Lie Symmetries of Klein-Gordon and Schrödinger Equations

In this paper, we investigate the Lie point symmetries of Klein-Gordon equation and Schrödinger equation by applying the geometric concept of Noether point symmetries for the below defined Lagrangian. Moreover, we organize a strong relationship among the Lie symmetries related to Klein-Gordon as well as Schrödinger equations. Finally, we utilize the consequences of Lie point symmetries of Poisson and heat equations within Riemannian space to conclude the Lie point symmetries of Klein-Gordon equation and Schrödinger equation within universal Riemannian space.


Introduction
According to Quantum Physics Klein-Gordon and Schrödinger equations are mainly two necessary equations, so it is compulsory that we can apply these equations and resolve their Lie point symmetries in the direction of discovering their invariant solutions by applying Lie point symmetries methods [1] [2] and [3].As we know, Klein Gordon equation is an appropriate case of Poisson equation and Schrödinger equation is an important form of the heat equation.Since Lie point symmetries of Poisson and heat equations within Riemannian space have been studied in [4] [5], however in this paper we utilize these consequences to conclude the Lie point symmetries of Klein-Gordon equation and Schrödinger equation within universal Riemannian space.
One of the most important topics of the current study is the approach of con-M.Iqbal, Y. F. Zhang DOI: 10.4236/am.2018.93025337 Applied Mathematics through potential as well as a metric guide in the equivalent equation; however, the dynamical variables are different.From every Lagrangian defining dynamical system, we determine a metric which is called kinematic metric defined by Lagrangian.Furthermore, Noether symmetries of the motion equations and conformal symmetries of that mentioned metric are a very strong connection.
Moreover, the mentioned kinematic metric determined Laplace operator, and therefore the Lie point symmetries of Poisson equation have described the conditions related to conformal symmetries of kinematic metric.Since these conclusions are continued to Yamabe operator as well as investigate Lie point symmetries related to the conformal Klein-Gordon equation.We define the Lagrangian in the universal Riemannian space such as And we can demonstrate with the intention of Noether symmetries of conformally Lagrangians be developed of metric ij f from the conformal algebra.
In this paper, we investigate the Lie symmetries of Klein- We'll examine the case of the kinematic metric which introduces an Homothetic Vector (HV) or Killing Vector (KV) such that it produced Noether symmetries for classical Lagrangian Equation (1) and display that Lie point symmetry is a nonlocal symmetry in both cases of Klein Gordon equation.The information of Lie point symmetries of Klein Gordon equation and Schrödinger equation in a classical Riemannian space create achievable decisions of solutions of the above equations such that they are invariant by a mention Lie point symmetry.Furthermore, to study the Wheeler De Witt equation [6], they are used in Quantum Cosmology [7] [8] [9] [10] and [11] in a Riemannian space.

Noether Point Symmetries Related to Conformal Lagrangian within Riemannian Paces
Let us consider Lagrangian in a Riemannian space containing the metric ij f moving with the reaction of a potential ( ) l W x , is given by ( ) According to the above reaction the equations of motion becomes Now we shift the variable t z → which is defined as Now according to the new coordinates ( ) , l z x the equations of motion becomes ( ) ( ) ( ) So the new Lagrangian becomes If we study the conformal transformation of a new potential function and the new metric ( ) , then according to the new coordinates ( ) Since the Lagrangian  in Equation ( 2) and the new Lagrangian  in Equation (7) of the same form therefore, the Lagrangian ( ) x x′  is called conformal.Moreover, the reaction will be the same i.e. the equations of motion in the original coordinates ( ) , i t x as well as the latest coordinates ( ) , l z x under the Lagrangian  will be the same.Proof: Let us examine the Lagrangian ( ) whereas Γ i jl called Christofell symbols for , , 1, 2, 3, , i j l m =  .Now the equivalent Hamiltonian is of the form ( ) For the new Lagrangian Now the equivalent Hamiltonian is of the form ( ) ( ) Now to prove that these two equations are the same we using the conformal transformation ( ) Now by substituting Equation (13) in Equation ( 10) we get ( ) Furthermore, by substituting Equation (11) in Equation ( 14) we get Hence, the new Euler Lagrange Equation ( 15) correspond with the original Euler Lagrange Equation ( 8) ⇔ 2 0 i.e. the Hamiltonian vanishes.
Obviously the converse is also true.

Lie Symmetries Related to Klein-Gordon Equation
According in the direction of Poisson equation ( ) , then we get the Klein Gordon equation which is of the form ( ) Hence, by implementing the theorem which is stated as: Theorem: Lie symmetries related to Poisson equation ( ) the ij f metric generated from the Conformal Killing Vectors (CKVs) describing the Laplace operator are the following 1) When 2 n > then we get the vector as whereas x φ is called a CKV having the conformal part ( ) is a solution of Equation ( 16) such as the following condition is satisfied 2) When 2 =  then the Lie symmetry vector become x φ is called a CKV having the conformal part ( ) is a solution of Equation ( 16) such as the following condition is satisfied If we take the comparison between the Lagrangian Equation ( 2) and the Klein Gordon equation Equation ( 16) and the conformal factor holds the conditions ; 0 ij ξ = , for some important functions CKV or KV or HV, then we analyze the following conclusion.Proposition 3.2: When 2 n > , the Lie symmetries related to Klein-Gordon equation in favor of the ij f metric are connected to Noether symmetries of Lagrangian equation having the equivalent metric and potential which describe by the Laplace operator as follows 1) If a proper CKV or special CKV of the ij f metric, produce a Lie symmetries of Klein Gordon Equation ( 16) and also satisfy the condition 0 ξ ∆ = , then it must generate a Noether symmetries of conformably Lagrangian Equation (2) so that the CKV convert into a HV or KV.
2) If an HV or KV of the ij f metric, produce Lie point symmetries related to Klein-Gordon Equation ( 16), subsequently the ij f metric, must generate Noether point symmetries related to Lagrangian Equation (2) with a constant gauge function.

