Feynman Path Integral and Landau Density Matrix in Probability Representation of Quantum States

Abstract

The quantizer–dequantizer method is employed. Using the construction of probability distributions describing density operators of a quantum system states, the connection between the Feynman path integral and the time evolution of the density operator (Landau density matrix) as well as the wave function of the stateconsidered. For single–mode systems with continuous variables, a tomographic propagator is introduced in the probability representation of quantum mechanics. An explicit expression for the probability in terms of the Green function of the Schrödinger equation is obtained. Equations for the Green functions defined by arbitrary integrals of motion are derived. Examples of probability distributions describing the evolution of state of a free particle, as well as states of systems with integrals of motion that depend on time (oscillator type) are discussed.

1. Introduction

The Feynman path integral formalism of quantum mechanics [1] relates to the formalism of density matrix introduced by Lev Landau [2] as well as the Schröedinger wave function equation approach [3] and the Dirac density operator $\widehat{\rho }$ acting in a Hilbert space $\mathcal{H}$ [4] with the classical-mechanics formalism. The states of a quantum system are commonly discussed using either a complex wave function $\psi (q,t)$, where q is the position of the particle satisfying the Schrödinger equation [3], or the Landau density matrix [2], or the density operator $\widehat{\rho }$ [4]. The classical states of the particle system are then described either by the position $q\left(t\right)$ and the velocity $\dot{q}\left(t\right)$ (or the momentum $p\left(t\right)=m\dot{q}\left(t\right)$, where and the dot denotes the time (t) derivative), satisfying the Newton equation, or by the combined probability distribution $f\left(q\right(t),p(t\left)\right)$ of the particle, satisfying the evolution equation determined by the Hamiltonian

$$H(q,p)={\displaystyle \frac{p^2}{2}}+V\left(q\right),m=1,$$

where the kinetic energy term depends on the momentum of the particle, and the potential energy term $V\left(q\right)$ determines the force acting onto the particle.

Recently, the probability representation of quantum states was proposed in ref. [5] and developed in refs. [6,7,8,9]. Within the framework of the probability representation of quantum mechanics, the states of particles are considered being determined by probability distributions. These probability distributions define quantum density operators $\widehat{\rho }$ both for mixed states and for pure states with the wave function $\psi \left(x\right)=\langle x\mid \psi \rangle$ and the density operator ${\widehat{\rho }}_{\psi }=\mid \psi \rangle \langle \psi \mid$. Different representations of quantum mechanics, in which quantum states are then determined through various functions, that are quasi-probabilities, were known earlier, for example, the Wigner function [10], the Husimi–Kano function [11,12], the Glauber–Sudarshan function [13,14], the Blokhintsev mixed-density matrix [15], and the Wootters quantum mechanics without probability amplitudes [16].

The languages of classical mechanics and quantum mechanics differ significantly. In classical mechanics, probability distribution functions are used to describe states. The properties of classical probability distributions are described by conventional probability theory [17]. Different aspects of the properties of quantum systems and methods of quantum–mechanical applications to various fields of science within the framework of probability theory are considered in refs. [18,19]. The entanglement phenomena have been discussed, for example, in refs. [20,21]. The phenomenon of entanglement in quantum mechanics, considered within the framework of the probability representation of quantum mechanics, makes it possible to introduce the concept of entangled probability distributions in classical probability theory [22,23]. Various processes in quantum and nonlinear optics have been discussed in refs. [24,25,26], and in the framework of symplectic and optical tomography schemes in refs. [27,28,29,30]. Some mathematical aspects of tomography and probability representation of quantum and classical states were considered in refs. [31,32,33,34].

The wave function formalism was introduced for classical oscillator systems in refs. [35,36]; in ref. [37] Hermitian operators were introduced to bring together the languages of classical mechanics and quantum mechanics. The groupoid approach to the evolution of the state of a system containing quantum and classical parts was considered in ref. [38], and the dynamical properties of classical, quantum and hybrid systems were discussed in ref. [39]. Various aspects of the theory of cosmology were considered in the framework of the probability representation of quantum mechanics in refs. [40,41]. Some applications of tomographic methods to different types of processes and experiments were discussed in refs. [42,43,44,45,46,47,48,49,50,51].

This study discusses the properties of probability representation of particle system states and the connection of this representation with known representation of states by wave functions [3], as well as the connection with the Feynman path integral method [1], with the evolution of the state of the system described by the Green function $\mathcal{G}(x,x^{\prime },t)$ of the Schrödinger equation expressed through the path integral, due to the Lagrangian of a classical particle system used. Examples of constructing probability distributions of free particle and oscillator in terms of the path integral (or the Green function $\mathcal{G}(x,x^{\prime },t)$) are found explicitly, as well as the connection of the method with time–dependent integrals of motion [52,53,54,55] associated with the Green function $\mathcal{G}(x,x^{\prime },t)$ are obtained.

