Some remarks about the time-dependent Schrödinger equation with damping

The missing derivation of the time-dependent Schrödinger equation following Schrödinger’s original description of the time-independent Schrödinger equation. Also, this description is extended to derive the Caldirola-Kanai, the Schuch-Schrödinger, and the Gisin-Schrödinger equation. In the second part, the Gisin-Schrödinger equation will be derived once more using the Ito formalism of stochastic differential equations. Furthermore, we discuss the extension to larger spin-system using the cluster mean-field theory.


Introduction
There are a few essential differential equations which describe the Universe and daily life. At the atomic level, this is the Dirac equation or if we ignore the effects of the special relativity the Schrödinger equation. The new quantum mechanics starts in 1925 with the work of W. Heisenberg, who has quantified the classical canonical variables q and p and introduced the corresponding matrix equations [1]. These equations are called now the Heisenberg equation and the formalism the matrix mechanics. However, this description of quantum mechanics, even if correct, is too complicated for practical purposes. For instance, Heisenberg itself was not able to solve the problem of the energy spectra of the Hydrogen atom using his theory. This was done one year later 1926 by Pauli [2]. The same year Schrödinger came up with another description of quantum mechanics which bases on the well-known formalism of differential equations: The Schrödinger equation [3]. However, and this is surprising, Schrödinger has not derived the time-dependent Schrödinger equation. Schrödinger has derived the time-independent version and wrote down a differential equation for the evolution in time of the wave function in phase space in full analogy to the classical electromagnetic wave function [4]. Later, Schrödinger noticed that this differential equation is wrong, and has replaced the second derivative in time ∂ 2 / ∂ t 2 by the first derivative t ¶ ¶ times the significant imaginary proportionality factor iÿ [5]. Not surprising is the order of the first two publications (communications) in Annalen der Physik [3,4] even if the second communication is undoubtedly in the timeline of the finding process before the first communication. This also explains, at least in parts, why Schrödinger has not derived the time-dependent version of his wave equation even if it is an easy task using the same way Schrödinger went to derive the time-independent Schrödinger equation.
It seems a more difficult task is to include friction or damping into quantum mechanics. The first steps in this direction go back to Caldirola in 1941 [6] and Kanai in 1948 [7]. The corresponding Schrödinger equation is called Caldirola-Kanai equation. However, neither Caldirola nor Kanai have seriously derived this differential equation. Caldirola shows how to include friction in the Lagrange formalism and thereby, his work is based on a description given by Levi-Civita in 1896 [8]. The result is what we call Caldirola-Kanai Lagrangian, respectively Caldirola-Kanai transformations nowadays. Bateman found the same transformation of the canonical variables in 1931 using another way [9]. In a second step, Caldirola wrote down the corresponding Schrödinger equation just using the two transformations: first, the transformation which includes the friction λ: and furthermore the standard transformation to derive the corresponding operators p i x x ,   - ˆ. Kanai has followed this example and has introduced the Heisenberg equations using transformed momenta p pe t CK = l and potential V Ve t It stays open from where Kanai has taken the idea of the transformed momenta p CK and potential V CK . However, in 1948, the idea of this transformation to include friction in the Lagrangian or Hamilton formalism was already known.
Then for a while, this topic was without interest, however, became a topic of interest again in the 1970s and '80s. During this period several non-linear Schrödinger equations describing quantum systems with friction respectively damping have been proposed by, e.g., Kostin, Haase, Schuch, and Gisin [10][11][12][13]. Then the focus shifted towards the description of decoherence processes. During this time Gisin and Percival proposed the description of quantum state diffusion (QSD) [14] and Mølmer the Monte Carlo wavefunction method (MCWF) [15].
Nowadays, the focus has shifted again, away from the description of Markovian towards Non-Markovian processes. However, this shall not be the topic of this publication. Within this paper, we stay with the description of friction and damping in quantum mechanics. This publication provides some remarks and derivations which might be trivial but to the best of our knowledge have not been described or published so far. This lack of missing information as simple it appears shall be closed.
Furthermore, we present a new way to derive the Schrödinger equation proposed by Gisin starting from the formalism of stochastic differential equations. This way might look strange in the first moment because of the Gisin-Schrödinger equation is not a stochastic differential equation. The Lindblad equation, which is a Liouvillevon-Neumann equation is also not a stochastic differential equation. However, the corresponding wave function is a stochastic differential equation. The corresponding Liouville-von-Neumann equation to the Gisin-Schrödinger equation is the quantum mechanical analog to the classical Landau-Lifshitz equation [16]. At this point, we can ask the question if the corresponding wave equation to this Liouville-von-Neumann equation is a stochastic differential equation or not. As we have mentioned it already, the corresponding Schrödinger equation is not a stochastic differential equation; this is easy to see without the complicated Ito formalism of stochastic differential equations. Nevertheless, strange or unusual ways offer new perspectives and are the basis for new ideas and descriptions. Therefore, additional descriptions to an already given one as strange they appear in the first moment should not be rejected immediately.
The publication is organized as follow: After a brief reminder of the variation calculus, we describe in detail the way Schrödinger has derived the time-independent Schrödinger equation and extend this description to derive as well the time-dependent Schrödinger equation. In the next step, we extend this description once more to get the Caldirola-Kanai equation. In section 4, we comment on the non-linear Schrödinger equations proposed by Schuch and N. Gisin and derive them in the before given context of the Hamilton-Jacobi equation. In section 5, we derive the Gisin-Schrödinger equation once more; however, this time using the theory of stochastic differential equations. The last section discusses the practical problems which occur when dealing with the Gisin-Schrödinger equation to describe a quantum mechanical spin system. Within this section, we give some ideas on how to overcome these problems, e.g., by using the cluster mean field theory (CMFT) [17].

