A Relationship between the Schrödinger and Klein–Gordon Theories and Continuity Conditions for Scattering Problems

Abstract

A rigorous analysis is undertaken based on the analysis of both Galilean and Lorentz (Poincaré) invariance in field theories in general in order to (i) identify a unique analytical scheme for canonical pairs of Lagrangians, one of them having Galilean, the other one Poincaré invariance; and (ii) to obtain transition conditions for the state function purely from Hamilton’s principle and extended Noether’s theorem applied to the aforementioned symmetries. The general analysis is applied on Schrödinger and Klein–Gordon theory, identifying them as a canonical pair in the sense of (i) and exemplified for the scattering problem for both theories for a particle beam against a potential step in order to show that the transition conditions that result according to (ii) in a ‘natural’ way reproduce the well-known ‘methodical’ continuity conditions commonly found in the literature, at least in relevant cases, closing a relevant argumentation gap in quantum mechanical scattering problems.

1. Introduction

In classical field theories, two-times continuously differentiable fields are usually assumed, but there are several field problems where discontinuities occur. In fluid dynamics of supersonic flow, for instance, shock waves may occur, requiring the formulation of transition conditions at the shock front, so–called Rankine–Hugoniot conditions [1,2,3]. In quantum theories, potentials of external forces with discontinuities are sometimes utilised [4], in particular for the modelling of particle scattering. Unlike the problem of shock waves in fluids, it is not the field to be determined that is discontinuous here, but an external field. Regardless of this, discontinuities occur in both cases in the field equations or in the Lagrangian.

In quantum mechanics, the continuity of the state function and its normal derivative at discontinuities of the potential are typically assumed, but while in fluid dynamics, transition conditions are based on physical principles such as the conservation of mass, energy, and momentum, the corresponding conditions in quantum mechanics lack a comparably convincing rationale and have been subject to debate; see, e.g., Branson [5], Andrews [6]. It is therefore a desirable aim to also trace the transition conditions in quantum mechanical systems back to the conservation of physical quantities.

Noether’s theorem [7] proves to be an elegant tool in this context, since it establishes the conservation of physical quantities from the symmetries of the Lagrangian of the physical system. It can conversely be used in an inverse manner for constructing an unknown Lagrangian from given physical balances while considering the respective symmetries (Galilean or Poincaré) [8], leading to a general scheme for Lagrangians [9] for arbitrary field theories being invariant with respect to the Galilean group, which includes Schrödinger’s theory. The question arises if the scheme developed in [9] can be extended to Lorentz–invariant field theories.

In [10,11], both Hamilton’s principle and Noether’s theorem were extended to Lagrangians with discontinuities, and applied to fluid mechanics and acoustics [12,13,14].

Two main objectives are pursued in the present work: Firstly, it is demonstrated that the two different Lagrangians related to Schrödinger and Klein–Gordon theory can be understood as a canonical pair in the sense that they result from a general scheme as the Galilei-invariant Lagrangian and its Lorentz-invariant counterpart. The second main objective is a rigorous derivation of continuity conditions for the state function at discontinuities of the potential, only based on symmetry considerations and Noether’s theorem. Both findings are applied to the standard problem of a particle beam against a potential frame, reproducing the result reported in open literature on scattering problems.

The paper is organised as follows: In Section 2, the existing mathematical methods are presented and extended where necessary; a general scheme for Lorentz-invariant Lagrangians, which can be extended in order to include conservative external forces. Transition and conservation conditions at discontinuities of the external potential are presented in general form. In Section 3 the methodology is applied to the Schrödinger and Klein–Gordon Lagrangians, revealing at first a relationship between both. The use of the transition and conservation conditions is demonstrated by the scattering problem of a particle beam against a step potential. Conclusions are drawn in Section 4, and perspectives for further research are also provided.

2. Materials and Methods

2.1. Systematic Approach to Lagrangians with Galilean Invariance

2.1.1. General Scheme for Lagrangians with Galilean Invariance

In [9], field theories are considered, the dynamics of which are deducible from Hamilton’s principle [15], where the associated Lagrangian $\ell \left({\psi }_i,{\partial }_t{\psi }_i,\nabla {\psi }_i\right)$ based on N independent fields ${\psi }_1,\cdots ,{\psi }_N$ fulfils Galilean invariance. If, additionally, equivalence $\overrightarrow{p}=\varrho \overrightarrow{u}$ of momentum density $\overrightarrow{p}$ and mass flux density $\varrho \overrightarrow{u}$ with mass density $\varrho$ and velocity field $\overrightarrow{u}$ is assumed, it is proven that by a certain transformation of the fundamental fields, ${\psi }_i=F_i\left(\phi ,{\vartheta }_j\right)$ a special representation of the Lagrangian,

$$\ell \left({\partial }_t\phi ,\nabla \phi ,{\vartheta }_j,{\partial }_t{\vartheta }_j,\nabla {\vartheta }_j\right)=L\left(\omega ,{\vartheta }_j,\stackrel{\odot }{{\vartheta }_j},\partial {\vartheta }_j\right),$$

(1)

in terms of field variables $\phi ,{\vartheta }_1,\cdots ,{\vartheta }_{N-1}$ can be obtained, with the replacement rules:

$$\begin{array}{cc}\hfill \omega & ={\partial }_t\phi +\frac{1}{2}{\left(\nabla \phi \right)}^2,\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill \stackrel{\odot }{{\vartheta }_j}& ={\partial }_t{\vartheta }_j+\nabla \phi ·\nabla {\vartheta }_j,j=1,\cdots ,N-1,\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill \partial {\vartheta }_j& =\nabla {\vartheta }_j.\hfill \end{array}$$

(4)

Galilean invariance is granted by the transformation rule $\phi \to \phi -{\overrightarrow{u}}_0·\overrightarrow{x}+{\overrightarrow{u}}_0^{2}t/2$ (with constant relative velocity ${\overrightarrow{u}}_0$) for the field $\phi$ with respect to Galilei boosts $\overrightarrow{x}\to \overrightarrow{x}-{\overrightarrow{u}}_{0}t$, while all other fields, ${\vartheta }_j$, are assumed to be invariant, implying invariance of $\omega ,\stackrel{\odot }{{\vartheta }_j}$, and $\partial {\vartheta }_j$ and therefore, invariance of the resulting Lagrangian.

