# Lagrangian form of Schrödinger equation

## Abstract

Lagrangian formulation of quantum mechanical Schrödinger equation is developed in general and illustrated in the eigenbasis of the Hamiltonian and in the coordinate representation. The Lagrangian formulation of physically plausible quantum system results in a well defined second order equation on a real vector space. The Klein–Gordon equation for a real field is shown to be the Lagrangian form of the corresponding Schrödinger equation.

## 1 Introduction

Our strategy and the main result could be schematically summarized as in Fig. 1. We shall start from the quantum Schrödinger equation on an abstract complex Hilbert space, which we consider as given. This is the box 1 in Fig.1. It is well known that the Schrödinger equation is equivalent to the appropriate Hamiltonian dynamical system on the corresponding phase space [8, 9, 10, 11]. This is the box 2 in Fig.1. In fact, the geometric Hamiltonian form of the Schrödinger evolution could be extended to an equivalent formulation of the quantum mechanics [10], and is quite suitable for treatment of problems like nonlinear constraints[12, 13], imbedding of classical into quantum mechanics [14, 15] and hybrid theories [16, 17, 18]. We shall apply the standard transition procedure from Hamiltonian to Lagrangian formulation in order to obtain the Lagrangian form of the Hamiltonian formulation of the Schrödinger equation. The result is the box 3 in Fig. 1. The final step is the transition from the Lagrangian dynamical system in box 3 to the Lagrangian form of the Schrödinger equation, which turns out to be a second order equation on a real Hilbert space. This is the box 4 in Fig. 1. The transitions between boxes, represented by the arrows in Fig. 1, are explained in detail, and in the general case, in the next section.

Of course, the Lagrangian form, when it exists, is equivalent to the Hamiltonian Schrödinger equation. Therefore, our main motivation for the derivation of the Lagrangian form of the general (linear) Schrödinger equation is purely formal. One further motivation is that the Lagrangian form of the Schrödinger equation is more suitable for Lorentz invariant models. In fact, we shall see that the Klein-Gordon equation [19] for the relativistic state vector is noting else but the Lagrangian formulation of the Schrödinger equation with the corresponding Hamiltonian. Also, symmetries are more naturally considered as morphisms of the Lagrangian form and not as time evolution, while the Hamiltonian formulation, i.e. the Schrödinger equation, on the other hand, offers the framework in which the time evolution is an automorphism of the relevant symplectic structure. However, one should bare on mind that the canonical coordinates and momenta in the Hamiltonian and velocities in the Lagrangian formulations of the Schrödinger equation are in fact the coefficients in a basis expansion of the quantum state vector.

The presentation in Sect. 2 is general and formal. The conditions for existence of the expression of the canonical momenta in terms of velocities will be briefly discussed in the general case near the end of Sect. 2. The possible problems will be further illustrated and discussed in Sect. 3, together with the presentation of particular examples obtained by rewriting the general formalism in particular bases. In this section we also treat the Klein-Gordon equation as an example of the Lagrangian formulation of the appropriate Schrödinger equation. Summary is given in Sect. 4.

## 2 Derivation of the Lagrangian Equations

The Lagrangian (14) with transition formulas (15) is the content of the box 3 and the arrow from the box 1 to the box 3.

Suppose that the operators \(\hat{L}_0\) and \(\hat{L}_1\) satisfying (18) exist for a given operator \(\hat{H}\) in (1). Then any orbit \(|q(t)\rangle \) of (19) in \(\mathcal{H}_R^N\) can be used to construct the corresponding orbit of the Schrödinger equation on \(\mathcal{H}_c^N\), and the Schrödinger orbit is given by \(|\psi (t)\rangle =|q(t)\rangle +i|p(t)\rangle \), where \(|p(t)\rangle \) is given in terms of \(|q(t)\rangle \) and \(|\dot{q}(t)\rangle \) by the formula (12). Also, the initial conditions for the Lagrangian formulation \(q(t_0),\dot{q}_(t_0)\) are related to the initial state of the Hamiltonian equations or of the Schrödinger equation essentially by the equation (12). Thus, the crucial question for the construction of the Lagrangian formulation is the existence of the operators \(\hat{L}_0, \> \hat{L}_1 \), which is essentially the question of the existence of the inverse operator \((\hat{H}^R)^{-1}\). The Hamiltonian operator \(\hat{H}\) in (1) uniquely determines the operators \(\hat{H}^R\) and \(\hat{H}^I\). However, a singular choice of basis in \(\mathcal{H}_c\) might imply that the operator \(\hat{H}^R\) (or \(\hat{H}^I\)) is represented by zero and \((H_{nm}^R)^{-1}\) does not exist (see example in 3.1). However, in a typical basis \( \hat{H}^R\) is represented by a nonzero operator. The existence of \((\hat{H}^R)^{-1}\) then depends on the physical problem. If \((\hat{H}^R)^{-1}\), for the system with the Hilbert space \(\mathcal{H}_c\) does not exist, then redefinition of the system by restriction on an appropriate subspace of \(\mathcal{H}_c\) would lead to a well defined \((\hat{H}^R)^{-1}\). Alternatively, in terms of the Hamiltonian and the corresponding Lagrangian formulations nonexistence of \((\hat{H}^R)^{-1}\) corresponds to constrained Hamiltonian and singular Lagrangian systems, and the relation between the two formalisms is treated by the appropriate methods [3].

## 3 Examples and Discussion

In this section we present a series of examples, in the order of increasing complexity, which are aimed to illustrate the problems that might occur in the construction of the Lagrangian formulation.

### 3.1 Discrete Finite Basis

### 3.2 Discrete Energy Eigenbasis

From these two examples we see that the construction of the Lagrangian formulation might fail if one makes a choice of the singular basis and/or includes nonphysical states.

