Nonrelativistic Quantum Mechanics with Fundamental Environment

Now it is obvious that quantum mechanics enters in the 21st century into a principally new and important phase of its development which will cardinally change the currently used technical facilities in the areas of information and telecommunication technologies, exact measurements, medicine etc. Indisputably, all this on the whole will change the production potential of human civilization and influence its morality. Despite unquestionable success of quantum physics in the 20th century, including creation of lasers, nuclear energy use, etc. it seems that possibilities of the quantum nature are not yet studied and understood deeply, a fortiori, are used. The central question which arises on the way of gaining a deeper insight into the quantum nature of various phenomena is the establishment of well-known accepted criteria of applicability of quantum mechanics. In particular, the major of them is the de-Broglie criterion, which characterizes any body-system by a wave the length of which is defined as p    , where  is the wavelength of the body-system, p is its momentum and  is the Plank constant. An important consequence of this formula is that it assigns the quantum properties only to such systems which have extremely small masses. Moreover, it is well known that molecular systems which consist of a few heavy atoms are, as a rule, well described by classical mechanics. In other words, the de-Broglie criterion is an extremely strong limitation for occurrence of quantum effects in macroscopic systems. Till now only a few macroscopic quantum phenomena have been known, such as superfluidity and superconductivity, which are not ordinary natural phenomena but most likely extremal states of nature. Thus, a reasonable question arises, namely, how much correct is the deBroglie criterion, or more precisely, how completely this criterion reflects the quantum properties of a system. In order to answer this essentially important question for development of quantum physics, it is necessary to expand substantially the concepts upon which quantum mechanics is based. The necessity for generalization of quantum mechanics is also dictated by our aspiration to consider such hard-to-explain phenomena as spontaneous transitions between the quantum levels of a system, the Lamb Shift of energy levels, EPR paradox, etc. within the limits of a united scheme. In this connection it seems important to realize finally the concept according to which any quantum system is basically an open


Introduction
Now it is obvious that quantum mechanics enters in the 21st century into a principally new and important phase of its development which will cardinally change the currently used technical facilities in the areas of information and telecommunication technologies, exact measurements, medicine etc. Indisputably, all this on the whole will change the production potential of human civilization and influence its morality. Despite unquestionable success of quantum physics in the 20th century, including creation of lasers, nuclear energy use, etc. it seems that possibilities of the quantum nature are not yet studied and understood deeply, a fortiori, are used. The central question which arises on the way of gaining a deeper insight into the quantum nature of various phenomena is the establishment of well-known accepted criteria of applicability of quantum mechanics. In particular, the major of them is the de-Broglie criterion, which characterizes any body-system by a wave the length of which is defined as p    , where  is the wavelength of the body-system, p is its momentum and  is the Plank constant. An important consequence of this formula is that it assigns the quantum properties only to such systems which have extremely small masses. Moreover, it is well known that molecular systems which consist of a few heavy atoms are, as a rule, well described by classical mechanics. In other words, the de-Broglie criterion is an extremely strong limitation for occurrence of quantum effects in macroscopic systems. Till now only a few macroscopic quantum phenomena have been known, such as superfluidity and superconductivity, which are not ordinary natural phenomena but most likely extremal states of nature. Thus, a reasonable question arises, namely, how much correct is the de-Broglie criterion, or more precisely, how completely this criterion reflects the quantum properties of a system.
In order to answer this essentially important question for development of quantum physics, it is necessary to expand substantially the concepts upon which quantum mechanics is based. The necessity for generalization of quantum mechanics is also dictated by our aspiration to consider such hard-to-explain phenomena as spontaneous transitions between the quantum levels of a system, the Lamb Shift of energy levels, EPR paradox, etc. within the limits of a united scheme. In this connection it seems important to realize finally the concept according to which any quantum system is basically an open www.intechopen.com Theoretical Concepts of Quantum Mechanics 162 system, especially when we take into account the vacuum's quantum fluctuations [1][2][3]. Specifically for a quantum noise coming from vacuum fluctuations we understand a stationary Wiener-type source with noise intensity proportional to the vacuum power  is the variance of the field frequencies averaged over some appropriate distribution (we assume 0    since  and −  must be considered as independent fluctuations). For example, in the cosmic background case where 2  TK we find, correspondingly, =1. 15 P pW . Calculation of 2    for quantum fluctuations is not trivial because vacuum energy density diverges as 3  [3] with uniform probability distribution denying a simple averaging process unless physical cutoffs at high frequencies exist. Thus, first of all we need such a generalization of quantum mechanics which includes nonperturbative vacuum as fundamental environment (FE) of a quantum system (QS). As our recent theoretical works have shown [4][5][6][7][8][9], this can be achieved by naturally including the traditional scheme of nonrelativistic quantum mechanics if we define quantum mechanics in the limits of a nonstationary complex stochastic differential equation for a wave function (conditionally named a stochastic Schrödinger equation). Indeed, within the limits of the developed approach it is possible to solve the above-mentioned traditional difficulties of nonrelativistic quantum mechanics and obtain a new complementary criterion which differs from de-Broglie's criterion. But the main achievement of the developed approach is that in the case when the de-Broglie wavelength vanishes and the system, accordingly, becomes classical within the old conception, nevertheless, it can have quantum properties by a new criterion. Finally, these quantum properties or, more exactly, quantum-field properties can be strong enough and, correspondingly, important for their studying from the point of view of quantum foundations and also for practical applications. The chapter is composed of two parts. The first part includes a general scheme of constructing the nonrelativistic quantum mechanics of a bound system with FE. In the second part of the chapter we consider the problem of a quantum harmonic oscillator with fundamental environment. Since this model is being solved exactly, its investigation gives us a lot of new and extremely important information on the properties of real quantum systems, which in turn gives a deeper insight into the nature of quantum foundations.

