operator p by the operator p − f , where f (x, t) is a real function.This function can describe external conditions in which the system moves and corresponds to the x-component of the vector potential.In standard quantum mechanics, systems with the infinite lifetime are usually considered.In such a case, the normalization condition for the probability density ∞ −∞ ρ dx = 1 is valid at all times and it does make sense to introduce the probability density in time analogous to the probability density in space.For this reason, time is taken as a parameter in standard quantum mechanics.In Section 14, systems with a finite lifetime are considered and a decaying probability to find the system anywhere in space ν(t) = ∞ −∞ ρ dx is introduced.It makes possible to define the mean lifetime and other quantities by analogy with those for the coordinate x.Similarly to Section 13, it is then possible to get the commutation relation for the operator ih(∂/∂t) and time t and to find mathematical arguments for the existence of the scalar potentials (Section 15).For systems with exponentially decaying wave functions, it possible to derive also the time-energy uncertainty relations (Section 16).Equations of motion are discussed in Section 17.To derive the equation of motion, the Fisher information I x defined for the space coordinate x is first generalized to two Fisher informations J x and J t in space-time in which the derivatives of the functions s 1 and s 2 with respect to x and t are taken into account.Then, the combined space-time Fisher information J t /c 2 ± J x is discussed.Further, we require that our theory is independent of the choice of the coordinate system in space-time and the concrete initial conditions.It yields the equation J t /c 2 − J x = const, where the signs of the space and time parts are different similarly to the signs in the metric in special relativity and const ≥ 0. Formulating this condition in the variational form, it leads to the equation of motion mathematically equivalent to the Klein-Gordon equation.The Schrödinger equation can be viewed as the non-relativistic approximation to the Klein-Gordon equation.The Dirac equation can be obtained in a similar way.It is shown also that the equations of motion in quantum mechanics should be linear.

Statistical description of results of measurement
In this section, we discuss probably the most important difference between classical and quantum mechanics -statistical description of results of measurement.We note that the measuring apparatus is not described in quantum mechanics on the microscopic level and the measured system interacts with the measuring apparatus.For this reason, results of measurement have to be described statistically.In agreement with experimental experience, we assume that results of repeated measurement of the coordinate x can be characterized by the mean values (1) and the corresponding mean square displacement Here, Δx = x − x (4) an ρ(x, t) ≥ 0 is a normalized probability density giving the probability of obtaining the value 207 Quantum Mechanics and Statistical Description of Results of Measurement For the sake of simplicity, we assume that ρ(x, t) fulfills the boundary conditions We assume also that in the limit of classical mechanics the mean coordinate x converges to the classical coordinate x cl = x cl (t).

Wave function
From the point of view of our statistical description, the wave function ψ can be introduced in the following simple way.First, we introduce a real function s 2 = s 2 (x, t) by the equation or equivalently where h denotes the reduced Planck constant, h = h/(2π).We note that the transition ρ(x, t) → δ(x − x cl ) can be formally performed for h → 0 + .Due to normalization condition ( 5), the probability density ρ has to obey the continuity equation ∂ρ ∂t where j = j(x, t) is the probability density current in one dimension and ∂j/∂x is the divergence in one dimension.Analogously to continuum mechanics, it is possible to express the probability density current j in terms of the "velocity" Further, by analogy with the expression v = p/m = (∂S/∂x)/m from the Hamilton-Jacobi theory we can write where m is the mass of the system, a real function s 1 = s 1 (x, t) corresponds to the Hamilton action S = S(x, t) and the function ∂s 1 /∂x represents the momentum in our statistical approach.In the limit of classical mechanics when the statistical description disappears, the function s 1 has to fulfill the condition s 1 (x, t) → S(x cl , t) and ∂s 1 /∂x → ∂S/∂x.It is seen that instead of two quantities ρ and j, the statistical state of the system can be described by two mutually independent real functions s 1 and s 2 or a new complex function depending on s 1 and s 2 (see also (Madelung, 1926)).Using this function, the probability density ρ and probability density current j given above can be rewritten in the form known from quantum mechanics ρ = |ψ| 2 (14) and where the star denotes the complex conjugate.The function ψ called the wave function in quantum mechanics is only a different way of representing the statistical state of the system described by two real functions ρ and j or s 1 and s 2 .We note that our expression for the wave function ( 13) is similar to that of Bohm (Bohm, 1952a;b).However, we do not assume the existence of hidden variables here.