The point of considering quantum states in terms of the probability distribution $G\left(X\right|\mu ,\nu )$ consist in constructing an invertible map of the density operators $\widehat{\rho }$ of quantum states acting in a Hilbert space $\mathcal{H}$, where the density operator $\widehat{\rho }$ and other operators, such as the position operator and the momentum operator, act. This signifies that one considers the map briefly shown as $\widehat{\rho }\to G_{\rho }\left(Y\right)$, $G_{\rho }\left(Y\right)\to \widehat{\rho }$. Similar maps have been constructed for the Wigner function and other quasi-probability distributions discussed, but were unknown for the case of probability distributions $G\left(Y\right)$. The general scheme consist of finding an invertible map of the operator $\widehat{A}$ onto the function $f_A\left(Y\right)$ for all the described cases, using the general scheme of constructing two sets of operators, called the quantizer $\widehat{D}\left(Y\right)$ and dequantizer $\widehat{U}\left(Y\right)$ operators. The paper discusses this method and the case of pairs defining functions $f_A\left(Y\right)$, which are probability distributions for the density operators $\widehat{A}=\widehat{\rho }$, is found.

2. Quantizer–Dequantizer Method

To introduce the concept of probability distributions describing quantum states, let us discuss the possibilities of describing the density operators $\widehat{\rho }$ of the states by functions f called symbols of the operators. It turned out [56,57] that there exist invertible maps of operators $\widehat{A}$ acting in a Hilbert space $\mathcal{H}$ onto the functions $f_A\left(\overrightarrow{x}\right)$, where $\overrightarrow{x}=x_1,x_2,\dots ,x_N$ and $x_i,i=1,\dots ,N$ are parameters. Some of those can be continuous and other can be discrete. The invertible map dicussed has the form (see, for example, [58])

$$f_A\left(\overrightarrow{x}\right)=\mathrm{T}\mathrm{r}\widehat{A}\widehat{U}\left(\overrightarrow{x}\right),$$

(1)

The inverse transform is then

$$\widehat{A}=\int f_A\left(\overrightarrow{x}\right)\widehat{D}\left(\overrightarrow{x}\right)d\overrightarrow{x}.$$

(2)

In the case when the dequantizer operator $\widehat{U}\left(\overrightarrow{x}\right)$ has the properties of density operators, i.e., $\widehat{U}\left(\overrightarrow{x}\right)={\widehat{U}}^{†}\left(\overrightarrow{x}\right)$ (where the symbol † denotes Hermitian conjugate), the trace $\mathrm{T}\mathrm{r}\widehat{U}\left(\overrightarrow{x}\right)=1$ and diagonal elements of the dequantizer operators are nonnegative, symbols of the density operator $\widehat{\rho }$ must have the properties of probability distribution functions. In connection with the relationship (1)–(2), all information contained in the state density operator $\widehat{\rho }$ is also available in the density operator symbol $f_{\rho }\left(\overrightarrow{x}\right)$.

For systems such as a single–mode harmonic oscillator or a free particle, it is known that the dequantizer operator $\widehat{U}\left(\overrightarrow{x}\right)$, where $\overrightarrow{x}$ has components $x_1=X,x_2=\mu ,\mathrm{and}{x}_3=\nu$, can be represented as

$$\widehat{U}{(X,\mu ,\nu )}_{x{x}^{\prime }}={\displaystyle \frac{1}{2\pi }}\int {\left[exp\left(ik\left(X\widehat{1}-\mu \widehat{q}-\nu \widehat{p}\right)\right)dk\right]}_{x{x}^{\prime }}=\delta {(X\widehat{1}-\mu \widehat{q}-\nu \widehat{p})}_{x{x}^{\prime }},$$

(3)

and the quantizer operator can be considered as

$$\widehat{D}{(X,\mu ,\nu )}_{x{x}^{\prime }}={\displaystyle \frac{1}{2\pi }}{\left[exp\left(i\left(X\widehat{1}-\mu \widehat{q}-\nu \widehat{p}\right)\right)\right]}_{x{x}^{\prime }}.$$

(4)