Subsequently, the function L is called the blueprint-Lagrangian, motivated by the fact that it can serve as a common starting point for establishing both a Lagrangian with Galilean invariance and an associated Lorentz-invariant Lagrangian.

2.1.2. Madelung’s Picture of Schrödinger’s Theory

Subsequently Schrödinger’s Lagrangian for a free particle [4],

$$\ell \left(\psi ,\overline{\psi },\partial \psi ,\partial \overline{\psi }\right)=\frac{\hslash }{2\mathrm{i}}\left(\psi {\partial }_t\overline{\psi }-\overline{\psi }{\partial }_t\psi \right)-\frac{{\hslash }^2}{2m}\nabla \psi ·\nabla \overline{\psi },$$

(5)

is considered in terms of the state function $\psi$ and its complex conjugate, $\overline{\psi }$. By applying polar decomposition to the complex state function,

$$\psi =\sqrt{\frac{\varrho }{m}}exp\left(\mathrm{i}\frac{m}{\hslash }\phi \right),\overline{\psi }=\sqrt{\frac{\varrho }{m}}exp\left(-\mathrm{i}\frac{m}{\hslash }\phi \right),$$

(6)

Madelung [16] discovered an alternative form of the Schrödinger theory, well known as the hydrodynamic picture or Madelung’s picture of quantum theory. In the context of this paper, the form of the Lagrangian that emerges from (5) by means of transformation (6),

$$\ell =-\varrho \left[\frac{\partial \phi }{\partial t}+\frac{1}{2}{\left(\nabla \phi \right)}^2\right]-\frac{{\hslash }^2}{8m^2}\frac{{\left(\nabla \varrho \right)}^2}{\varrho },$$

(7)

is of particular interest, having obviously the analytical form (1) with the blueprint-Lagrangian:

$$L\left(\omega ,\varrho ,\partial \varrho \right)=-\varrho \omega -\frac{{\hslash }^2}{8m^2}\frac{{\left(\partial \varrho \right)}^2}{\varrho }.$$

(8)

2.2. Modified Scheme for Lagrangians with Lorentz/Poincaré Invariance

As stated above, Galilean invariance of scheme (1) results from the invariance of the expressions defined in the replacement rules (2–4). The idea is to modify the replacement rules in a minimalistic way in order to achieve invariance with respect to Lorentz boosts and, moreover, Poincaré invariance. An obvious replacement for the gradient is:

$$\nabla \to \left\{\begin{array}{cc}-{\partial }^{\alpha },\hfill & \text{contra-variant},\hfill \\ +{\partial }_{\alpha },\hfill & \mathrm{covariant},\hfill \end{array}\right.$$

(9)

implying, for instance, $\nabla \phi ·\nabla {\vartheta }_j\to -{\partial }_{\alpha }\phi {\partial }^{\alpha }{\vartheta }_j$ for the inner product of two gradients. In contrast, it is less obvious how to deal with a single time derivative, e.g., ${\partial }_t{\vartheta }_i$, the solitary occurrence of which in (2, 3) is contrary to Lorentz invariance. Motivated by Scholle [17], the following substitution of the phase field:

$$\phi =\varphi +\mathrm{\Xi },$$

(10)

with a certain expression $\mathrm{\Xi }$, termed as ’generating field’ in [17], allows one to re-write (2, 3) as Lorentz-scalars on the proper choice of $\mathrm{\Xi }$: by applying (9) and (10) to (3), the general form

$$\begin{array}{cc}\hfill \stackrel{\odot }{{\vartheta }_j}& \to {\partial }_t{\vartheta }_j-{\partial }^{\alpha }\left(\varphi +\mathrm{\Xi }\right){\partial }_{\alpha }{\vartheta }_j\hfill \\ \hfill & =c{\partial }_0{\vartheta }_j-{\partial }^{\alpha }\mathrm{\Xi }{\partial }_{\alpha }{\vartheta }_j-{\partial }^{\alpha }\varphi {\partial }_{\alpha }{\vartheta }_j\hfill \\ \hfill & =\left[c{\delta }_0^{\alpha }-{\partial }^{\alpha }\mathrm{\Xi }\right]{\partial }_{\alpha }{\vartheta }_j-{\partial }^{\alpha }\varphi {\partial }_{\alpha }{\vartheta }_j\hfill \end{array}$$

results, turning the expression to a Lorentz scalar if the square bracket vanishes, implying ${\partial }^{\alpha }\mathrm{\Xi }=c{\delta }_0^{\alpha }$, and therefore

$$\mathrm{\Xi }=c{x}_0+\mathrm{const}=c^{2}t+\mathrm{const},$$

(11)