Momentum operator
By analogy with Eq. ( 1) and our discussion in the preceding section, the mean momentum can be defined as (see also (Skála, Čížek & Kapsa, 2011)) It follows from conditions (6) that the integral equals zero.Using this result it is easy to verify that Eq. ( 16) can be also written as where the momentum operator equals Equations ( 16) and ( 18) yield the same result and representation of the momentum by the function ∂s 1 /∂x and the operator p is in this case equivalent.

Mean value of xp
In this section, we investigate the mean value of the product of the coordinate and momentum which is important in the uncertainty relations (see also (Skála, Čížek & Kapsa, 2011)).
As it is known, the mean value of the product of the coordinate and momentum is in quantum mechanics given by the expression Using Eq. ( 13) we get 209 Quantum Mechanics and Statistical Description of Results of Measurement By using integration by parts in the last integral and Eqs. ( 5) and ( 6) we get The resulting formula agrees with the expression analogous to Eqs. ( 1) and ( 16).Summarizing results of the last two sections we see that contribution of the function ∂s 2 /∂x to the mean values p and ( x p + px )/2 equals zero and the momentum operator can be in these cases represented either by the function p = ∂s 1 /∂x or the operator p = −ih(∂/∂x).
However, as it will be seen in the following sections, it it not true in more complicated cases.
We start with normalization condition (5) for the probability density ρ in which we perform integration by parts where a is an arbitrary real number.Taking into account Eq. ( 6) we get the starting point of the following discussion Now we make use of the Schwarz inequality for the inner product 210 Theoretical Concepts of Quantum Mechanics in inequality (28) and using Eq. ( 27) we get Here, the second integral is the well-known quantity from mathematical statistics called the Fisher information Inequality ( 31) is usually written in the form (Fisher, 1925) This result is very general and does not depend on the concrete meaning of the variable x.
Interpretation of the last inequality is similar to that of the uncertainty relations in quantum mechanics: For given I x the integral (x − a) 2 cannot be smaller than 1/I x and vice versa.
The minimum of the integral (x − a) 2 is obtained for a = x .We note that inequality (33) in a more general form is known in mathematical statistics as the Rao-Cramér inequality (Cover & Thomas, 1991;Cramér, 1946a;b;Rao, 1945;1992).Hence, any correctly formulated statistical theory has to lead to inequality (33) or an analogous one.Using Eq. ( 8) for the probability density the Fisher information can be written in the equivalent form which will appear in the following discussion.

Kinetic energy
Now we discuss the kinetic energy T in quantum mechanics where q denotes the charge, m the mass and A is the vector potential in one dimension (see also (Skála, Čížek & Kapsa, 2011)).Using Eq. ( 13) for the wave function and Eq. ( 19) for the momentum operator we get and Therefore, the kinetic energy can be written as a sum of two terms where and The first term T 1 is statistical generalization of the kinetic energy known from classical mechanics.The second part of the kinetic energy T 2 depending on ∂s 2 /∂x is proportional to the Fisher information I x and does not have its counterpart in classical mechanics.Therefore, in contrast to classical mechanics, the Fisher information is an important part of the kinetic energy in quantum mechanics.