Derivation of the time-dependent Schrödinger equation
Within this section, we close the lack of derivation of the time-dependent Schrödinger equation. The way will be the same as Schrödinger has used to derive the time-independent Schrödinger equation. However, we will extend the description to derive the time-dependent Schrödinger equation and several non-linear Schrödinger equations which take into account friction and damping. Schrödinger has started his description with the Hamilton-Jacobi equation where he inserted a wave ansatz for the action S. Then, he derived the timeindependent Schrödinger equation by performing a calculus of variation. Therefore, the first step is to give a short reminder of the variational calculus.

Calculus of variation
The main idea of the calculus of variation is to find the extrema of a given functional, e.g., the Lagrangian  or the action S. Therefore, the calculus of variation is strongly connected to the principle of least action of the Hamilton mechanics, but also to other principles, e.g., Fermat, Euler, Maupertuis, or Dirichlet. Central element of the calculus of variation is the Euler-Lagrange equation which derivation shall briefly be described in the following.
Let us assume; we are searching for the extrema of the action S which is defined after Hamilton as the integral of the Lagrangian t q q , , (˙) between time t 1 and t 2 : The Lagrangian itself is a functional of the generalized coordinates q=(q 1 , q 2 , K q N ), and their time derivatives t q q d d = .
During the variational calculus, we vary the functions, in this case, q and q , within the functional t q q , , (˙) and search for the δ q and q d˙which fulfill the following condition: Caused by the fact that we assume small changes we can Taylor expand the first functional: and find the following condition which has to be fulfilled: We can further write the second summand, using partial integration, as: The term in the middle is zero when we assume that we do not change the Lagrangian at the starting and final time δq(t 1 )=δq(t 2 )=0. Then, all together leads to: Here, the expression in the round brackets, if set equal to zero, is the Euler-Lagrange equation.
There are some points we have to mention here. Most textbooks give the impression that the Euler-Lagrange equation is a fundamental equation of mechanics where we vary the Lagrangian, which is a functional of the functions q(t) and t q ( ). This idea is not wrong but also not the whole truth. The Euler-Lagrange equation is the second-order partial differential equation whose solutions are the functions for which a given functional is stationary. This means the functional has not necessarily to be the Lagrangian t q q , , (˙) and the included functions not necessarily q and q or restricted to just two functions as in the example above. The Euler-Lagrange equation, and therefore the calculus of variation is quite general.
The second remark is concerning the integral. For the principle of the least action the integral is onedimensional. However, the calculus of variation is not restricted to one-dimensional integrals, as we will see for the derivation of the Schrödinger equation. In that case, the boundary condition is not the Lagrangian t q q , , (˙) at times t 1 and t 2 . Here, we set the wave function ψ equal to zero. The wave function itself is not necessarily oneand can be three-dimensional, and so the boundary condition and the integrals. This gives us a new challenge. Within the description above, the integral over time t is one-dimensional, and therefore the partial integration is easy to handle. However, what if the integral is of higher dimension? The general concept of partial integration is for one-dimensional integrals and cannot be used for two-or three-dimensional integrals. However, in that case, there are theorems, e.g., in the three-dimensional case, the theorem given by G. Green: n is the normal vector of the surface corresponding to the boundary condition. u and v functions of x y z r , , T = ( ) and the integrals are over the total volume Ω respectively the surface ∂ Ω. This theorem replaces the partial integration within the derivation of the Schrödinger equation.