Formulation of the problem
We will consider the nonrelativistic quantum system with random environment as a closed united system QS and FE within the limits of a stochastic differential equation (SDE) of Langevin-Schrödinger (L-Sch) type: In equation ( where  denotes a Laplace operator,   ,; {} Vt xf describes the interaction potential in a quantum system which has regular and stochastic terms. We will suppose that when {} 0  f , the system executes regular motion which is described by the regular nonstationary interaction potential    f. In this case the quantum system will be described by the equation: We also assume that in the limit t  the QS passes to an autonomous state which mathematically equals to the problem of eigenvalues and eigenfunctions: where in the (in) asymptotic state E  designates the energy of the quantum system and, correspondingly, the interaction potential is defined by the limit: In the (out) asymptotic state when the interaction potential tends to the limit: designates an array of quantum numbers. Further we assume that the solution of problem (2.4) leads to the discrete spectrum of energy and wave functions which change adiabatically during the evolution (problem (2.3)). The latter implies that the wave functions form a full orthogonal basis: where the symbol  means complex conjugation. Finally, it is important to note that an orthogonality condition similar to (2.5) can be written also for a stochastic wave function: designates random field (definition see below ).

The equation of environment evolution
The solution of (2.1) can be represented, Now substituting (2.6) into (2.1) with taking into account (2.3) and (2.5), we can find the following system of complex SDEs: where the following designations are made:   where the following designations are made:  Assuming that random forces satisfy the conditions of white noise: Now, using the system of equations (2.10) and correlation properties (2.11), it is easy to obtain the Fokker-Planck equation for the joint probability distribution of fields {} ξ (see in particular [6,10] The joint probability in (2.12) is defined by the expression: where the function () Nt is the term which implements performing of the normalization condition to unit, defined by the expression:

Stochastic density matrix method
We consider the following bilinear form (see representation (2.6)): where the symbol ""  means complex conjugation. After integrating (2.15) by the coordinates 3  xR and n  ξΞ with taking into account the weight function (2.13), we can find: Now, using (2.16) we can construct an expression for a usual nonstationary density matrix [12]: www.intechopen.com

Thus, equation (2.19) differs from the usual von Neumann equation for the density matrix.
The new equation (2.19), unlike the von Neumann equation, considers also the exchange between the quantum system and fundamental environment, which in this case plays the role of a thermostat.