Heisenberg uncertainty relations
For the sake of simplicity, we assume that the potential A equals zero.The Heisenberg uncertainty relation (Heisenberg, 1927) for the coordinate x and momentum p has the form where and Using Eqs. ( 13), ( 14) and ( 17) we get where and We see that, analogously to the kinetic energy T, the mean square deviation of the momentum (Δp) 2 can be split into two parts (see also (Kapsa & Skála, 2011;Skála & Kapsa, 2011;Skála, Čížek & Kapsa, 2011)).
The first part (Δp 1 ) 2 corresponds to the representation of the momentum by the function p = ∂s 1 /∂x and the first part of the kinetic energy T 1 .
According to the well-known result of mathematical statistics, the product of variances of two quantities is greater than or equal to the square of their covariance (Cramér, 1946b).In the following cases, it is equivalent to Schwarz inequality (28) with a suitable choice of the functions u and v. First, we put Then, the Schwarz inequality yields the first uncertainty relation As it follows from section 5, the function ∂s 1 /∂x in the last integral represents the momentum and this relation has the usual above mentioned meaning known from mathematical statistics.Depending on the functions s 1 and s 2 , the square of the covariance of the coordinate and momentum at the right-hand side of this relation can have arbitrary values greater than or equal to zero.
The second uncertainty relation can be obtained in an analogous way for with the result It follows from Eq. ( 17) that the right-hand side of this relation can be simplified Then, Eq. ( 23) leads to the final form of the second uncertainty relation This uncertainty relation follows from the Schwarz inequality in a similar way as the first one, however, the covariance (u, v) is in this case constant and equals h/2 > 0 independently of the concrete form of the functions s 2 or ρ.We note also that relation ( 55) is for x = a equivalent to inequality (33) for the Fisher information.It confirms again that inequality (33) is in quantum mechanics correctly respected.
Analogous uncertainty relations can be derived also in the multidimensional case (Skála & Kapsa, 2008;2009) and for the mixed states described by the density matrix (Skála & Kapsa, 2009).The sum of uncertainty relations ( 50) and ( 55) gives the relation Heisenberg uncertainty relation ( 42) can be obtained from this relation by neglecting the first term on its right-hand side.Therefore, uncertainty relations ( 50) and ( 55) are stronger than the corresponding Heisenberg uncertainty relation (42).
For two linear hermitian operators Â and B, the Robertson-Schrödinger uncertainty relation can be written in the form where Â = ψ| Âψ is the mean value of the operator Â in the state described by the wave function ψ, Δ Â = Â − Â , { Â, B} = Â B + B Â denotes the anticommutator and [ Â, B] = Â B − B Â the commutator of the operators Â and B.
For the operators x = x and p = −ih(∂/∂x) the straightforward calculation yields Further, taking into account the commutation relation [x, p] = ih, relation (57) leads to Eq. ( 56).Therefore, relations (50) and ( 55) are stronger than both the Heisenberg and Robertson-Schrödinger relations ( 42) and ( 56) and yield more detailed information in terms of the mean square deviations (Δx) 2 , (Δp 1 ) 2 and (Δp 2 ) 2 .For the momentum represented by the function p = ∂s 1 /∂x, the mean value [Δx, Δp] equals zero and the Heisenberg and Robertson-Schrödinger uncertainty relations ( 42) and ( 56) do not contain the term h2 /4.It shows again that this representation of the momentum is not, except for the cases discussed in sections 4 and 5, correct.The equality sign in Schwarz inequality (28) is obtained if the functions u and v are collinear, i.e. for u = const v, where const is a complex number.However, since the functions s 1 , s 2 and ρ are real, the corresponding functions u and v are also real.Therefore, const must be a real number or a real function of t.It follows from the conditions u = const v for the functions s 1 and s 2 that these functions have to be quadratic functions of x of the form p(t)x 2 + q(t)x + r(t), where real coefficients p(t), q(t) and r(t) can depend on time.
It is worth to notice that the condition for the equality sign in relation ( 55) is independent of the form of the function s 1 .Therefore, the equality sign in this relation can be achieved for much larger class of the wave functions than in case of the Heisenberg or Robertson-Schrödinger uncertainty relations.It is interesting not only from the theoretical point of view but also from the point of view of some applications.
We assume that the wave function of a free particle is at time t = 0 described by the gaussian wave packet with the energy where a > 0 and k are real constants.By solving the time Schrödinger equation we get The corresponding functions s 1 and s 2 and their derivatives equal and As it could be anticipated, the mean momentum and the mean coordinate equal p = ∂s 1 ∂x = hk (66) and The mean square deviations of the coordinate and momentum are given by the equations and The left-hand side of relation ( 50) equals Calculating the right-hand side of this relation we get the same result Therefore, uncertainty relation ( 50) is fulfilled with the equality sign.
Calculating the left-hand side of uncertainty relation (55) we obtain and see that uncertainty relation ( 55) is fulfilled with the equality sign, too.The corresponding Robertson-Schrödinger uncertainty relation has the form and is fulfilled with the equality sign for all t ≥ 0. The Heisenberg uncertainty relation ( 42) for our wave packet can be obtained if the first term on the right-hand side of the last equation is neglected.

Linear harmonic oscillator
The second example of application of uncertainty relations ( 50) and ( 55) is the linear harmonic oscillator in the coherent state described at time t = 0 by the gaussian wave packet (Skála, Kapsa & Lužová, 2011) where and x 0 is the center of the packet.The corresponding energy E equals By solving the time Schrödinger equation we get The corresponding functions s 1 and s 2 equal and The mean momentum and the mean coordinate have the same form as in classical mechanics and x = x 0 cos(ωt).
The mean square deviations of the coordinate and momentum from their mean values are given by the equations It means that uncertainty relations ( 50) and ( 55) have the form It is seen that uncertainty relation (50) has in this case very simple form 0 = 0.It follows from equation ( 88) that the left-hand side of relation (55) achieves for this example its minimum h2 /4.