### 3.3 Coordinate Representation

Construction of the Lagrangian formulation for the Schrödinger equation in the coordinate representation follows the same steps as in the general case, or can be obtained by applying the general formulas (18) and (19) written in the coordinate basis. The only potential problem is, like in the general case, non-existence of the inverse of the real part of the Hamiltonian operator \((\hat{H}^R)^{-1}\), appearing in (12). This operator typically has non-diagonal elements in almost all bases and, of course, this fact is irrelevant for the question of its existence. In the coordinate representation, the non-diagonal character of \((\hat{H}^R)^{-1}\) appears as non-locality of the relevant differential operator, and this fact, like its analog in the general case, is irrelevant for the existence of the Lagrangian formulation.

### 3.4 Klein–Gordon Equation

## 4 Summary

In summary, we have developed the Lagrangian formalism for the abstract linear Schrödinger equation on a complex Hilbert space of a quantum system. For a given Hamiltonian operator \(\hat{H}=\hat{H}^R+i\hat{H}^I\) the Lagrangian system is expressed in terms of the operators \(\hat{H}^R,\>\hat{H}^I\) and \((\hat{H}^R)^{-1}\), and is given by a second order equation on a real space. The operator \((\hat{H}^R)^{-1}\), which is crucial for construction of the Lagrangian formulation, exists provided that the spectrum of \(\hat{H}\) is bounded away from zero. If this is not the case, than the Schrödinger equation is in fact equivalent to a constrained Hamiltonian system, with the corresponding singular Lagrangian formulation. A simple example illustrating the failure of the procedure that might occur in a singularly chosen basis is provided. The general formulation of the Lagrangian system is also illustrated in the eigenbasis of a Hamiltonian with a discrete spectrum and in the coordinate representation. The Klein–Gordon equation is seen as the Lagrangian system corresponding to the Schrödinger equation of a relativistic free particle.

## Notes

### Acknowledgments

This work was supported in part by the Ministry of Education and Science of the Republic of Serbia, under project No. 171017, 171028 and 171006. and by COST (Action MP1006).

### References

- 1.Dirac, P.A.M.: The Principles of Quantum Mechanics, 4th edn. Claredon Press, Oxford (1958)MATHGoogle Scholar
- 2.Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1978)CrossRefMATHGoogle Scholar
- 3.Deriglazov, A.: Classical Mechanics Hamiltonian and Lagrangian Formalism. Springer, Berlin (2010)MATHGoogle Scholar
- 4.Schrödinger, E.: Quantisierung als eigenwertproblem (Vierte Mitteilung). Ann. Physik
**81**, 109 (1926)CrossRefMATHGoogle Scholar - 5.Deriglazov, A.: Analysis of constrained theories without use of primary constraints. Phys. Lett. B
**626**, 243 (2005)ADSCrossRefMATHMathSciNetGoogle Scholar - 6.Dirac, P.A.M.: The lagrangian in quantum mechanics. Phys. Zeits. Sowjetunion
**3**, 64 (1933)Google Scholar - 7.Feynman, R.P., Hibbs, A.R.: Quantum Mechanics and Path Integrals. McGraw-Hill, New York (1965)MATHGoogle Scholar
- 8.Kibble, T.W.B.: Relativistic models of nonlinear quantum mechanics. Commun. Math. Phys.
**64**, 73 (1978)ADSCrossRefMathSciNetGoogle Scholar - 9.Heslot, A.: Quantum mechanics as a classical theory. Phys. Rev. D
**31**, 1341 (1985)ADSCrossRefMathSciNetGoogle Scholar - 10.Ashtekar, A., Schilling, T.A.: Geometrical formulation of quantum mechanics. In: Harvey, A. (ed.) On Einstein’s Path. Springer, Berlin (1998)Google Scholar
- 11.Brody, D.C., Hughston, L.P.: Geometric quantum mechanics. J. Geom. Phys.
**38**, 19 (2001)ADSCrossRefMATHMathSciNetGoogle Scholar - 12.Burić, N.: Hamiltonian quantum dynamics with separability constraints. Ann. Phys. (NY)
**233**, 17 (2008)ADSGoogle Scholar - 13.Brody, D.C., Gustavsson, A.C.T., Hughston, L.P.: Symplectic approach to quantum constraints. J. Phys. A
**41**, 475301 (2008)ADSCrossRefMathSciNetGoogle Scholar - 14.Radonjić, M., Prvanović, S., Burić, N.: System of classical nonlinear oscillators as a coarse-grained quantum system. Phys. Rev. A
**84**, 022103 (2011)ADSCrossRefGoogle Scholar - 15.Radonjić, M., Prvanović, S., Burić, N.: Emergence of classical behavior from the quantum spin. Phys. Rev. A
**85**, 022117 (2012)ADSCrossRefGoogle Scholar - 16.Elze, H.-T.: Linear dynamics of quantum-classical hybrids. Phys. Rev. A
**85**, 052109 (2012)ADSCrossRefGoogle Scholar - 17.Radonjić, M., Prvanović, S., Burić, N.: Hybrid quantum-classical models as constrained quantum systems. Phys. Rev. A
**85**, 064101 (2012)ADSCrossRefGoogle Scholar - 18.Burić, N., Mendaš, I., Popović, D.B., Radonjić, M., Prvanović, S.: Statistical ensembles in the Hamiltonian formulation of hybrid quantum-classical systems. Phys. Rev. A
**86**, 034104 (2012)ADSCrossRefGoogle Scholar - 19.Sakurai, J.J.: Advanced Quantum Mechanics. Addison Wesley, Reading, MA (1967)Google Scholar