Entropy of the quantum subsystem
For a quantum ensemble, entropy was defined for the first time by von Neumann [11]. In the considered case where instead of a quantum ensemble one united system QS + FE, the entropy of the quantum subsystem is defined in a similar way: In connection with this , there arises an important question about the behavior of the entropy of a multilevel quantum subsystem on a large scale of times. It is obvious that the relaxation process can be nontrivial (for example, absence of the stationary regime in the limit t  ) and, hence, its investigation will be a difficult-to-solve problem both by analytic methods and numerical simulation. A very interesting case is when the QS breaks up into several subsystems. In particular, when the QS breaks up into two fragments and when these fragments are spaced far from each other, we can write for a reduced density matrix of the subsystem the following expression: .
Recall that the vectors y and z describe the first and second fragments, correspondingly. Now, substituting the reduced density matrix (, ,) t   xx into the expression of the entropy of QS (2.20), we obtain: Since at the beginning of evolution the two subsystems interact with each other, it is easy to show that 1 (;)1 Jt  λ and 2 (;)1 Jt  λ , moreover, they can be fluctuated depending on the time. The last circumstance proves that the subsystems of the QS are in the entangled state. This means that between the two subsystems there arises a new type of nonpotential interaction which does not depend on the distance and size of the subsystems. In the case when subsystems 1 and 2 have not interacted, 12 1 JJ   and, correspondingly, 1 S and 2 S are constants denoting entropies of isolated systems.

Conclusion
The developed approach allows one to construct a more realistic nonrelativistic quantum theory which includes fundamental environment as an integral part of the quantum system. As a result, the problems of spontaneous transitions (including decay of the ground state) between the energy levels of the QS, the Lamb shift of the energy levels, ERP paradox and many other difficulties of the standard quantum theory are solved naturally. Equation (2.12) -(2.13') describes quantum peculiarities of FE which arises under the influence of the quantum system. Unlike the de-Broglie wavelength, they do not disappear with an increase in mass of the quantum subsystem. In other words, the macroscopic system is obviously described by the classical laws of motion; however, space-times structures can be formed in FE under its influence. Also, it is obvious that these quantum-field structures ought to be interpreted as a natural continuation and addition to the considered quantum (classical) subsystem. These quantum-field structures under definite conditions can be quite observable and measurable. Moreover, it is proved that after disintegration of the macrosystem into parts its fragments are found in the entangled state, which is specified by nonpotential interaction (2.22), and all this takes place due to fundamental environment. Especially, it concerns nonstationary systems, for example, biological systems in which elementary atom-molecular processes proceed continuously [13]. Note that such a conclusion becomes even more obvious if one takes into account the well-known work [14] where the idea of universal description for unified dynamics of micro-and macroscopic systems in the form of the Fokker-Planck equation was for the first time suggested. Finally, it is important to add that in the limits of the developed approach the closed system QS + FE in equilibrium is described in the extended space 3 n R   , where n  can be interpreted as a compactified subspace in which FE in equilibrium state is described.

The quantum one-dimensional harmonic oscillator (QHO) with FE as a problem of evolution of an autonomous system on the stochastic space-time continuum
As has been pointed out in the first part of the chapter, there are many problems of great importance in the field of non-relativistic quantum mechanics, such as the description of the Lamb shift, spontaneous transitions in atoms, quantum Zeno effect [15] etc., which remain unsolved due to the fact that the concept of physical vacuum has not been considered within the framework of standard quantum mechanics. There are various approaches for investigation of the above-mentioned problems: the quantum state diffusion method [16], Lindblad density matrix method [17,18], quantum Langevin equation [19], stochastic Schrödinger equation method (see [12]), etc. Recall that representation [17,18] describes a priori the most general situation which may appear in a non-relativistic system. One of these approaches is based on the consideration of the wave function as a random process, for which a stochastic differential equation (SDE) is derived. However, the consideration of a reduced density matrix on a semi-group [20] is quite an ambiguous procedure and, moreover, its technical realization is possible, as a rule, only by using the perturbation method. For investigation of the inseparably linked closed system QSE, a new mathematical scheme has been proposed [5][6][7][8] which allows one to construct all important parameters of the quantum system and environment in a closed form. The main idea of the developed approach is the following. We suppose that the evolution time of In particular, the method of stochastic density matrix (SDM) is developed, which permits to construct all thermodynamic potentials of the quantum subsystem analytically, in the form of multiple integrals from the solution of a 2D second-order partial differential equation.