Derivation of the time-independent Schrödinger equation
Within this subsection, we describe Schrödinger's derivation of the time-independent Schrödinger equation. It is a known fact that Schrödinger has first tried to find a full-relativistic wave equation but failed. Schrödinger found the Klein-Gordon equation which is correct in case of spin-less particles but was not able to give a complete relativistic description. This has been done by Dirac in 1928 (Dirac equation) [18]. Therefore, Schrödinger started to look for a non-relativistic wave equation and found the two differential equations named after him. As in many cases, Schrödinger has not started from zero and was able to build his description on the work of de Broglie [19]. de Broglie mentions two things in his Ph.D. thesis: First that the particle-wave duality of light holds in general. Moreover, second, the principles like the one of Maupertuis or Fermat shall also hold for matter waves. These ideas have been the fuel for Schrödinger, and the starting point of his derivation is the following ansatz for the action S in the framework of the Hamilton-Jacobi formalism: K is a constant and has the dimension of an action. In principle, it is clear that K i = -. It is also clear that the mysterious ansatz which Schrödinger has not further commented in the first communication is S K exp y = ( ) [3]. ψ describes a wave with the action S as the phase. This information, we get in the following (second) communication [4]. At this moment, it also becomes clear that K is imaginary. However, within this publication, we follow the description of Schrödinger and handle K as a constant and assume ψ as a real function.
As already mentioned Schrödinger uses the framework of the Hamilton-Jacobi theory to find the wave equation which describes the central-potential problem of hydrogen. Caused by the fact that Schrödinger was looking for stationary orbits Schrödinger used the time-independent Hamilton-Jacobi equation: Within this differential equation, H is the Hamilton function and E the energy of the system. We get equation (10) from the time-dependent Hamilton-Jacobi equation if we assume that we can separate the time-dependent action W(t) in two parts W t S Et = -( ) : the time-independent action S and the time-dependence Et. The idea here is that the action W in three-dimensional space is described by a wave function with iso-surfaces of constant S moving with constant velocity v. This means especially that we have t )and K i =defined before. In other words, this is the same separation we deal with in case of time-independent quantum mechanical problems where the time-dependence occurs just like an additional phase which can be separated.
Then, if we insert the ansatz for S, equation (9), in the time-independent Hamilton-Jacobi equation (10), we find: the kinetic energy and V r e r 2 = -( ) the potential of the hydrogen atom. Thereby, the momentum p is replaced by p=∇S, which comes from the canonical transformation F 2 , which we need to derive the Hamilton-Jacobi formalism from the Hamilton theory. For completeness, e is the elementary charge and r r = | |.
The modified Hamilton-Jacobi equation (11) is the functional r r ,  y y  [ ( ) ( )] which we have to treat with the variational calculus to find the stationary solutions for the wave function ψ(r). This means we can use the description given in section 2.1. For that purpose, we replace the one-dimensional integral over t by the threedimensional integral over the volume Ω, q by ψ, q by ∇ψ, and the Lagrangian  by the functional  . Then, equation (4) becomes: As before in section 2.1, we modify the second summand using Green's theorem As said before, equation (11) is the functional  . Therefore, we can write: and therefore:

Derivation of the time-dependent Schrödinger equation (TDSE)
Within this subsection, we derive the time-dependent Schrödinger equation. The procedure is the same as for the time-independent Schrödinger equation. Here, we start with the Hamilton-Jacobi equation where we have not separated the time as before in the previous subsection. Here, the action S=S(t) is timedependent, the same is true for the wave function ψ=ψ(r, t). Then, with H=T+V and m p T 2 2 = , as well as p=∇S, we get: Some readers might have already noticed the problem with the given description in section 2.2. Within this subsection, we have followed the original description of Schrödinger, which is inaccurate in one point. The wave function has been treated as a real function. The same is true for the action S. Correct is that the wave function ψ and also the action S are complex functions. This means we have not only ψ but also the complex conjugate  y and the same for the action S. And this means, we have to replace equation (20) by the following two differential equations: )( ). Then, as ansatz for the wave functions, we take S K ln y = respectively S K ln where the dot stands for the time derivative and the star in parentheses  ( ) stands for complex conjugate or not, we find for equation (21b) Cause the second and third term is not handy, we treat them as before with Green's theorem. The resulting surface integral with a now time-dependent surface corresponds to the boundary condition. We set these integrals equal to zero and focus on the remaining volume integral: we find from equation (24): If we vary δ J, which means we deal with equation (21a), we find the corresponding complex conjugate timedependent Schrödinger equation:

Derivation of the Caldirola-Kanai equation
So far, we have derived the time-independent and the time-dependent Schrödinger equations starting from the corresponding Hamilton-Jacobi equations. The main idea here is that there is a wave mechanics which corresponds to the point mechanics. A similar behavior we know from the optics where we have the wave and ray picture. Both types of the Schrödinger equation are quite successful. However, they describe systems without friction or damping. In principle, there are two ways to include these effects: microscopic or phenomenological. Microscopic means next to the process we want to describe, we deal with the process of friction or damping. This way is possible but increases the complexity of the treatment. In some cases, we also have no idea about the underlying friction or damping process. Alternatively, we are just interested in the effect of additional friction or damping without taking care of the underlying friction or damping processes and to describe the primary process. In such cases, our choice is the phenomenological description. Now the question is, how can we describe such scenarios? First in classical mechanics: Here, we follow the description given by P. Caldirola [6]: In the manner of Lagrange, we can write This is the Euler-Lagrange equation introduced in section 2.1 with the Lagrangian V T  = -, however for a nonconservative system and therefore with an additional external force F which describes the friction.
The differential equations that govern the movement of a material system when individual points meet resistances proportional to the respective speed, can be obtained from the equations related to the free movement of the same by the change of independent variable t. In other words, we perform the transformation: and search for the function j(t) which transforms equation (29) to: Furthermore, T¢ is the abbreviation for t T T 2 j ¢ = ( ), which follows from q T 2 ¢ µ ¢ (˙) . Then, we can write the first term on the left-hand side as: . Moreover, the second term as: Then, if we compare equation (34) with equation (29), we get V V 2 j ¢ = and and can be easily solved: The result is the general transformation. However, this transformation is quite complex, and in many cases it is already enough to assume that f t Then, with this in mind, we can write the transformed Hamilton-Jacobi equation as: and, with the ansatz S i ln  y = -, we get:

Derivation of the Schuch-and Gisin-Schrödinger equation
Up to now, we have derived the time-independent and the time-dependent Schrödinger equation and extended the description to include friction. However, there are even more proposals of Schrödinger equations with the goal to include friction and damping in the dynamics of quantum mechanics. Within this section, we will derive and discuss two of these Schrödinger equations. We start with the Schrödinger equation proposed by Schuch, followed by the proposal of Gisin.

