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In this paper nonconservative systems are investigated within the framework of Euler Lagrange equations. The solutions of these equations are used to find the principal function
**S**
, this function is used to formulate the wave function and then to quantize these systems using path integral method. One example is considered to demonstrate the application of our formalism.

The path integral formulation of quantum mechanics is a description of quantum theory that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral over the canonical phase space. The basic idea of the path integral formulation can be traced back by [

The canonical formalism for investigating singular systems was developed by [

This paper is mainly concerned with nonconservative systems which are characterized by irregular Lagrangian using canonical approach and the quantization of these systems using the canonical path integral method.

Our present work is organized as follows. In Section 2, the principal function formulation and path integral formulation of nonconservative systems for irregular Lagrangian are discussed. In Section 3, the definition of the wave function for irregular Lagrangian is explained. In Section 4, illustrative example is examined in detail. The work closes with some concluding remarks in Section 5.

The Euler Lagrange equation for conservative systems is given by [

d d t ( ∂ L ∘ ∂ q ˙ ) − ∂ L ∘ ∂ q = 0 (1)

In this work we would like to find the principal function for nonconservative systems using Euler Lagrange equation. We Start with the Lagrangian L = L ∘ ( q , q ˙ ) e λ t [

S ( q , q ˙ , t ) = ∫ t 1 t 2 L ∘ ( q , q ˙ ) e λ t d t = ∫ t 1 t 2 L ( q , q ˙ , t ) d t = ∫ t 1 t 2 ( p q ˙ − H ) d t (2)

where H is the Hamiltonian function and p is the canonical momentum.

The path integral representation by using Hamiltonian and Lagrangian mechanics may be written as

D ( q , p ) = ∫ [ exp ( i S ) ] d p d q = ∫ [ exp i { ∫ t 1 t 2 ( p q ˙ − H ) d t } ] d p d q = ∫ [ exp i { ∫ t 1 t 2 L ( q , q ˙ , t ) d t } ] d p d q (3)

In the semi classical expansion (WKB) of the Hamilton Jacobi function of regular systems has been studied [

ψ ( q i , t ) = [ ∏ i = 1 N ψ 0 i ( q i ) ] [ e i S ( q , t ) ℏ ] (4)

where ψ 0 i ( q i ) is the amplitude of the wave function, which is defined as

ψ 0 i ( q i ) = 1 p ( q i ) . (5)

We use the principal function to formulate the wave function.

Let us consider one-dimensional Lagrangian of a free particle of mass m in the presence of damping [

The Lagrangian is

L = 1 2 m q ˙ 2 e λ t (6)

From Euler Lagrange equation

d d t ( ∂ L ∂ q ˙ ) − ∂ L ∂ q = 0 (7)

The equation of motion is

q ¨ + λ q ˙ = 0 (8)

and the solution of Equation (8) is

q ˙ = C − λ q (9)

where C is the constant of integration.

Integrating Equation (9) and choosing q ∘ = 0 this gives

q ( t ) = C λ ( 1 − e − λ t ) (10)

Taking the first time derivative of Equation (10)

q ˙ = C e − λ t (11)

and

q ˙ 2 = C 2 e − 2 λ t (12)

Substituting Equation (12) into Equation (6) we get

L = 1 2 m C 2 e − λ t (13)

Using Equation (2)

S = ∫ 0 t 1 2 m C 2 e − λ t d t (14)

Thus, the principal function takes the following form

S = m C 2 2 λ [ 1 − e − λ t ] = m C 2 q (15)

Also, the conjugate momentum is

p = ∂ S ∂ q = m C 2 (16)

So that using Equations (15) and (16) the principal function can be written in the final form as following

S ( q , p ) = p q (17)

Using Equation (5), we obtain the amplitude of the wave function as

ψ 0 i ( q i ) = 1 p ( q i ) = 1 m C / 2 (18)

Making use of Equation (4) the wave function can be written as

ψ ( q i , t ) = ψ 0 i ( q i ) e i S ( q , t ) ℏ = 1 m C / 2 e i m C q 2 ℏ = 1 m C / 2 e i S ( q , p ) ℏ = 1 m C / 2 e i q p ℏ (19)

Finally, the path integral representation may be written as

D ( q , p ) = ∫ [ exp ( i S ) ] d p d q = ∫ [ exp ( i p q ) ] d p d q = − i ln | q | − i ∑ n = 1 ∞ ( i p q ) n n . n ! + A ( q ) + B ( p ) (20)

This paper is mainly concerned with path integral quantization of nonconservative systems. Nonconservative systems were studied within framework of Euler Lagrange equation. The corresponding complete integral, or principal function S, was determined using the method of time integral of the given Lagrangian. The momentum p was calculated from S in the usual manner. The appropriate wave function was then determined. The path integral for the nonconservative systems is obtained as an integration over the canonical phase space coordinates. One illustrative example is considered to demonstrate the application of our formalism.

The authors declare no conflicts of interest regarding the publication of this paper.

Jarab’ah, O.A. and Nawafleh, K.I. (2018) Path Integral Quantization of Nonconservative Systems. Journal of Applied Mathematics and Physics, 6, 1637-1641. https://doi.org/10.4236/jamp.2018.68139