Description of the problem
We will consider that the 1D QHO+FE closed system is described within the framework of the L-Sch type SDE (see equation (2.1)), where the evolution operator has the following form: In expression (3.1) the frequency   f is a random function of time where its stochastic component describes the influence of environment. For the analysis of a model of an environment a set of harmonic oscillators [21][22][23][24][25] and quantized field [26,27] are often used. For simplicity, we will assume that frequency has the following form: where () f t is an independent Gaussian stochastic process with a zero mean and  is a shaped correlation function: www.intechopen.com (see [28]). It has the following form: where the function (,) y   describes the wave function of the Schrödinger equation: for a harmonic oscillator on the stochastic space-time {,} y  continuum. In (3.6) the following designations are made: Taking into account (3.5) and the well-known solution of autonomous quantum harmonic oscillator (3.6) (see [28]) for stochastic complex processes which describe the 1D QHO+FE closed system, we can write the following expression: where the symbol ""  means complex conjugation. So, the initial L-Sch equation (

The mean values of measurable parameters of 1D QHO
For investigation of irreversible processes in quantum systems the non-stationary density matrix representation based on the quantum Liouville equation is often used. However, the application of this representation has restrictions [11]. It is used for the cases when the system before switching on the interaction was in the state of thermodynamic equilibrium and after switching on its evolution is adiabatic. Below, in the frames of the considered model the new approach is used for the investigation of the statistical properties of an irreversible quantum system without any restriction on the quantities and rate of interaction change. Taking (1) R with taking into account (3.9), we obtain the normalization condition for weight functions: Below we define the mean values of various operators. Note that at averaging over the extended space  the order of integration is important. In the case when the integral from the stochastic density matrix is taken at first in the space, 1 R and then in the functional space, {} R  the result becomes equal to unity. This means that in the extended space  all conservation laws are valid, in other words, the stochastic density matrix in this space is unitary. In the case when we take the integration in the inverse order, we get another www.intechopen.com The mean value of the operator  

,| { }
Axt  over all quantum states, respectively, will be: Note that the operation Tr  in (3.12) and (3.13) denotes functional integration: where () D  designates the measure of functional space which will be defined below. If we wish to derive an expression describing the irreversible behavior of the system, it is necessary to change the definition of entropy. Let us remind that the von Neumann nonstationary entropy (the measure of randomness of a statistical ensemble) is defined by the following form: After substitution of (3.17) into (3.7) we can define the following nonlinear SDE: 22 0 0 00 00 0 The second equation in (3.18) expresses the condition of continuity of the function () t  and its first derivative at the moment of time 0 tt  . Using the fact that the function () t describes a complex-valued random process, the SDE (3.18) may be presented in the form of two SDE for real-valued fields (random processes). Namely, introducing the real and imaginary parts of () t : The pair of fields 12 (,) uu in this model is not independent because their evolution is influenced by the common random force () f t . This means that the joint probability distribution of fields can be represented by the form: which is a non-factorable function. After differentiation of functional (3.21) with respect to time and using SDEs (3.18) and correlation properties of the random force (3.3), as well as making standard calculations and reasonings (see [29,30]), we obtain for a distribution of fields the following Fokker-Planck equation: with the initial condition:

Entropy of the ground state of 1D QHO with fundamental environment
For simplicity we will suppose that (0) 1 w  and, correspondingly, () 0 m w  for all quantum numbers 1 m  (see expression (3.10) ). In this case the SDM (3.10) with consideration of expressions (3.8), (3.14) and (3.16) may be represented by the following form: where the following designation (0) Now, we can calculate the reduced density matrix: After conducting integration in the space 1 R in (3.33), it is easy to find the expression: where the following designations are made: Similarly, as in the case with (3.29), using expressions ( It is simple to show that in the limit   entropy aspires to zero.
Thus, at the reduction information in a quantum subsystem is lost, as a result of which the entropy changes, too. Let us remind that usually the entropy of a quantum subsystem at environment inclusion grows, however, in the considered case the behavior of the entropy depending on the interaction parameter  can be generally nontrivial.