Lie Point Symmetries Related to Conformal Klein-Gordon Equation
According to conformal Poisson equation ( ) , then we get the conformal Klein Gordon equation (Yamabe KG) equation which is of the form ( ) Hence, by implementing the theorem which is stated as Theorem: Lie point symmetries related to conformal Poisson equation ( ) of the ij f metric generated from the CKVs describing the conformal Laplace operator as follows is called a CKV having the conformal part ( ) which is a solution of equation ( ) such as the following condition is satisfied ( ) Then we obtained a conclusion such as

Lie Symmetries of Schrödinger Equation
According to the heat equation ( ) ( ) , then we get Schrödinger equation which is the form Theorem 5.1: Lie symmetries of Schrödinger equation Equation ( 24) by flux in n-dimensional Riemannian spaces be composed of homothetic algebra by ij f metric are the following 1) When i Z be non-gradient KV or an HV Then Lie symmetry vector become where ( )

Klein Gordon Equation and sl(2, R) Algebra
According to the preceding considerations for a constant gauge function, we proved that the Lie symmetries of Klein

Oscillator Systems
We regard as the situation such that a metric acknowledge the gradient KV such that it produces Lie symmetries of the standard Lagrangian which can be written in the form as The potential is follow as The acknowledged Noether symmetries produce as of a gradient HV, ,i L containing the vectors

Corollary 2 . 1 :Lemma 2 . 2 :
In the conformably connected Lagrangians Equation (2) and Equation (7) the Noether point symmetries are included in the conformal algebra of ij f , ij f  metrics.For the two conformal Lagrangians transform the Euler Lagrange equations covariant the relating Lagrangians under the conformal transformation ⇔ the Hamiltonian disappear.

1 :
Lie symmetries related to Klein-Gordon equation Equation(16) of the ij f metric generated from the CKVs describing the Laplace operator are the following 1) When 2 n > then we get vector as

Theorem 4 . 1 :
Lie point symmetries related to conformal Klein-Gordon equation Equation (21) of the ij f metric generated from the CKVs describing Laplace operator are the following 1) When 2 n > then we get vector as

Proposition 4 . 2 : 1 ) 2 )
of Equation (21) such as the following condition is satisfied 2 n = , then Equation (21) is called the Laplace Klein Gordon equation of Equation (16), and the Lie symmetry vector become If we take the comparison between the Lagrangian Equation (2) and the conformal Klein Gordon equation Equation (21) and the conformal factor holds the conditions ; 0 ij ξ = , for some important functions CKV or KV or HV , then we analyze the following conclusion.If an HV or KV of the ij f metric, produce Lie point symmetries related to Klein-Gordon Equation(16), subsequently the ij f metric, must generate Noether point symmetries related to Lagrangian Equation (2) with a constant gauge function.M. Iqbal, Y. F. Zhang DOI: 10.4236/am.2018.93025342 Applied Mathematics If a proper CKV or special CKV of the ij g metric (dim 2 ij f ≥ ) which describe the conformal Laplace operator and produce Lie symmetries related to conformal Klein-Gordon Equation (21), subsequently it must generate a Noether symmetries of conformably Lagrangian Equation (2) so that the CKV convert into a HV or KV.
the comparison between the Lagrangian Equation (2) and the Schrödinger equation Equation (24), then we analyze the following conclusion.Proposition 5.2: when an HV/KV of the ij f metric, produces Lie symmetries related to Schrödinger equation Equation (24), then it must produce a Noether symmetries of Lagrangian Equation (2) with same metric ij f and same potential ( ) l W x , obviously the converse also holds.
tensor such that ordinary to the KV.According to the above coordinate, a Lagrangian can be written of the form Klein-Gordon equation described through the Equation (31) as well as Equation (33) is of the form Equation (35) doesn't acknowledge a Lie symmetry for the combut, it is called independent through x when a result be capable of in the type as ˆ0 p I = i.e.The Klein Gordon equation Equation (35) occupy Lie Bäcklund symmetry of[12] [13], containing the generator ( ) the above equation doesn't introduce Lie symmetry, but it is separate in the logic such that ( ) ( ) ( ) Gordon equation stimulate Noether symmetries used for the standard Lagrangian.Moreover, in this part, we study the case for a no constant gauge function while the stimulated Noether symmetries used for the standard Lagrangian, whereas, the stimulated Noether point symmetries to come from a comprehensive Lie point symmetries of Klein-Gordon equation.Obviously, if HV or KV produces Noether symmetries for the standard Lagrangian satisfy the condition i.e.