Commutation relations and vector potential
To introduce potentials, we make use of Eq. ( 27) with a = 0 Using Eq. ( 14) we get Performing integration by parts in the first term and taking into account conditions (6) we have Multiplying this equation by −ih we obtain the equation indicating validity of the commutation relation known from quantum mechanics.Further, it is seen that Eq. ( 92) is valid also in case that the momentum operator p is replaced by p − f , where f = f (x, t) is a real function.This function can describe external conditions in which the system moves.In physics, such functions are usually denoted as the vector potential.For example, the function f can equal qA in Eq. ( 35) (see also (Kapsa & Skála, 2011;Skála & Kapsa, 2005a;b;2007a)).

Time
Systems investigated in standard quantum mechanics are supposed to have infinite lifetime.Therefore, normalization condition ( 5) is for such systems valid at all times t from the preparation of the system in a state described by ψ at time t = t 1 to the subsequent measurement at later time t 2 .Therefore, the probability to find the measured system anywhere in space equals one for all times t 1 ≤ t ≤ t 2 .For this reason, it does not make sense to introduce the probability density in time analogous to the probability density in space and time is taken as a parameter in standard quantum mechanics.Rather different situation is obtained if we assume that the investigated state has a finite lifetime and the probability to find the system anywhere in space decays in time (see also (Kapsa & Skála, 2011;Skála & Kapsa, 2005a;b;2007a)).Normalization of ν is given by the equation 218 Theoretical Concepts of Quantum Mechanics expressing the fact that, after its preparation at time t = t 1 , the investigated state decays with the probability equal to one.This generalization includes standard quantum mechanics with the infinite lifetime as a limit case.By analogy with the coordinate x, it is then possible to define the mean lifetime τ the mean value of the square of time and the corresponding mean square deviation

Scalar potential
Similarly to Eq. ( 26), we perform integration by parts with respect to time in Eq. ( 95) and get By analogy with Eq. ( 6) we can assume validity of conditions lim t→t 1 and lim Using Eqs. ( 14), ( 94), ( 100) and ( 101) we get from Eq. (99 Performing integration by parts in the first term and taking into account Eqs. ( 100) and ( 101) we have Multiplying this equation by −ih we obtain the equation This result indicates that for systems with a finite lifetime the operator ih(∂/∂t) has analogous mathematical properties as the momentum operator p. Further, it is seen that Eq. ( 99) remains valid even in case when the operator ih(∂/∂t) is replaced by the operator ih(∂/∂t) − g, where g = g(x, t) is a real function.Analogously to the function f , the function g can describe external conditions in which the system moves.For example, the function g can equal qV, where q is the charge and V the scalar potential of the electromagnetic field (see also (Skála & Kapsa, 2005a;b;2007a)).

Time-energy uncertainty relations
To derive the "time-energy" uncertainty relation, we start from the equation that is a bit more general than Eq. ( 99) and is analogous to Eq. ( 27) (see also (Skála & Kapsa, 2007a) By using Schwarz inequality (28) for where and we get the inequality analogous to inequality (31).It is a general form of the "time-energy" uncertainty relation.
As an example, we assume now that the probability ν(t) to find the system in state ψ decays exponentially in time where τ denotes the lifetime.The corresponding mean values t − t 1 , (t − t 1 ) 2 and [Δ(t − t 1 )] 2 equal Further, we assume that the wave function describing the state with a finite lifetime has the following simple form where E 1 and E 2 > 0 are the real and imaginary part of the energy, respectively, and ψ 0 (x) is the space part of the wave function.Then, using Eq. ( 114), we calculate the second integral in Eq. ( 109) and get The resulting time-energy uncertainty relation has the form 220 Theoretical Concepts of Quantum Mechanics This relation shows that the lifetime and imaginary part of the energy are not independent and obey the well-known time-energy uncertainty relation.
To determine the shape of the corresponding spectral line it is necessary to calculate the Fourier transform of the function (110).As a result, the Lorentz form of the spectral line is obtained.

Equations of motion
As mentioned above, to describe motion in space both the probability density ρ and probability density current j or the functions s 1 and s 2 have to be used.To describe time evolution, integration in the Fisher information should be obviously performed not only over the space coordinates but also over time.For these reasons and in agreement with the last three sections, we define a generalized space Fisher information in the form (see also (Kapsa & Skála, 2011;Skála & Kapsa, 2005a;b;2007a)) Analogously, we define a generalized time Fisher information where ψ = exp[(is 1 − s 2 )/h] and ρ = |ψ| 2 .Both generalized Fisher informations depend on the space and time derivatives of the functions s 1 and s 2 in a similar way.Since there are no potentials in the last two equations, they correspond to a free motion.
To find equations of motion, we need an additional physical principle.To describe physical phenomena in a way independent of the choice of the concrete coordinate system and the state of the investigated system, we require that the combined generalized space-time Fisher information equals a real constant const Here, c is the speed of light and the sign in front of the generalized spatial Fisher information J x can be either plus or minus.First we notice that the space initial conditions for the wave function ψ at t = 0 can be from the mathematical point of view chosen arbitrarily and J x can have arbitrary values greater than or equal to zero.In contrast to it, the wave function ψ at later times is given by the evolution consistent with Eq. ( 119).It makes possible to derive the equation of motion from this equation.Further, to determine the sign in Eq. ( 119), we consider a free particle which is at rest in a given coordinate system.It follows from Eq. ( 38) with A = 0 that it is obtained for very small values of |∂s 1 /∂x| and |∂s 2 /∂x|.In such a case, the Fisher information J x is close to zero and Eq. ( 119) yields const ≥ 0. (120) Then, we consider a particle having a large kinetic energy T and a large Fisher information J x > const.In such a case, it is impossible to obey Eq. ( 119) with the plus sign.Therefore, we can conclude that the sign in Eq. ( 119) must be negative It is seen that this combination of the Fisher informations J t and J x is Lorentz invariant.
The last equation can be rewritten into the form This equation must be valid for arbitrary initial conditions at t = t 1 , i.e., it has to be independent of ψ.Therefore, its variation must equal zero where δψ denotes the variation of ψ.Performing integration by parts with respect to t in the first term and with respect to x in the second one and assuming that variations δψ and δψ * equal zero at the borders of the integration region we get This equation has to be obeyed for arbitrary values of δψ and δψ * .It leads to the equation of motion We see that except for the number of space dimensions and the constant const, this equation has the same mathematical form as the Klein-Gordon equation known from quantum mechanics where m 0 is the rest mass.We note that another derivation of the Klein-Gordon equation is given in (Frieden, 1998;2004;Frieden & Soffer, 1995;Reginatto, 1998;1999).As it is known, the Schrödinger equation for a free particle can be obtained from the Klein-Gordon equation in the non-relativistic approximation for the function ψ = e m 0 c 2 t/(ih) ϕ, (128 where ϕ is the wave function appearing in the Schrödinger equation. The Dirac equation for a free particle can be also obtained in a similar way (see also (Frieden, 1998;2004;Frieden & Soffer, 1995;Skála & Kapsa, 2005a;b;2007a)).
Potentials can be included into the theory by the method described in sections 13 and 15.
It worth to notice that the equations of motion discussed above are linear and the superposition principle is for them valid.This property can be traced back to the expression (32) for the Fisher informations I x .By using the substitution ρ = exp(−2s 2 /h), I x can be written in terms of the square of the function ∂s 2 /∂x (see Eq. ( 34)).Similar approach is used in Eqs. ( 117) and (118) for J x and J t , too.Then, using Eq. ( 121) and performing the variations and integration by parts in Eq. ( 122), the squares of the functions ∂ψ/∂x and ∂ψ/∂t disappear and the second partial derivatives of ψ with respect to the coordinates and time are obtained.Therefore, the resulting equations of motion are linear.The role of the operator ih(∂/∂t) is different from the role of the energy operatorhamiltonian.In agreement with discussion in this section, the operator ih(∂/∂t) is important for describing the time evolution of the wave function given by the equations of motion.We note also that condition (121) of the constant value of the generalized space-time Fisher information expressed in the variational form yields in the limit of classical mechanics the Hamilton principle (Kapsa & Skála, 2009).
Finally we note that quantization known from quantum mechanics is consequence of the statistical description of results of measurement and boundary conditions applied to the wave function ψ.As it is known, only some solutions of equations of motion obey these conditions and possible states of quantum systems can be quantized.

Conclusion
We have shown that the basic mathematical structure of quantum mechanics can be understood as generalization of classical mechanics in which the statistical character of results of measurement is taken into account and the most important general properties of statistical theories known from mathematical statistics are correctly respected.It is not therefore surprising that quantum mechanics yields correct description of physical reality in agreement with experiments.This work was supported by the MSMT grant No. 0021620835 of the Czech Republic.