Energy spectrum of a quantum subsystem
The energy spectrum is an important characteristic of a quantum system. In the considered case we will calculate the first two levels of the energy spectrum in the limit of thermodynamic equilibrium. Taking into account expressions (3.12) and (3.28) for the energy of the «ground state», the following expression can be written: where the operator: 22    As obviously follows from expressions (3.42)-(3.46), the relaxation effects lead to infringement of the principle of equidistance between the energy levels of a quantum harmonic oscillator Fig.1. In other words, relaxation of the quantum subsystem in fundamental environment leads to a shift of energy levels like the well-known Lamb shift.

Spontaneous transitions between the energy levels of a quantum subsystem
The question of stability of the energy levels of a quantum subsystem is very important. It is obvious that the answer to this question may be received after investigation of the problem of spontaneous transitions between the energy levels. Taking

Uncertainty relations, Weyl transformation and Wigner function for the ground state
According to the Heisenberg uncertainty relations, the product of the coordinate and corresponding momentum of the quantum system cannot have arbitrarily small dispersions. This principle has been verified experimentally many times. However, at the present time for development of quantum technologies it is very important to find possibilities for overcoming this fundamental restriction.
As is well-known, the dispersion of the operator ˆi A is determined by the following form: In the considered case the dispersion of the operator at the arbitrary time t in the «ground state» can be calculated by the following expression: where average values of operators ˆ( ) x  and ˆ( ) p  can be found from (3.53) and (3.54) in the limit t . It is obvious that expressions for operator dispersions (3.53)-(3.54) are different from Heisenberg uncertainty relations and this difference can become essential at certain values of the interaction parameter  . The last circumstance is very important since it allows controlling the fundamental uncertainty relations with the help of the  parameter. Definition 4. We will refer to the expression: Wm p x t  as partial stochastic Wigner function. In particular, for the partial stochastic Wigner function the following expression may be found: whole is described by a SDE for the wave function. However, in this case there arises a nonsimple problem of how to find a measure of the functional space, which is necessary for calculating the average values of various parameters of the physical system. We have explored the possibility of building the non-relativistic quantum mechanics of a closed system QS+FE within the framework of one-dimensional QHO which has a random frequency. Mathematically, the problem is formulated in terms of SDE for a complex-valued probability process (3.1) defined in the extended space 1 {} RR  ξ .The initial SDE for complex processes is reduced to the 1D Schrödinger equation for an autonomous oscillator on a random space-time continuum (3.6). For this purpose the complex SDE of Langevin type has been used. In the case when random fluctuations of FE are described by the white noise correlation function model, the Fokker-Plank equation for conditional probability of fields is obtained (3.22)-(3.23) using two real-valued SDE for fields (3.20). With the help of solutions of this equation, a measure of the functional space {} R ξ is constructed (3.27) on infinitely small time intervals (3.24).In the context of the developed approach representation of the stochastic density matrix is introduced, which allows perform an exact computation scheme of physical parameters of QHO (of a quantum subsystem) and also of fundamental environment after relaxation under the influence of QS. The analytic formulas for energies of the «ground state» and for the first excited state with consideration of shift (like the Lamb shift) are obtained. The spontaneous transitions between various energy levels were calculated analytically and violation of symmetry between elementary transitions up and down, including spontaneous decay of the «ground state», was proved. The important results of the work are the calculation of expressions for uncertainty relations and Wigner function for a quantum subsystem strongly interacting with the environment. Finally, it is important to note that the developed approach is more realistic because it takes into account the shifts of energy levels, spontaneous transitions between the energy levels and many other things which are inherent to real quantum systems. The further development of the considered formalism in application to exactly solvable manydimensional models can essentially extend our understanding of the quantum world and lead us to new nontrivial discoveries.