in accordance with [17]. Considering this subsequently, (2) becomes

$$\begin{array}{cc}\hfill \omega & \to {\partial }_t\left(\varphi +\mathrm{\Xi }\right)-\frac{1}{2}{\partial }^{\alpha }\left(\varphi +\mathrm{\Xi }\right){\partial }_{\alpha }\left(\varphi +\mathrm{\Xi }\right)\hfill \\ & =\underset{0}{\underbrace{c{\partial }_0\varphi -{\partial }^{\alpha }\mathrm{\Xi }{\partial }_{\alpha }\varphi }}+\underset{c^2/2}{\underbrace{c{\partial }_0\mathrm{\Xi }-\frac{1}{2}{\partial }^{\alpha }\mathrm{\Xi }{\partial }_{\alpha }\mathrm{\Xi }}}-\frac{1}{2}{\partial }^{\alpha }\varphi {\partial }_{\alpha }\varphi ,\hfill \end{array}$$

and therefore takes the form of a Lorentz scalar, too. Summarising the above, by applying the altered replacement rules

$$\begin{array}{cc}\hfill \omega & =\frac{c^2}{2}-\frac{1}{2}{\partial }^{\alpha }\varphi {\partial }_{\alpha }\varphi ,\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill \stackrel{\odot }{{\vartheta }_j}& =-{\partial }^{\alpha }\varphi {\partial }_{\alpha }{\vartheta }_j,j=1,\cdots ,N-1,\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill \partial {\vartheta }_j& =\left\{\begin{array}{cc}-{\partial }^{\alpha }{\vartheta }_j,\hfill & \text{contra-variant},\hfill \\ +{\partial }_{\alpha }{\vartheta }_j,\hfill & \mathrm{covariant},\hfill \end{array}\right.\hfill \end{array}$$

(14)

to a blueprint-Lagrangian according to (1), the resulting Lagrangian is automatically a Lorentz scalar. Since it does not depend on the coordinates $x^{\alpha }$ explicitly, it is also invariant with respect to the Poincaré group.

The above procedure allows for the definition of a canonically associated relativistic form for any Lagrangian with Galilean invariance.

2.3. Inclusion of External Forces

The formalism presented above assumes Galilean or Poincare invariance, and thus the absence of external force fields. Nevertheless, the following considerations show how the general scheme (1) can be extended with at least regard to conservative forces. By exemplary consideration of Schrödinger’s Lagrangian for a particle in a conservative external force field and its transformation into the Madelung picture, one obtains

$$\ell =-\varrho \left[\frac{\partial \phi }{\partial t}+\frac{1}{2}{\left(\nabla \phi \right)}^2+V\right]-\frac{{\hslash }^2}{8m^2}\frac{{\left(\nabla \varrho \right)}^2}{\varrho },$$

(15)

where $V=V\left(x^{\alpha }\right)$ denotes the specific potential energy of the force. By comparing (15) with its free-particle form (7), it becomes apparent that $\omega =\partial \phi /\partial t+{(\nabla \phi )}^2/2$ is effectively replaced by $\omega +V$, giving motivation for the extend scheme:

$$\ell \left({\partial }_t\phi ,\nabla \phi ,{\vartheta }_j,{\partial }_t{\vartheta }_j,\nabla {\vartheta }_j;V\right)=L\left(\omega +V,{\vartheta }_j,\stackrel{\odot }{{\vartheta }_j},\partial {\vartheta }_j\right),$$

(16)

where the external field V appears as a control parameter. We remark that the occurrence of the external field V may cause a symmetry breaking, since it implies explicit dependences on the coordinates $x^{\alpha }$.

2.4. Noether Balances for Mass, Particle Number or Charge, and Energy

Noether’s theorem [7,18,19,20] assigns to each parameter of a symmetry of the Lagrangian represented by a Lie group a homogeneous balance equation of the form

$$\frac{\partial {\varrho }_Q}{\partial t}+\nabla ·{\overrightarrow{j}}_Q=0,$$

(17)

with a density ${\varrho }_Q$ and an associated flux density ${\overrightarrow{j}}_Q$. Physically, this corresponds to the conservation of the global quantity:

$$Q:=\underset{V}{\int }{\varrho }_Q\mathrm{d}V,$$

(18)

with V being the entire volume of the continuum. The focus of the present work is on two symmetries and their related balances:

• Lagrangians being in accordance to the general scheme (1) or its extended form (16) are invariant with respect to the phase shift $\psi \to \psi exp\left(-\mathrm{i}\frac{m}{\hslash }\epsilon \right)$. The related density $\varrho =j^0/c$ and flux density $\overrightarrow{j}=\left(j^1,j^2,j^3\right)$ result from Noether’s theorem for the relativistic case as

  $$j^{\alpha }=-\frac{\partial \ell }{\partial \left({\partial }_{\alpha }\varphi \right)},$$

  (19)

  and in Galilean case, the same formula with $\phi$ in place of $\varphi$, respectively. The physical interpretation of $j^{\alpha }$ is different in both cases: in Schrödinger’s theory, $\varrho$ is the mass or particle density, while in Klein–Gordon theory, it is interpreted as charge density.
• In general, translation invariance is violated if an external potential V is considered according to (16). However, if V depends only on spatial coordinates, i.e., $V=V\left(x^1,x^2,x^3\right)$, then the time translation $t\to t+\tau$ remains as a symmetry transformation of the Lagrangian. The energy density $w=S^0/c$ and associated energy flux density (Poynting vector) $\overrightarrow{S}=\left(S^1,S^2,S^3\right)$ result from Noether’s theorem as

  $$\frac{S^{\alpha }}{c}=\frac{\partial \ell }{\partial \left({\partial }_{\alpha }{\psi }_i\right)}{\partial }_0{\psi }_i-{\delta }_0^{\alpha }\ell .$$

  (20)

2.5. Transition Conditions at Discontinuities

In [10,11], special attention is paid to Lagrangians with discontinuities. Along the interfaces where discontinuities occur, the Euler–Lagrange equations are amended by the transition conditions [10]

$$\overrightarrow{n}·⟦\frac{\partial \ell }{\partial \nabla {\psi }_i},-,\frac{\partial \ell }{\partial \left({\partial }_t{\psi }_i\right)},{\overrightarrow{v}}_s⟧=0,$$

(21)

where $\overrightarrow{n}$ is the normal vector of the interface and ${\overrightarrow{v}}_s$ its velocity. The double square brackets denote the jump of the respective quantity at the discontinuity interface. Additionally Noether’s theorem can be extended to discontinuous Lagrangians, delivering the continuity conditions [11]

$$\overrightarrow{n}·⟦{\overrightarrow{j}}_Q,-,{\varrho }_Q,{\overrightarrow{v}}_s⟧=0,$$

(22)

stating that the flux of the associated quantity Q across the interface has to be steady.

In the present work, a static discontinuity occurs when the external potential V is assumed to be a discontinuous function, e.g., a step potential. In this case ${\overrightarrow{v}}_s=\overrightarrow{0}$ is given.

3. Results

3.1. Relationships between Schrödinger’s and Klein–Gordon’s Theory

3.1.1. Discovery of a Common ‘Blueprint’ of Schrödinger’s and Klein–Gordon’s Lagrangian

By applying the replacement rules (12)–(14) to the blueprint Lagrangian (8) obtained from Madelung’s picture of Schrödinger’s theory, the Poincaré invariant Lagrangian

$$\ell =\frac{1}{2}\varrho {\partial }^{\alpha }\varphi {\partial }_{\alpha }\varphi +\frac{{\hslash }^2}{8m^2}\frac{{\partial }^{\alpha }\varrho {\partial }_{\alpha }\varrho }{\varrho }-\frac{\varrho c^2}{2}$$

(23)

is obtained. By inversion of transformation (6), a corresponding form in terms of a complex state function $\psi$ and its complex conjugate results as:

$$\ell =\frac{{\hslash }^2}{2m}{\partial }^{\alpha }\overline{\psi }{\partial }_{\alpha }\psi -\frac{m{c}^2}{2}\overline{\psi }\psi ,$$

(24)

which is obviously the Klein–Gordon Lagrangian [21], apart from a factor 2. This identifies the Schrödinger and Klein–Gordon Lagrangians as a canonical pair of Lagrangians that can be derived from a common blueprint-Lagrangian, and therefore a close relationship between both theories.

3.1.2. Noether Density and Flux Associated to a Phase Shift

Here and subsequently, the classical notation is preferred to the tensor notation for the purpose of better comparability of the expressions resulting in the Galilean case with those resulting in the relativistic case.

By applying (19) to the general scheme (16), the associated Noether density and flux density result as:

$$\begin{array}{cc}\hfill \varrho & =\left\{\begin{array}{cc}-\frac{\partial L}{\partial \omega }\hfill & ,\mathrm{Galilean},\hfill \\ \frac{1}{c^2}\frac{\partial L}{\partial \omega }{\partial }_t\varphi +\frac{1}{c^2}\frac{\partial L}{\partial \stackrel{\odot }{{\vartheta }_j}}{\partial }_t{\vartheta }_j\hfill & ,\mathrm{relativistic},\hfill \end{array}\right.\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill \overrightarrow{j}& =\left\{\begin{array}{cc}-\frac{\partial L}{\partial \omega }\nabla \phi -\frac{\partial L}{\partial \stackrel{\odot }{{\vartheta }_j}}\nabla {\vartheta }_j\hfill & ,\mathrm{Galilean},\hfill \\ -\frac{\partial L}{\partial \omega }\nabla \varphi -\frac{\partial L}{\partial \stackrel{\odot }{{\vartheta }_j}}\nabla {\vartheta }_j\hfill & ,\mathrm{relativistic}.\hfill \end{array}\right.\hfill \end{array}$$

(26)

While the canonical density $\varrho$ has a remarkably different form in the Galilean and in the relativistic case, the flux density $\overrightarrow{j}$ comes out to be identical in both cases.

For the example Schrödinger Lagrangian (5) and Klein–Gordon Lagrangian (24), one obtains for the canonical density (conveniently in terms of complex fields):

$$\varrho =\left\{\begin{array}{cc}m\overline{\psi }\psi \hfill & ,\text{Schrödinger},\hfill \\ \frac{\hslash }{2\mathrm{i}{c}^2}\left(\psi {\partial }_t\overline{\psi }-\overline{\psi }{\partial }_t\psi \right)\hfill & ,\text{Klein–Gordon},\hfill \end{array}\right.$$

(27)

while the associated canonical flux density results in

$$\overrightarrow{j}=\frac{\hslash }{2\mathrm{i}}\left(\overline{\psi }\nabla \psi -\psi \nabla \overline{\psi }\right),$$

(28)

as a unified form for both theories. This feature is used subsequently in a beneficial way in order to formulate and solve a scattering problem for both theories simultaneously.

3.1.3. Energy Density and Poynting Vector

Assuming a time–independent potential V in (16), the Noether expressions (20) for the energy density and the associated flux density (Poynting vector) result in

$$\begin{array}{cc}\hfill w& =\left\{\begin{array}{cc}-\varrho {\partial }_t\phi +\frac{\partial L}{\partial \stackrel{\odot }{{\vartheta }^j}}{\partial }_t{\vartheta }^j-L\hfill & ,\mathrm{Galilean},\hfill \\ -\varrho {\partial }_t\varphi -\frac{1}{c^2}\frac{\partial L}{\partial \stackrel{\odot }{{\vartheta }_j}}{\partial }_t\varphi {\partial }_t{\vartheta }_j+\frac{1}{c}\frac{\partial L}{\partial \left({\partial }_0{\vartheta }_j\right)}{\partial }_t{\vartheta }_j-L\hfill & ,\mathrm{relativistic},\hfill \end{array}\right.\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill \overrightarrow{S}& =\left\{\begin{array}{cc}-\overrightarrow{j}{\partial }_t\phi +\left(\frac{\partial L}{\partial \stackrel{\odot }{{\vartheta }_j}}\nabla \phi +\frac{\partial L}{\partial \nabla {\vartheta }_j}\right){\partial }_t{\vartheta }_j\hfill & ,\mathrm{Galilean},\hfill \\ -\overrightarrow{j}{\partial }_t\varphi +\left(\frac{\partial L}{\partial \stackrel{\odot }{{\vartheta }_j}}\nabla \varphi +\frac{\partial L}{\partial \nabla {\vartheta }_j}\right){\partial }_t{\vartheta }_j\hfill & ,\mathrm{relativistic},\hfill \end{array}\right.\hfill \end{array}$$

(30)

with the density $\varrho$ and the flux density $\overrightarrow{j}$ already determined by (25), (26). We again notice that the energy density w resulting from the Galilean and from the relativistic Lagrangian differ relevantly from each other, while the Poynting vector $\overrightarrow{S}$ takes again the same form for both Lagrangians.

Especially for Schrödinger’s Lagrangian (5) and the Klein–Gordon Lagrangian (24), the Poynting vector takes the unified form (again preferably in complex representation):

$$\overrightarrow{S}=-\frac{{\hslash }^2}{2m}\left({\partial }_t\overline{\psi }\nabla \psi +{\partial }_t\psi \nabla \overline{\psi }\right).$$

(31)

3.1.4. Discussion

Madelung’s picture of quantum mechanics is hardly known in the scientific community compared to other approaches like Schrödinger’s picture (static operators and time evolution of the state function), Heisenberg’s picture (time evolution of operators and static state function), or Dirac’s picture. A reason for this may be that by transformation (6), Schrödinger’s equation is transformed into two real-valued but nonlinear PDEs [16], inhibiting the finding of solutions significantly. The historical benefit of this approach is primarily to show a connection between quantum mechanics and fluid mechanics, rather than using it for the construction of solutions. In the context of the current work, however, the Madelung picture has been proven to be a powerful tool for identifying (8) as the blueprint form of Schrödinger’s Lagrangian (5).

After applying the replacement rules (12)–(14) and backward transformation to the complex state function, the Klein–Gordon Lagrangian is directly obtained. Although it has been known for a long time that Schrödinger’s equation can be obtained in the limit case $c\to \infty$ of the (transformed) Klein–Gordon equation [21], the reverse direction of obtaining the Klein–Gordon Lagrangian from Schrödinger’s Lagrangian by means of a well–defined canonical procedure is a novelty, revealing a yet-unknown relationship between both theories.

The computation of the Noether fluxes for the phase shift $\psi \to \psi exp\left(-\mathrm{i}\frac{m}{\hslash }\epsilon \right)$ and the time translation $t\to t+\tau$ revealed identical formulas (28) and (31) for the Schrödinger and Klein–Gordon theories, which is a consequence of both Lagrangians constituting a canonical pair.

3.2. Continuity Conditions at Discontinuities of the Potential

Subsequently, transition and conservation conditions at discontinuities of the potential are derived by applying the general formulas from Section 2.5 to the Schrödinger and Klein–Gordon Lagrangians.

3.2.1. Transition Conditions Resulting Directly from Hamilton’s Principle

Transition conditions according to (21) are implied at discontinuities of the the specific potential energy V. In case of the canonical pair of Lagrangians, (5) and (24), variation with respect to $\overline{\psi }$ delivers the condition

$$\overrightarrow{n}·⟦\nabla \psi ⟧=0,$$

(32)

while the respective condition for $\psi$ turns out to be its complex conjugate,

$$\overrightarrow{n}·⟦\nabla \overline{\psi }⟧=0.$$

(33)

Thus, Equations (32) and (33) are effectively one complex condition.

3.2.2. Conservation Conditions Resulting from Extended Noether’s Theorem

Conservation conditions of the form (22) are implied by Noether’s theorem at discontinuities. For the canonical pair of Lagrangians, (5) and (24), these result in the continuity of the canonical flux density (28) and the Poynting vector (31), and can written after simple manipulations as

$$\begin{array}{cc}\hfill \overrightarrow{n}·⟦\overline{\psi }\nabla \psi -\psi \nabla \overline{\psi }⟧& =0,\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill \overrightarrow{n}·⟦{\partial }_t\overline{\psi }\nabla \psi +{\partial }_t\psi \nabla \overline{\psi }⟧& =0.\hfill \end{array}$$

(35)

Some useful relations are obtained via the ‘product rule’:

$$⟦XY⟧=〈X〉⟦Y⟧+⟦X⟧〈Y〉,$$

(36)

which relates the jump of a product $XY$ along the interface to jumps $⟦·⟧$ and mean values $<·>$ of the expressions X and Y. This allows (34) and (35) to be written as:

$$\begin{array}{cc}\hfill ⟦\overline{\psi }⟧〈\overrightarrow{n}·\nabla \psi 〉+〈\overline{\psi }〉\overrightarrow{n}·⟦\nabla \psi ⟧-⟦\psi ⟧〈\overrightarrow{n}·\nabla \overline{\psi }〉-〈\psi 〉\overrightarrow{n}·⟦\nabla \overline{\psi }⟧& =0,\hfill \\ \hfill ⟦{\partial }_t\overline{\psi }⟧〈\overrightarrow{n}·\nabla \psi 〉+〈{\partial }_t\overline{\psi }〉\overrightarrow{n}·⟦\nabla \psi ⟧+⟦{\partial }_t\psi ⟧〈\overrightarrow{n}·\nabla \overline{\psi }〉+〈{\partial }_t\psi 〉\overrightarrow{n}·⟦\nabla \overline{\psi }⟧& =0,\hfill \end{array}$$

and after considering the matching condition (32):

$$\begin{array}{cc}\hfill ⟦\overline{\psi }⟧〈\overrightarrow{n}·\nabla \psi 〉-⟦\psi ⟧〈\overrightarrow{n}·\nabla \overline{\psi }〉& =0,\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill ⟦{\partial }_t\overline{\psi }⟧〈\overrightarrow{n}·\nabla \psi 〉+⟦{\partial }_t\psi ⟧〈\overrightarrow{n}·\nabla \overline{\psi }〉& =0.\hfill \end{array}$$

(38)

These are two real–valued conditions.

3.2.3. Discussion

We remark that the above conservation conditions (37) and (38) result solely from symmetries of the respective Lagrangians via the variation extended Noether’s theorem, and the transition condition transition conditions (32) directly form Hamilton’s principle without any additional ad hoc assumptions. These continuity conditions are referred as ‘natural transition conditions’ (NTC) subsequently, and differ at first glance in part from that applied in the classical literature [4], where simply the continuity of the state function and its normal derivative is assumed, $⟦\psi ⟧=0$ and $\overrightarrow{n}·⟦\nabla \psi ⟧=0$, subsequently referred as ‘methodical continuity conditions’ (MCC). Although frequently used, the latter lack a convincing derivation from physical principles; see discussion in [5,6].

Obviously by the transition condition (32), a reproduction of the MCC $\overrightarrow{n}·⟦\nabla \psi ⟧=0$ is achieved. In contrast, it is yet unclear how the two NTCs (37) and (38) implied by extended Noether’s theorem are related to the remaining MCC $⟦\psi ⟧=0$. This can be shown at least exemplarily for the scattering problem in the following Section 3.3.

3.3. Scattering of a Bimodal Particle Beam at a Step Potential

A particle beam consisting of particles of mass m and momentum $\overrightarrow{p}=\hslash \overrightarrow{k}$ is running against a potential step of height $V_0$, i.e., it passes through a spatial region in which the specific potential energy V has the form of a Heaviside function, see Figure 1. A fraction of the particles passes the threshold, while another fraction is reflected.

3.3.1. State Function for Bimodal Particle Beam

It is assumed that the incoming particle beam according to

$$\psi \left(x,t\right)=u_1\left(x\right)exp\left(-\mathrm{i}{\omega }_{1}t\right)+u_2\left(x\right)exp\left(-\mathrm{i}{\omega }_{2}t\right)$$

(39)

consists of particles with two different energies, $E_n=\hslash {\omega }_n$ with $n=1,2$, where the two eigenfunctions are given as

$$u_n\left(x\right)=\left\{\begin{array}{cc}{A}_{n+}exp\left(\mathrm{i}{k}_{n}x\right)+A_{n-}exp\left(-\mathrm{i}{k}_{n}x\right)\hfill & ,x<0,\hfill \\ B_{n}exp\left(\mathrm{i}{k}_n^{\prime }x\right)\hfill & ,x>0.\hfill \end{array}\right.$$

(40)

3.3.2. Relationships between $k_n,k_n^{\prime }$, and ${\omega }_n$

Schrödinger’s equation is solved by (39) if and only if the dispersion relations

$$\begin{array}{cc}\hfill \frac{{\hslash }^2{k}_n^2}{2m}& =\hslash {\omega }_n,\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill \frac{{\hslash }^2{k}_n^{\prime 2}}{2m}& =\hslash {\omega }_n-V_0\hfill \end{array}$$

(42)

are fulfilled. On elimination of ${\omega }_n$, the direct relationship

$$k_n^{\prime }=\sqrt{k_n^2-\frac{2m{V}_0}{{\hslash }^2}}$$

(43)

between $k_n^{\prime }$ and $k_n$ results. Different from (41) and (42), insertion of (39) into the Klein–Gordon equation, the relativistic dispersion relations

$$\begin{array}{cc}\hfill m{c}^2+\frac{{\hslash }^2{k}_n^2}{m}& ={\displaystyle \frac{{\hslash }^2{\omega }_n^2}{m{c}^2}},\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill m{c}^2+\frac{{\hslash }^2{k}_n^{\prime 2}}{m}& ={\displaystyle \frac{{\hslash }^2{\omega }_n^2}{m{c}^2}}-2V_0\hfill \end{array}$$

(45)

are obtained. Although these differ from the non-relativistic ones, elimination of ${\omega }_n$ leads to the same relationship (43) as in the case of Schrödinger’s equation.

3.3.3. Transition Conditions and Solutions

By inserting (39), the NTCs (37) and (38) take the form:

$$\begin{array}{cc}\hfill 0& =⟦\overline{\psi }⟧〈\overrightarrow{n}·\nabla \psi 〉-\mathrm{c}.\mathrm{c}.\hfill \\ \hfill & =⟦{\overline{u}}_1⟧〈\overrightarrow{n}·\nabla u_1〉-⟦u_1⟧〈\overrightarrow{n}·\nabla {\overline{u}}_1〉+⟦{\overline{u}}_2⟧〈\overrightarrow{n}·\nabla u_2〉-⟦u_2⟧〈\overrightarrow{n}·\nabla {\overline{u}}_2〉\hfill \\ \hfill & +\left(⟦{\overline{u}}_1⟧〈\overrightarrow{n}·\nabla u_2〉-⟦u_2⟧〈\overrightarrow{n}·\nabla {\overline{u}}_1〉\right)exp\left(-\mathrm{i}\left[{\omega }_2-{\omega }_1\right]t\right)\hfill \\ & +\left(⟦{\overline{u}}_2⟧〈\overrightarrow{n}·\nabla u_1〉-⟦u_1⟧〈\overrightarrow{n}·\nabla {\overline{u}}_2〉\right)exp\left(\mathrm{i}\left[{\omega }_2-{\omega }_1\right]t\right),\hfill \\ \hfill 0& =⟦{\partial }_t\overline{\psi }⟧〈\overrightarrow{n}·\nabla \psi 〉+\mathrm{c}.\mathrm{c}.\hfill \\ & =\mathrm{i}\left({\omega }_1⟦{\overline{u}}_1⟧〈\overrightarrow{n}·\nabla u_1〉-{\omega }_1⟦u_1⟧〈\overrightarrow{n}·\nabla {\overline{u}}_1〉+{\omega }_2⟦{\overline{u}}_2⟧〈\overrightarrow{n}·\nabla u_2〉-{\omega }_2⟦u_2⟧〈\overrightarrow{n}·\nabla {\overline{u}}_2〉\right)\hfill \\ & +\mathrm{i}\left({\omega }_1⟦{\overline{u}}_1⟧〈\overrightarrow{n}·\nabla u_2〉-{\omega }_2⟦u_2⟧〈\overrightarrow{n}·\nabla {\overline{u}}_1〉\right)exp\left(-\mathrm{i}\left[{\omega }_2-{\omega }_1\right]t\right)\hfill \\ \hfill & +\mathrm{i}\left({\omega }_2⟦{\overline{u}}_2⟧〈\overrightarrow{n}·\nabla u_1〉-{\omega }_1⟦u_1⟧〈\overrightarrow{n}·\nabla {\overline{u}}_2〉\right)exp\left(\mathrm{i}\left[{\omega }_2-{\omega }_1\right]t\right),\hfill \end{array}$$

delivering after Fourier decomposition the four independent conditions:

$$\begin{array}{cc}\hfill ⟦{\overline{u}}_1⟧〈\overrightarrow{n}·\nabla u_1〉-⟦u_1⟧〈\overrightarrow{n}·\nabla {\overline{u}}_1〉+⟦{\overline{u}}_2⟧〈\overrightarrow{n}·\nabla u_2〉-⟦u_2⟧〈\overrightarrow{n}·\nabla {\overline{u}}_2〉& =0,\hfill \\ \hfill {\omega }_1\left(⟦{\overline{u}}_1⟧〈\overrightarrow{n}·\nabla u_1〉-⟦u_1⟧〈\overrightarrow{n}·\nabla {\overline{u}}_1〉\right)+{\omega }_2\left(⟦{\overline{u}}_2⟧〈\overrightarrow{n}·\nabla u_2〉-⟦u_2⟧〈\overrightarrow{n}·\nabla {\overline{u}}_2〉\right)& =0,\hfill \\ \hfill ⟦{\overline{u}}_1⟧〈\overrightarrow{n}·\nabla u_2〉-⟦u_2⟧〈\overrightarrow{n}·\nabla {\overline{u}}_1〉& =0,\hfill \\ \hfill {\omega }_1⟦{\overline{u}}_1⟧〈\overrightarrow{n}·\nabla u_2〉-{\omega }_2⟦u_2⟧〈\overrightarrow{n}·\nabla {\overline{u}}_1〉& =0,\hfill \end{array}$$

which via linear combinations simplify to

$$\begin{array}{cc}\hfill ⟦{\overline{u}}_1⟧〈\overrightarrow{n}·\nabla u_1〉-⟦u_1⟧〈\overrightarrow{n}·\nabla {\overline{u}}_1〉& =0,\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill ⟦{\overline{u}}_2⟧〈\overrightarrow{n}·\nabla u_2〉-⟦u_2⟧〈\overrightarrow{n}·\nabla {\overline{u}}_2〉& =0,\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill ⟦{\overline{u}}_1⟧〈\overrightarrow{n}·\nabla u_2〉& =0,\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill ⟦u_2⟧〈\overrightarrow{n}·\nabla {\overline{u}}_1〉& =0.\hfill \end{array}$$

(49)

Since it is assumed that both $\hslash {\omega }_1$ and $\hslash {\omega }_2$ are greater than $V_0$, and the particle current does not vanish across the potential threshold, so that both normal derivatives must be non-zero, $〈\overrightarrow{n}·\nabla u_1〉\ne 0$ and $〈\overrightarrow{n}·\nabla u_2〉\ne 0$, it obviously follows from Equations (48) and (49) that $⟦u_1⟧=0$ and $⟦u_2⟧=0$. Considering $⟦\psi ⟧=⟦u_1⟧exp\left(-\mathrm{i}{\omega }_{1}t\right)+⟦u_2⟧exp\left(-\mathrm{i}{\omega }_{2}t\right)$, one finally ends up with

$$⟦\psi ⟧=0,$$

(50)

which is a reproduction of the well-known MCC used in the literature [4,5], together with the NTC (32) for the normal derivative. Both conditions could be obtained here in a consistent manner from Hamilton’s principle and the extended Noether theorem.

Due to the linearity of all relevant equations, the solution for the coefficients of the scattering problem can be taken from the already known solution for a mono-modal particle beam [4]:

$$\begin{array}{cc}\hfill B_n& =\frac{2k_n}{k_n+k_n^{\prime }}{A}_{n+},\hfill \end{array}$$

(51)

$$\begin{array}{cc}\hfill A_{n-}& =\frac{k_n-k_n^{\prime }}{k_n+k_n^{\prime }}{A}_{n+},\hfill \end{array}$$

(52)

for prescribed amplitudes $A_{n+}$ of the incoming beam.

3.3.4. Considerations about Non-Bimodal Particle Beams

The above considerations can also be applied to more general cases, e.g., on particle beams consisting of particles with N different energies and also on such with a continuous energy spectrum, like coherent states. In any of such cases, a reproduction of the condition (50) can be achieved on an analogous treatment.

In case of a mono-modal beam, however, a different situation is encountered: the state function is then an eigenfunction of the energy operator, fulfilling

$${\partial }_t\psi =-\mathrm{i}\omega \psi ,$$

(53)

with $\hslash \omega =E$ being the energy of one particle. Inserting this into (35) delivers the condition

$$\mathrm{i}\omega \overrightarrow{n}·⟦\overline{\psi }\nabla \psi +\psi \nabla \overline{\psi }⟧=0,$$

which gives just a reproduction of the condition (34), and therefore does not contain any additional information. As a consequence, there is one condition missing, and the scattering problem is therefore underdetermined! This issue occurs in general if the state function is an eigenfunction of the energy operator, but can be resolved, e.g., by considering the eigenstate as limiting case of the bimodal problem solved above.

3.3.5. Discussion

The NTC (32) resulting directly by variation recovers the well-known MCC for the normal derivative of the state function, while the two NTCs (37) and (38) resulting from extended Noether’s theorem recover the MCC (50) if the state function is not an eigenfunction of the energy operator, as demonstrated explicitly for a bimodal particle beam against a potential step. By these results, a coherent rationale for the continuity conditions at discontinuities of the potential [4,5,6], derived from system symmetries via extended Noether’s theorem, is given. However, the argumentative gap is only partially but not yet fully closed due to the necessary conditions for the state function not being an eigenfunction of the energy operator. Further investigations on this question are required.

4. Conclusions

By suitable adaptations, the previously established general scheme (1) for Lagrangians of field theories with Galilean invariance could successfully be modified toward Lorentz-invariant field theories, applying the altered replacement rules (12–14). By this, a canonical procedure is defined in order to assign to any Galilei-invariant Lagrangian a canonical Lorentz-invariant analogue. The general scheme allows also for the consideration of the potential of external forces. The latter decreases the symmetry, but the gauge invariance with respect to the phase shift $\psi \to \psi exp\left(-\mathrm{i}\frac{m}{\hslash }\epsilon \right)$ remains as a Lie symmetry. Furthermore, if the specific potential energy of the external force does not depend on time, the time translation $t\to t+\tau$ remains also as a symmetry. Remarkably, the two associated Noether fluxes, $\overrightarrow{j}$ and $\overrightarrow{S}$, prove to be identical for the Lagrangian with Galilean invariance and for the canonically associated relativistic Lagrangian. As an example, it has been demonstrated that the Schrödinger and Klein–Gordon Lagrangians are a canonical pair. In this context, the transformation (6) originally used by Madelung [16] proves to be beneficial in identifying the common blueprint Lagrangian.

The second objective, to trace the continuity conditions commonly used in the literature back to physical principles by obtaining them from Hamilton’s principle itself and from the extended Noether theorem, could not be achieved for the general case, but could at least be achieved for the standard problem of a particle beam against a potential step assuming a bimodal (or multimodal) state.

The derivation of continuity conditions at discontinuities of the potential in the Schrödinger and Klein–Gordon theories in a natural way from the respective Lagrangian and its symmetries, and therefore from basic physical principles, is primarily of fundamental interest. However, the results obtained above from the extended Noether’s theorem are equally useful for the development of semi-analytical and numerical methods and their applications: obvious applications of scattering theory are imaging methods based on particle beams, such as electron backscatter diffraction (EBSD) [22] and others, the use of which could be in the elucidation of periodic crystal structures [23] with and without micro-cracks for non-destructive material examinations, as well as media with inner boundaries, e.g., inhomogeneous materials [24]. The contribution of the present work could be in particular the development of new variational-based simulation methods, such as the Ritz’s direct method [25], which can be implemented both as a semi-analytical method and as a numerical FE method; due to the realisation that continuity conditions are naturally contained in the Lagrangian density on which the method is based, there is no need to explicitly take these into account in the implementation, which may lead to an increase in the efficiency of the respective method.

Other future research paths related to fundamental questions about other field theories are conceivable, e.g., finding a common blueprint-Lagrangian of the form (1) for Pauli’s and Dirac’s equation or identifying a Galilean analogue of Proca’s equation.

One can even go further by using the analogy between quantum theory and fluid mechanics revealed by Madelung’s picture in order to apply the concept of canonical pairs to Lagrangians for classical fluid dynamics known in the literature [10,26,27,28,29], in order to construct corresponding canonically associated Lagrangians also for relativistic fluid dynamics [30]. This could also be extended to non–Newtonian fluids, e.g., viscoelastic ones [31]. A considerable benefit of using the canonical procedure developed in Section 2.2 is the avoidance of the limit $c\to \infty$ from relativistic to Newtonian regimes, which is known to be not without complications [32]. Again, variational formulations found in this way could be the starting point for developing new semi-analytical or FE methods to complement existing ones, e.g., finite difference methods [33].