Let us use here matrix elements of the quantizer and dequantizer operators in the position representation. In this case, the symbol of density operators $\widehat{\rho }$ of the system states $G\left(X\right|\mu ,\nu )$ has the form

$$G\left(X\right|\mu ,\nu )=\mathrm{T}\mathrm{r}\widehat{\rho }\delta (X\widehat{1}-\mu \widehat{q}-\nu \widehat{p})$$

(5)

and is called a state tomogram. Also, tomograms are conditional probability distributions of random variable (the position X) with condition parameters $\mu$ and $\nu$ such that for determining the axes of position q and momentum p in the phase space of the system one has

$$X=\mu q+\nu p,\mu =scos\theta ,\nu =s^{-1}sin\theta .$$

(6)

The parameter $\theta$ defines the angle between different pairs of positions for these axes in system phase space, and s is the scaling parameter used for those axes. The conditional probability $G\left(X\right|\mu ,\nu )$ in the case of a classical oscillator is determined by the Radon transform [59] (see English translation in ref. [60]) of the probability distribution function $f(q,p)$. The symplectic tomogram of the classical system was introduced in ref. [6].

As it was found [61] for the pure state with the wave function $\psi \left(y\right)$, the state tomogram is

$$G_{\psi }\left(X\right|\mu ,\nu )={\displaystyle \frac{1}{2\pi \left|\nu \right|}}{\left|\int \psi \left(y\right)exp\left({\displaystyle \frac{i\mu y^2}{2\nu }}-{\displaystyle \frac{iXy}{\nu }}\right)dy\right|}^2.$$

(7)

For the ground state of the harmonic oscillator, there is the conditional probability distribution of the form

$$G_0\left(X\right|\mu ,\nu )={\displaystyle \frac{1}{\sqrt{\pi ({\mu }^2+{\nu }^2)}}}exp\left(-{\displaystyle \frac{X^2}{{\mu }^2+{\nu }^2}}\right).$$

(8)

Note that all known descriptions of states, such as the Wigner function or the Huzimi-Kano function, are determined by quasi-probability distributions that are associated with the tomogram (8) by integral transform, with certain pairs of quantizer and dequantizer operators.

3. Green Function and Feynman Path Integral

The Schrödinger equation for the wave function $\psi (y,t)$ reads

$$i{\displaystyle \frac{\partial \psi (y,t)}{\partial t}}=\left({\displaystyle \frac{{\widehat{p}}^2}{2}}+V\left(y\right)\right)\psi (y,t),\widehat{p}=-i{\displaystyle \frac{\partial }{\partial y}},$$

(9)

where y is the position, $\widehat{p}$ is the momentum operator, and $V\left(y\right)$ is the potential energy in the position representation. It can be seen as an evolution for the initial wave function $\psi (y,t=0)$, defined by the Green function $\mathcal{G}(y,y^{\prime },t)$, which is the matrix element of the evolution operator $\langle y|\widehat{u}\left(t\right)|y^{\prime }\rangle ={\left[\widehat{u}\left(t\right)\right]}_{y{y}^{\prime }}$ in the position representation. Here $\widehat{u}\left(t\right)$ is the evolution operator. Thus one has

$$\psi (y,t)=\int \mathcal{G}(y,y^{\prime },t)\psi (y^{\prime },t=0)d{y}^{\prime }.$$

(10)

This matrix element was expressed in terms of the Feynman path integral [62] (see, also [1])

$$\mathcal{G}(y,y^{\prime },t)=\langle q_f|e^{-i\widehat{H}t}|q_I\rangle ={\int }_{y\left(t\right)=q_f,y\left(0\right)=q_I}Dqexp\left[i{\int }_0^{t}d{t}^{\prime }L(q,\dot{q})\right],$$

(11)

where $L(q,\dot{q})$ is the Lagrangian of the classical system

$$L(q,q^{\prime })={\displaystyle \frac{{\dot{q}}^2}{2}}-V\left(q\right).$$

Let us assume that along with an assumption for the mass $m=1$, reduced Planck’s constant is set $\hslash =1$. Also let us define

$$Dq=\underset{N\to \infty }{lim}{\left({\displaystyle \frac{N}{2it\pi }}\right)}^{N/2}\prod _{n=1}^{N-1}d{q}_n.$$

The current study finds a connection between the properties of the Green function with the probability distribution-tomogram $G\left(X\right|\mu ,\nu ,t)$, which provides the dependence of the wave function $\psi (x,t)$ of the form (10). Then, Equation (7) gives a tomographic probability representation of the state

$$G_{\psi }\left(X\right|\mu ,\nu ,t)={\displaystyle \frac{1}{2\pi \left|\nu \right|}}{\left|\int \psi (y,t)exp\left({\displaystyle \frac{i\mu y^2}{2\nu }}-{\displaystyle \frac{iXy}{\nu }}\right)dy\right|}^2.$$

(12)

Expressing $\psi (y,t)$ in terms of the Green function of the system, one obtains

$$G_{\psi }\left(X\right|\mu ,\nu ,t)={\displaystyle \frac{1}{2\pi \left|\nu \right|}}{\left|\int \int \mathcal{G}(y,y^{\prime },t)\psi (y^{\prime },0)d{y}^{\prime }exp\left({\displaystyle \frac{i\mu y^2}{2\nu }}-{\displaystyle \frac{iXy}{\nu }}\right)dy\right|}^2.$$

(13)

Then, one has the tomogram of the form

$$G_{\psi }\left(X\right|\mu ,\nu ,t)={\displaystyle \frac{1}{2\pi \left|\nu \right|}}{\left|\int g(y^{\prime },X,\mu ,\nu ,t)\psi (y^{\prime },0)d{y}^{\prime }\right|}^2,$$

(14)

where

$$g(y^{\prime },X,\mu ,\nu ,t)=\int \mathcal{G}(y,y^{\prime },t)exp\left({\displaystyle \frac{i\mu y^2}{2\nu }}-{\displaystyle \frac{iXy}{\nu }}\right)dy.$$

(15)

The function $g(y^{\prime },X,\mu ,\nu ,t)$ (15) is called here a tomographic propagator in the probability representation of quantum mechanics.

4. Example of the Evolution of Free Particle Tomogram

Let us consider an example of the probability representation of the Green function for a free particle. Consider the situation when, at the initial moment of time $t=0$, one prepares the oscillator system in the ground state. Then, at time $t>0$, the oscillator spring is cut, and system under consideration evolves as a free particle. Thus, one considers the free particle system with a given initial state as the ground state of the oscillator. Using the known propagator for a free particle, one can see how the initial wave function evolves. Then, the wave function of the ground states of the harmonic oscillator reads

$${\psi }_0(y,o)={\displaystyle \frac{1}{{\pi }^{1/4}}}exp\left(-{\displaystyle \frac{y^2}{2}}\right).$$

(16)

If the oscillator spring is removed, then the evolution of this state becomes a free evolution with the Green function (see, for example, [63,64])

$$\mathcal{G}(y,y^{\prime },t)={\displaystyle \frac{1}{\sqrt{2\pi it}}}exp\left({\displaystyle \frac{i{(y-y^{\prime })}^2}{2t}}\right).$$

(17)

Thus, the wave function ${\psi }_0(y,t)$ of the system at time moment t becomes

$${\psi }_0(y,t)=\int {\displaystyle \frac{1}{{\pi }^{1/4}\sqrt{2\pi it}}}exp\left({\displaystyle \frac{i{(y-y^{\prime })}^2}{2t}}-{\displaystyle \frac{y^{\prime 2}}{2}}\right)d{y}^{\prime }.$$

(18)

and finally,

$${\psi }_0(y,t)={\displaystyle \frac{1}{{\pi }^{1/4}\sqrt{1+it}}}exp\left(-{\displaystyle \frac{y^2}{2i(t-i)}}\right).$$

(19)

Due to the unitarity of the evolution operator,

$$\int |{\psi }_0{(y,t)|}^{2}dy=1.$$

(20)

The density operator of the state (16) is equal to

$${\widehat{\rho }}_0\left(t\right)=|{\psi }_0\left(t\right)\rangle \langle {\psi }_0\left(t\right)|$$

(21)

and the corresponding tomogram reads

$$G_0\left(X\right|\mu ,\nu ,t)=\mathrm{T}\mathrm{r}{\widehat{\rho }}_0\left(t\right)\delta \left(X\widehat{1}-\mu \widehat{q}-\nu \widehat{p}\right).$$

(22)

This tomogram can be calculated by using the wave function ${\psi }_0(y,t)$. Since ${\left({\widehat{\rho }}_0\left(t\right)\right)}_{y{y}^{\prime }}={\psi }_0(y,t){\psi }_0^{*}(y^{\prime },t)$ and according to the approach developed for systems with quadratic Hamiltonian (like free particle motion), the calculation of this trace can be performed using integrals of the motion of the system [52,53,54,55]

$${\widehat{x}}_0\left(t\right)=\widehat{q}-\widehat{p}t$$

and

$${\widehat{p}}_0\left(t\right)=\widehat{p}.$$

Then, the Heisenberg operators have the form ${\widehat{x}}_H\left(t\right)=\widehat{q}+\widehat{p}t,{\widehat{p}}_H\left(t\right)=\widehat{p}$. After calculating the trace (22), obtains

$$G_0\left(X\right|\mu ,\nu ,t)=\mathrm{T}\mathrm{r}{\widehat{\rho }}_0(t=0)\delta \left(X\widehat{1}-{\mu }_H\left(t\right)\widehat{q}-{\nu }_h\left(t\right)\widehat{p}\right).$$

(23)

For ${\widehat{\rho }}_0(t=0)$ the function ${\psi }_0(y,t=0)$ is determined by Equation (16). Then, taking into account (7), one finds

$$G_0\left(X\right|\mu ,\nu ,t=0)={\displaystyle \frac{1}{\sqrt{\pi ({\mu }^2+{\nu }^2)}}}exp\left(-{\displaystyle \frac{X^2}{{\mu }^2+{\nu }^2}}\right).$$

(24)

This points that Equation (24) allows us calculating an explicit expression for $G\left(X\right|\mu ,\nu ,t)$ by replacing the parameters $\mu \to \mu$ and $\nu \to \nu +\mu t$; one obtains

$$G_0\left(X\right|\mu ,\nu ,t)={\displaystyle \frac{1}{\sqrt{\pi ({\mu }^2+{(\nu +\mu t)}^2)}}}exp\left(-{\displaystyle \frac{X^2}{{\mu }^2+{(\nu +\mu t)}^2}}\right).$$

(25)

Thus, the expression (25) using the calculated wave function ${\psi }_0(y,t)$ (19), the formula for the propagator in the tomographic representation $g(y^{\prime },X,\mu ,\nu ,t)$ (15), and the connection of the propagator with the initial tomogram are obtained.

5. Integrals of Motion and the Green Function (Path Integral)

For a given Schrödinger equation

$$i{\displaystyle \frac{\partial \psi (x,t)}{\partial t}}=\widehat{H}\left(t\right)\psi (x,t),\mathit{ħ}=\omega =m=1,$$

(26)

the evolution operator $\widehat{u}\left(t\right)$ such that

$$\psi (x,t)=\widehat{u}\left(t\right)\psi (x,0),$$

(27)

or in the Dirac notation for a state with $\psi (x,t)$,

$$\langle x\left|\psi \right(t)\rangle =\psi (x,t);$$

here $|\psi \left(t\right)\rangle =\widehat{u}\left(t\right)|\psi \left(0\right)\rangle$. Since one has Equation (27) and $\widehat{u}(t=0)=\widehat{1}$, there are integrals of motion $\widehat{I}\left(t\right)$, such that the mean value $\langle \widehat{I}\left(t\right)\rangle$ has the form

$$\langle \psi \left(t\right)|\widehat{I}\left(t\right)|\psi \left(t\right)\rangle =\langle \psi \left(0\right)|\widehat{I}\left(0\right)|\psi \left(0\right)\rangle .$$

(28)

The integral of motion $\widehat{I}\left(t\right)$ has the form of a relation of the operator $\widehat{I}\left(0\right)$ and the evolution operator $\widehat{u}\left(t\right)$, such that

$$\widehat{I}\left(t\right)=\widehat{u}\left(t\right)\widehat{I}\left(0\right){\widehat{u}}^{†}\left(t\right)$$

(29)

for the Hermitian Hamiltonians and unitary evolution operators $\widehat{u}\left(t\right)$. In position representation, the position operator $\widehat{q}\psi (x,t)=x\psi (x,t)$ and the momentum operator $p\psi (x,\widehat{t})=-i\partial \psi (x,t)/\partial x$. Also, the Green function $\mathcal{G}(x,x^{\prime },t)$ of the Schrödinger equation acts as a matrix element of the evolution operator, namely

$$\mathcal{G}(x,x^{\prime },t)=\langle x|\widehat{u}\left(t\right)|x^{\prime }\rangle =u_{x{x}^{\prime }}\left(t\right)$$

(30)

and

$$\mathcal{G}(x,x^{\prime },0)=\delta (x-x^{\prime }),$$

(31)

with $\delta (a-b)$ the Dirac delta function. The operator $\widehat{u}\left(t\right)$ is unitary, that is ${\widehat{u}}^{†}\left(t\right)={\widehat{u}}^{-1}\left(t\right)$. Equation (29) ponts that an arbitrary integral of motion satisfies the condition

$$\widehat{I}{\left(t\right)}_{\left(x\right)}\mathcal{G}(x,x^{\prime },t)={\left[\widehat{u}\left(t\right)\widehat{I}\left(0\right){\widehat{u}}^{†}\left(t\right)\right]}_{\left(x\right)}\mathcal{G}(x,x^{\prime },t),$$

(32)

and this signifies that the Green function satisfies the equation

$$\widehat{I}{\left(t\right)}_{\left(x\right)}\mathcal{G}(x,x^{\prime },t)={\widehat{I}}^T{\left(0\right)}_{\left(x^{\prime }\right)}\mathcal{G}(x,x^{\prime },t).$$

(33)

The superscript T signifies that the operator ${\widehat{I}}^T$ is a transposed operator with respect to the operator $\widehat{I}$. It is considered that, for a given matrix of the operator $\widehat{I}$ in the position representation given by the formula

$$\langle x\left|I\right|x^{\prime }\rangle =I_{x{x}^{\prime }},$$

the transposed operator ${\widehat{I}}^T$ satisfies the condition

$${\left(I^T\right)}_{x{x}^{\prime }}=I_{x^{\prime }x}.$$

For example, the integral of motion

$${\widehat{q}}_0\left(t\right)=\widehat{u}\left(t\right)\widehat{q}{\widehat{u}}^{†}\left(t\right)$$

preserves the initial values of the particle position and provides the equation for the Green function

$${\widehat{q}}_0{\left(t\right)}_{\left(x\right)}\mathcal{G}(x,x^{\prime },t)=x^{\prime }\mathcal{G}(x,x^{\prime },t).$$

(34)

Thus, there are relations for the Green function $\mathcal{G}(x,x^{\prime }t)$, where ${\widehat{q}}_0\left(t\right)$, acting on the Green function, provides equality for the action of the operator ${\left({\widehat{q}}_0\left(t\right)\right)}^T$ on the second variable of the Green function. The integral of motion ${\widehat{p}}_0\left(t\right)=\widehat{u}\left(t\right)\widehat{p}{\widehat{u}}^{†}\left(t\right)$ preserves the initial value of the particle momentum and gives an equation for the Green function

$${\widehat{p}}_0{\left(t\right)}_{\left(x\right)}\mathcal{G}(x,x^{\prime },t)=i{\displaystyle \frac{\partial \mathcal{G}(x,x^{\prime },t)}{\partial x^{\prime }}}$$

(35)

for arbitrary Hamiltonian systems. A similar relation holds for the Green function, given the action of the integral of motion ${\widehat{p}}_0\left(t\right)$ on the first and second variables of the Green function.

For example, for the motion of a free particle, the momentum operator has the form ${\widehat{p}}_0\left(t\right)=\widehat{p}=-i\partial /\partial x$, and the operator of the preserved initial position has the form ${\widehat{q}}_0\left(t\right)=\widehat{q}-t\widehat{p}=x+it\partial /\partial x$.

This signifies that the Green function of free motion satisfies the equations

$$-i{\displaystyle \frac{\partial }{\partial x}}\mathcal{G}(x,x^{\prime },t)=i{\displaystyle \frac{\partial }{\partial x^{\prime }}}\mathcal{G}(x,x^{\prime },t)$$

(36)

and

$$\left(x+it{\displaystyle \frac{\partial }{\partial x}}\right)\mathcal{G}(x,x^{\prime },t)=-x^{\prime }\mathcal{G}(x,x^{\prime },t).$$

(37)

Moreover, $\mathcal{G}(x,x^{\prime },0)$ must be equal to the matrix element of the identity operator, that is, $\mathcal{G}(x,x^{\prime },0)=\langle x|\widehat{1}|x^{\prime }\rangle =\delta (x-x^{\prime })$. Then one obtains an explicit Gaussian form of the Green function of a free particle (see, e.g., [63,64])

$$\mathcal{G}(x,x^{\prime },t)={\displaystyle \frac{1}{\sqrt{2\pi it}}}exp\left({\displaystyle \frac{i{(x-x^{\prime })}^2}{2t}}\right),$$

(38)

satisfying the formulated relations with the integrals of motion. In the probability representation of quantum mechanics, the propagator (15) has the form

$$g(y^{\prime },X,\mu ,\nu ,t)=\int {\displaystyle \frac{dy}{\sqrt{2\pi it}}}exp\left(i{\displaystyle \frac{{(y-y^{\prime })}^2}{2t}}+{\displaystyle \frac{i\mu y^2}{2\nu }}-{\displaystyle \frac{iXy}{\nu }}\right),$$

and its explicit form is

$$g(y^{\prime },X,\mu ,\nu ,t)={\displaystyle \frac{\sqrt{\nu }}{\sqrt{\mu +\nu t}}}exp\left({\displaystyle \frac{-i{(y^{\prime }\nu +Xt)}^2}{2\nu t(\nu +\mu t)}}+{\displaystyle \frac{i{y}^{\prime 2}}{2t}}\right).$$

Similarly, there are integrals of motion for the harmonic oscillator

$${\widehat{q}}_0\left(t\right)=\widehat{q}cost-\widehat{p}sint,$$

$${\widehat{p}}_0\left(t\right)=\widehat{q}sint+\widehat{p}cost.$$

The harmonic oscillator Green function satisfies the equations

$$xcost\mathcal{G}(x,x^{\prime },t)+isint{\displaystyle \frac{d\mathcal{G}(x,x^{\prime },t)}{dx}}=x^{\prime }\mathcal{G}(x,x^{\prime },t),$$

(39)

$$-icost{\displaystyle \frac{d\mathcal{G}(x,x^{\prime },t)}{dx}}+xsint\mathcal{G}(x,x^{\prime },t)=i{\displaystyle \frac{d\mathcal{G}(x,x^{\prime },t)}{d{x}^{\prime }}};$$

(40)

it is (see, for example, [63,65])

$$\mathcal{G}(x,x^{\prime },t)=\sqrt{{\displaystyle \frac{1}{2\pi isint}}}exp\left({\displaystyle \frac{i}{2sint}}\left[\left(x^2+x^{\prime 2}\right)cost-2x{x}^{\prime }\right]\right).$$

(41)

The same formulas for the Green function in the case of a damped oscillator were obtained, for example, in ref. [66].

The Green functions (38) and (41) under discussion satisfy the equations with integrals of motion (36) and (37) and (39) and (40). The Green functions also satisfy the commonly known equation determined by the Hamiltonian.

6. Three Methods of Calculating Tomograms

The tomogram of the state with the wave function ${\psi }_0\left(t\right)$, obtained for the free motion of the initial state of the oscillator ${\psi }_0(x,t=0)={\displaystyle \frac{1}{{\pi }^{1/4}}}{e}^{-x^2/2}$, can be obtained in three different ways. Namely, one can find the tomogram of the initial state of the oscillator using the density matrix of the state of the oscillator ${\rho }_0(x,x^{\prime },t=0)={\psi }_0(x,t=0){\psi }^{*}(x^{\prime },t=0)$, where the asterisk denotes the complex conjugate. There is a rule for calculating the trace of the product of several operators ${\widehat{a}}_i,i=1,\dots n$, which signifies the equality

$$\mathrm{T}\mathrm{r}\left(f({\widehat{a}}_1{\widehat{a}}_2\dots {\widehat{a}}_n)\right)=\mathrm{T}\mathrm{r}\left(f({\widehat{a}}_2{\widehat{a}}_3\dots {\widehat{a}}_n{\widehat{a}}_1)\right).$$

(42)

Let us calculate the product of the density operator $\widehat{\rho }\left(t\right)$ at time t and the dequantizer operator $\delta (X\widehat{1}-\mu \widehat{q}-\nu \widehat{p})$; this product is the tomogram of the state. The relationship between the density operator at time t and the density operator at the initial time $t=0$, namely, $\widehat{\rho }\left(t\right)=\widehat{u}\widehat{\rho }\left(0\right){\widehat{u}}^{†}\left(t\right)$ is to be used. Then, taking into account Equation (42), one obtains the relations $\mathrm{T}\mathrm{r}\left(\rho \left(t\right)\delta (X\widehat{1}-\mu \widehat{q}-\nu \widehat{p})\right)=\mathrm{T}\mathrm{r}\left(\widehat{u}\left(t\right)\widehat{\rho }(t=0){\widehat{u}}^{†}\left(t\right)\delta (X\widehat{1}-\mu \widehat{q}-\nu \widehat{p})\right)=\mathrm{T}\mathrm{r}\widehat{\rho }(t=0){\widehat{u}}^{†}\left(t\right)\delta (X\widehat{1}-\mu \widehat{q}-\nu \widehat{p})\widehat{u}\left(t\right))=\mathrm{T}\mathrm{r}(\widehat{\rho }(t=0)\delta ({\widehat{u}}^{†}\left(t\right)X\widehat{1}\widehat{u}\left(t\right)-\mu {\widehat{u}}^{†}\left(t\right)\widehat{q}\widehat{u}\left(t\right)-\nu {\widehat{u}}^{†}\left(t\right)\widehat{p}\widehat{u}\left(t\right))=\mathrm{T}\mathrm{r}\left(\rho (t=0)\delta (X\widehat{1}-\mu {\widehat{q}}_H-\nu {\widehat{p}}_H)\right)$. ${\widehat{q}}_H$ and ${\widehat{p}}_H$ are the Heisenberg position and momentum operators expressed in terms of the integrals of motion ${\widehat{q}}_0\left(t\right)$ and ${\widehat{p}}_0\left(t\right)$. For a free particle, one can take the tomogram of the state $\psi (x,t=0)$ and just replace ${\widehat{q}}_0\left(t\right)$ and ${\widehat{p}}_0\left(t\right)$ with the Heisenberg operators in the tomogram. In this case, the rule for replacing the parameters $\mu \to {\mu }_H\left(t\right)$ and $\nu \to {\nu }_H\left(t\right)$ in the tomogram is obtained. For a free particle, this leads to the rule $\mu \to \mu +\nu t$ and $\nu \to \nu$ for the tomogram of the ground state of a harmonic oscillator.

The second method for calculating a tomogram is to use a wave function. The same result can be obtained by calculating the wave function of the free particle motion, which begins in the oscillator state $\psi (x,t=0).$ After this, the tomogram can be calculated using Equation (12) [61].

The third way to calculate the tomogram is to use the propagator $g(y^{\prime },X,\mu ,\nu ,t)$ in the probability representation (15). Here, the tomogram calculation is performed in two steps. First, one calculates the propagator $g(y^{\prime },X,\mu ,\nu ,t)$, which was introduced in this paper; this propagator acts on the initial wave function. But this contains the result of the action of evolution on the system, taking into account the influence of the parameters $\mu$ and $\nu$. Then, the second step is to calculate the action of the propagator $g(y^{\prime },X,\mu ,\nu ,t)$ on the wave function $\psi \left(0\right)$. The combination of both actions using two steps yields a final result that is equivalent to the result of the first and second methods.

The methods just discussed correspond to the application of the path integral method, associated with the Green function expressed through a path integral. For example, for a free particle motion in the initial state, which is the ground state of the oscillator, the wave function is given by the expression (19). The tomogram of this state has the integral form (7) and is explicitly represented by the expression (25).

7. Conclusions

In conclusion, let us point out the main results presented in this paper.

The Green function of the Schrödinger equation for the state wave function was considered. Since the Green function is related to the Feynmann path integral, this property was used for specifying the probability representation of the quantum state wave function introduced in ref. [6], for applying tomographic probability distributions to the Feynman path integral, and for introducing the corresponding propagator (the tomographic propagator), for example, the tomographic propagator of free motion (the Green function) given by Equation (15). The paper discussed explicit forms of the new equations for the Green functions given by integrals of motion operators, and these Green functions were used in examples of an oscillator system and free motion of a particle. Other systems, such as an inverted oscillator and its description using the path integral, are be considered for future studies.

Let us summarize the main results of this paper. A new relation was introduced being associated the Green function of Schrödinger equation $\mathcal{G}(x,x^{\prime },t)$ and the symplectic tomogram $G\left(X\right|\mu ,\nu ,t)$, which is the conditional probability distribution of the random position X. The parameters $\mu$ and $\nu$ define conditions that provide information about scales and orientations of the axes that specify the position q and momentum p for the phase space of the system. The tomogram associated with the Green function of the system is determined by the wave function of the system $\psi \left(y\right)$ or the density operator $\widehat{\rho }$ of the state of the system.

The new function (propagator $g(y^{\prime },X,\mu ,\nu ,t)$ (15)) was constructed which was then used to obtain the connection of the wave function $\psi (y,t)$ of a system with the probability representation $G_{\psi }\left(X\right|\mu ,\nu ,t)$ of the form

$$G_{\psi }\left(X\right|\mu ,\nu ,t)=\int g(y^{\prime },X,\mu ,\nu ,t)\psi (y^{\prime },0)d{y}^{\prime };$$

here, for arbitrary systems, the introduced propagator (15) is determined by the Green function of the system $\mathcal{G}(y,y^{\prime },t)$ (similar to that used for free motion and oscillator). For an arbitrary integral of motion, the equation of the Green function and the Feynman path integral was derived. The new probability distributions can be obtained determining the system states with Hamiltonians, which are nonquadratic forms of the position and momentum. New equations for Green functions $\mathcal{G}(y,y^{\prime },t)$ are obtained using arbitrary integrals of motion $\widehat{I}\left(t\right)$; these equations are considered to be used for examples with nonquadratic Hamiltonians in the forthcoming studies. In addition, the examples of two–dimensional systems with wave function $\psi (x_1,x_2,t)$ (like Landau levels problem; see [55]) are considered for future studies.