The Schuch-Schrödinger equation
Motivated by the descriptions of Caldirola and Kanai, primarily inspired by equation (41), we start with the following Hamilton-Jacobi equation: , and similar S Se t = l -. As usual, H is the Hamiltonian, S the action, and λ is the friction constant. This ansatz is in principle the reversal of the final result of Cladirola and Kanai. Now, we can write the right-hand side of equation (42) as: and therefore: This means the official starting point to derive the Schuch-Schrödinger equation is the modified Hamilton-Jacobi equation: Here, an additional term S l -á ñhas been added on the right-hand side to guarantee later that the norm of the wave function ψ is conserved. Different to the Cladirola-Kanai equation the norm is not conserved, and the additional term needed. The brackets á¼ñ, here, mean that we deal with the expectation value.
The last step of the procedure is to insert the ansatz S i ln  y =in equation (45) which immediately leads to the following non-linear Schrödinger equation: Schuch has proposed this differential equation [12,21]. However, there are also similar looking non-linear Schrödinger equations proposed by other physicists, e.g., Kostin [10].

The Gisin-Schrödinger equation
While Schuch had in mind to find a differential equation which describes the motion of a particle with friction which also fulfills the Ehrenfest theorem, the intention of Gisin was more general. The question which Gisin has tried to answer in his work was how does a quantum mechanical description with damping look like, in general. His ansatz is quite unusual for textbook quantum mechanics because Gisin starts with a non-Hermitian Now, we make the ansatz S i ln  y =and insert it in the Hamilton-Jacobi equation: The result of this step is the following Schrödinger equation As before, during the derivation of the Schuch-Schrödinger equation the additional term f is included to guarantee the norm. At this point, we switch to the Dirac bracket description, which means, we replace ψ by yñ | . The bracket description makes the calculation of the norm n y y = á ñ | easier to follow because we do not need to write out the integrals. From equation (48), we have the two differential equations for the bra-and the ketvector: As before, the star å stands for the complex conjugate. The Hamilton operator H and the operator Γ itself are assumed to be Hermitian. Then, we can write:

Comment on the practical use
Until this point, we have derived several Schrödinger equations with friction, respectively damping. Within this subsection we discuss the practical usability of these Schrödinger equations.
In case of the equations proposed by Caldirola-Kanai and Schuch, we deal with wave functions ψ(r, t), which depend on the space coordinate r and the time t. In that case, the system can be discretized, and we can perform finite element or finite difference calculations using several methods to solve the Schrödinger equation. Here, we are restricted by computational power. However, we can calculate reliable system sizes and several problems like [22,33,34].
The situation is different in case of a spin system. The actual limit in case of a spin S=1/2 system is 40 spins using exact diagonalization. In comparison to the 40 spins, we have approximately 4 million spins in case of the classical description [35][36][37][38][39]. The reason for the restriction to a small number of quantum spins are the large matrices we have to deal with. In general, for a spin system with N spins with spin quantum number S the dimension of the Hamilton operator matrix is S S ) . However, within one of our previous publications, we have shown how to extend the description of 40 spin S=1/2 up to 200×200=40000 spins S=1 [23]. Furthermore, this system size is not even the computational limit. In principle, with the given method, we can treat system sizes similar to sizes of classical spin dynamics simulations.
In our previous publication, we have described the system with the aid of the quantum mean-field theory. This means the spins do not interact directly as usual in case of the Heisenberg or Ising model. Instead, the spins experience only a mean-field additional to the external field. This means, we have individual spins, and we can write the Hamilton operator of our system as: