Natural Extension of the Schr&ouml;dinger Equation to Quasi-Relativistic Speeds

A Schrodinger-like equation for a single free quantum particle is presented. It is argued that this equation can be considered a natural relativistic extension of the Schrodinger equation for energies smaller than the energy associated to the particle’s mass. Some basic properties of this equation: Galilean invariance, probability density, and relation to the Klein-Gordon equation are discussed. The scholastic value of the proposed Grave de Peralta equation is illustrated by finding precise quasi-relativistic solutions for the infinite rectangular well and the quantum rotor problems. Consequences of the non-linearity of the proposed equation for the quantum superposition principle are discussed.


Introduction
Since the discovery of the quantum wave mechanics by Erwin Schrödinger in 1925, the Schrödinger equation has been often used for introducing the fundamentals of quantum mechanics [1] [2] [3] [4] [5]. The one-dimensional Schrödinger equation for a free particle with mass m is given by the following equation [ where ℏ is the Plank constant (h) divided by 2π. However, the Schrödinger equation is not Lorentz invariant but Galilean invariant [6]; therefore, a relativistic How to cite this paper: Grave de Peralta, L.
(2020) Natural Extension of the Schrödinger quantum mechanics cannot be based on Equation (1). A fully relativistic quantum theory requires to be funded on equations that are valid for any two observers moving respect to each other at constant velocity. In contrast, the Galilean invariance of Equation (1) means that two such observers will only agree in the adequacy of Equation (1) for describing the movement of a massive free quantum particle when the relative speed between the observers (V o ) is much smaller than the speed of the light in the vacuum (c). In practice, this is not a terrible limitation of the Schrödinger equation because up to today humans have been only able to travel at speeds much smaller than c. This is one of the principal reasons why the Schrödinger equation is still relevant almost 100 years after its discovery. However, as it will be discussed in Section 2, there is another important limitation of Equation (1): it describes a particle in which linear momentum (p) and kinetic energy (K) are related by a classical relation that is not valid at relativistic speeds [1] [2] [3] [6]. The famous relativistic equation E m = mc 2 , where E m is the energy associated to the mass of a particle [7] [8], implies the equivalence between mass and energy. This equivalence has profound implications for the formulation of any relativistic quantum mechanics theory. When the kinetic energy of a free particle with mass m equals the energy associated to the mass of the particle, i.e., K = mc 2 , a second particle with the same mass can be created from the kinetic energy of the original particle; therefore, the number of particles may not be conserved in a fully relativistic quantum theory [2] [8] [9]. A common argument used for guiding the search for the correct Lorentz invariant basic equation of a relativistic quantum mechanics is that in such equation the time and spatial variables should appear on equal footing as it happens in the Lorentz transformations [8] [9]. For instance, in contrast to Equation (1), in the Lorentz invariant Klein-Gordon equation does not appear the first partial derivative respect to time but the second one as shown in Equation (2), which is the Klein-Gordon equation for free particle [8] Unfortunately, Equation (2) does not formally look at all like Equation (1), thus masking how the Klein-Gordon equation becomes the Schrödinger equation when the particle moves at speeds (V) much smaller than c. Moreover, there are solutions of Equation (2) with unwanted properties like superluminal phase velocity, negatives energies, and associated with negative probabilities [8] [9]. In Section 2, the consequences of an intriguing natural extension of the Schrödinger equation to quasi-relativistic speeds are explored. The term "quasi-relativistic" is used in this work as meaning a particle moving at so large speeds that it is necessary to use the correct relativistic relation between p and K but still the number of particles is constant because K < mc 2 . The following equation is the center of attention here: Clearly, the Grave de Peralta equation (Equation (3)) exactly coincides with the Schrödinger equation (Equation (1)) when V c  . As it will be discussed in Section 3, the formal similitude between Equation (3) and Equation (1) The plane waves ψ and ψ KG in Equation (6) are solutions of Equations (3) and (2), respectively. E, K and p are the relativistic total and kinetic energy and the linear momentum of a free particle, respectively [7] [8]. It is worth noting that two solutions of Equation (3) corresponding to two different particle's speeds are not simultaneously solution of the same equation but solutions of two slightly different equations only differing in the value of γ v . Even when the full discussion of this topic is outside of the scope of this work, due to its relevance, the implications of the non-linearity of Equation (3) for the quantum mechanics superposition principle are briefly discussed in Section 5. Finally, the conclusions of this work are given in Section 7.

Schrödinger Equation Extension to Quasi-Relativistic Speeds
Formally, Equation (1) can be obtained from the classical relation between K and p for a free particle when V c Then, Equation (1) is obtained by substituting K and p by the following energy and momentum quantum operators [1] [2] [3]: By analogy, Equation (3) can be simply obtained combining Equation (8) with the relation between the relativistic expressions of the kinetic energy and the linear momentum of a free particle traveling at quasi-relativistic speeds: Equation (9) can be easily obtained from the following well-known relativistic equations [7] [8]: The Klein-Gordon equation can formally be obtained from the first expression of Equation (10) by assigning the temporal partial derivative operator in Equation (8) to the total relativistic energy (E) of the free particle, which is the sum of its kinetic energy plus the energy associated to the mass of the particle [7] [8]. However, if one chooses to assign this operator to K, as it is done when obtaining the Schrödinger equation, then from Equations (9) and (8) follows Equation (3). This is not the customary choice, but in this work instead of simply discharging this option, it is explored the consequences of this natural choice.
For instance, a simple substitution of ψ(x, t) given by Equation (6) in Equation (3) results in Equation (9), thus demonstrating that ψ(x, t) given by Equation (6) is a plane wave solution of Equation (3), which phase velocity V ph = K/p is related to the velocity of the particle by the following expression: Consequently, V ph < V < c; i.e., the plane wave ψ(x, t) given by Equation (6) is subluminal and, as happen for a plane wave solution of the Schrödinger equation, V ph ~ V/2 when V c  . In contrast, the substitution of ψ KG (x, t) given by Equation (6) in Equation (2) results in Equation (10), thus demonstrating that ψ KG (x, t) given by Equation (6) is a plane wave solution of Equation (2), which phase velocity V KG = E/p is given by the following expression: Consequently, ψ KG (x, t) is superluminal because V KG > c. Equations (5) and (6) suggest that the time-independent equations corresponding to Equations (2) and (3) are equal. In fact, looking for solutions of the form X(x)T(t) of Equations (1), for Equation (3), produces the same time-independent equation in the three cases: As it will be illustrated below, often X(x) and κ are determined solving Equation (14) under adequate boundary conditions; then the possible values of p are determinate from the possible values of κ. However, the relation between K and p are different for non-relativistic and quasi-relativistic speeds; therefore, the solutions of Equations (1) and (3) have equal spatial dependences but different values of K. Also, the relation between E, K, and p are different for quasi-relativistic speeds; therefore, the solutions of Equations (2) and (3) have equal spatial dependences but different values of K and E. Equations (9) and (10) can be obtained from each other using Equation (11); however, Equation (10) admits solutions with positive and negative energies but K only can be positive in Equation (9). This is in correspondence to the presence of a second-order temporal partial derivative in Equation (2), which determines that Equation (2) has solutions with positive and negative energies [8] [9]. In contrast, there is a first-order temporal partial derivative in Equations (1) and (3). This determines that Equations (1)

Probability Density and Galilean Invariance
Due to the formal similitude between Equation (3) and Equation (1) [4]. In short, one can associate a probability density ρ(x, t) to a normalized solution of Equation (3) in the following way: The probability density corresponding to the Schrödinger equation is well-defined when both ψ/(2m) and ψ * /(2m) tend to zero when |x| is very large [1]. Similarly, provided that V 2 and γ V are constant, it can be shown than ρ(x, t) defined by Equation (15)  tend to zero when |x| is very large, which is a less restrictive condition when 1 V γ  than the one required for the Schrödinger equation. The rate of the temporal variation of ρ(x, t) is then given by the following expression: The temporal derivatives of ψ and ψ * in Equation (16) can be substituted by expressions containing spatial derivatives of ψ and ψ * by using Equation (3) and its complex conjugate equation. In this way Equation (16) can be transformed in the following one: Then using Equation (18) permits to rewrite Equation (17) as the one-dimensional (1D) probability continuity equation [1]: Like for the Schrödinger equation [1], it is easy to show that Equation (19) can be generalized to three dimensions (3D). The absence of negative values of ρ and J is a consequence of the absence of a second time derivative in the Equation (3) [8] [9]. This concludes the demonstration that a probability continuity equation can be associated to the solutions of Equation (3) as it is done for the Schrödinger equation. In what follows a qualitative discussion about the Galilean invariance of Equation (3) is presented. A more formal discussion about this topic is presented in Annex A. At a first sight, Equation (3) does not look neither Galilean nor Lorentz invariant. Equation (3) should not be Lorentz invariant because in Equation (3) the temporal and spatial partial derivatives do not have the same order [8] [9]. In contrast, it is well known that Equation (2) is Lorentz invariant [9]. The formal similitude between Equations (3) and (1) suggests that Equation (3) may be Galileo invariant, but there is a problem. A well-defined Equation (3) requires a constant value of V 2 and γ V . As it will be illustrated in Sections 4 and 6, there are very interesting problems where this requirement is fulfilled. For instance, one of these problems is the description of the movement of a massive quantum particle confined in a 1D box, which is at rest respect to an inertial reference frame S. An observer at rest respect S may think about the particle as moving with constant quasi-relativistic speed (V) but changing direction each time the particle bounced in the box's walls. However, a second observer moving parallel to the box with velocity +V o respect to the first observer, but at rest respect to a second inertial reference frame S', would see the particle moving Thus, the second observer would not find well-defined the value of V' 2 and V γ ′ that should be introduced in Equation (3). However, at quasi-relativistic particle's speeds Consequently, at quasi-relativistic particle's speeds when o V V  , both observers will see the particle moving with (almost) the same values of V 2 and γ V . Moreover, in this quasi-relativistic limit p' ~ p and K' ~ K. Consequently, both observers will agree in that they should solve Equation (3) for finding the possible quantum states of the massive particle moving at quasi-relativistic speeds inside of the 1D box. i.e., Equation (3) is Galilean invariant. Nevertheless, as it will be shown below, Equation (3) can be used for solving quasi-relativistic quantum problems.

Infinite Rectangular Well
An important but simple problem often solved in quantum mechanics textbook is a particle moving inside an infinite rectangular well at speeds much smaller   infinite rectangular well can easily be extended to the 3D infinite rectangular well as it is done for the Schrödinger equation [4] [5]. Consequently, Equation (3) establishes a fundamental connection between quantum mechanics and especial theory of relativity: no single particle with mass can be confined in a volume much smaller than (λ C ) 3 because when this occurs, K > mc 2 and the number of particles may not be constant anymore; therefore, a single point-particle with mass cannot exist. Point-particles with mass can only exist in fully relativistic quantum field theories where the number of particles is not constant. This is true for an electron, a quark, and probably may also be true for a black hole and the whole universe at the beginning of the Big Bang. This is consistent, for instance, with the confinement of an electron in the Hydrogen atom because for an electron λ Ce ~ 2.4 × 10 −3 nm, which is ~20 times smaller than the radius of the Hydrogen atom, r B ~ 5.   [5]. However, in general, the values of K n calculated using Equation (27) are smaller than the ones calculated using the Schrödinger equation. This in excellent correspondence with more involved numerical results obtained solving the Dirac equation for the 1D infinite rectangular well [10]. Moreover, and more significant for experiments, the differences in energies between different energy levels are slightly different when obtained using Equations (1) and (3).

Superposition Principle
Besides allowing to obtain precise quasi-relativistic solutions of several interesting problems, like tunneling through a barrier and other problems with piecewise constant potentials, following similar procedures than in the quantum mechanics textbooks for V c  represents a legitime possible state of a particle in an infinite well. The superposition state represented by ψ Sch (x, t) is often interpreted as a state where the particle is neither in the state ψ Sch1 (x, t) where the kinetic energy is K 1 nor in the state ψ Sch2 (x, t) where the kinetic energy is K 2 , but somehow the particle is simultaneously in both states. The existence of superposition states like ψ Sch (x, t) is then a fundamental consequence of the linearity of Equation (1) with no classical counterpart. This exemplifies the weirdness of quantum mechanics [6] [11].
Moreover, the superposition state ψ Sch (x, t) represent a qubit, concept that is at the heart of current attempts to demonstrate a practical quantum computer [11] [12]. In contrast to the Schrödinger equation, Equation (3) is not linear. If ψ 1 (x, t) and ψ 2 (x, t) are two solutions of Equation (3)  Alternatively, the non-linearity of Equation (3) suggests that ψ 1 (x, t) and ψ 2 (x, t) could be understood as corresponding to two different phases-of-a-system which are described by a different equation each. ψ(x, t), which is not a solution of Equation (3), describes them a state of the system where no one of these two phases exists but where somehow, when a set of identical measurements is done on a system which is prepared in the state ψ(x, t) each time, then a fast transition of the system is induced by the measurement and the system randomly transits either to the phase represented by ψ 1 (x, t) with probability |a| 2 or to the phase represented by ψ 2 (x, t) with probability |b| 2 . In this description, the state of the system represented by ψ(x, t) must be different from a state where a mixture of the phases ψ 1 (x, t) and ψ 2 (x, t) actually exist. For instance, let's assume that ψ 1 (x, t) and ψ 2 (x, t) are two solutions, of the set of Equations (3) for the infinite rectangular well, with kinetic energies given by K 1 and K 2 , respectively. Loosely borrowing the words "superheated" and "supercooled" from possible phase transitions in liquids, one could say that the state ( ) ( ) ( ) responds to a state of the system which is not a solution of any Equation (3). The state ψ(x, t) could be formed by superheating the state ψ 1 (x, t) or by supercooling the state ψ 2 (x, t). When a set of N measurements is done on the system in the state ψ(x, t), |a| 2 N times a fast system's transition occurs from the superheated state to the state ψ 1 (x, t), and |b| 2 N times the transition occurs from the supercooled state to the state ψ 2 (x, t). This point of view may motivate the search for the unknown equation for which ψ(x, t) is a solution.

Quasi-Relativistic Quantum Rotor
Courses of Quantum Mechanics often include how to solve the Schrödinger equation for a quantum rigid rotator, which in general is a quantum particle moving with constant speed in a sphere. Therefore, all the kinetic energy of a quantum rigid rotator is rotational. Instances of the quantum rigid rotor appear when describing the relative movement between two particles forming a system like a diatomic molecule (neglecting vibrations) [5]. The 3D Schrödinger equation for a particle moving in a central potential Φ(r) is given by the following equation [ The rotational kinetic energy of a quantum rigid rotator is given by the second term in the right size of Equation (32); therefore, the first term in the right size of Equation (32) vanishes for a quantum rigid rotator [5]. In addition, r = r S and Φ(r) = Φ(r S ) are constants because the radius of the sphere containing the particle trajectory (r S ) is constant; therefore, choosing Φ(r S ) = 0, and introducing the moment of inertia of a rotating mass  [5]. Therefore, the quasi-relativistic kinetic energy is given by the following expression: where L C = 2πr S is the maximum length of a circle contained in the sphere where the particle moves. From Equation (37) follows that the spatial dependence of ψ l,m coincide with the spatial dependence of the wavefunction calculated using the Schrödinger equation [5]. As expected, Equation (38) gives the know values of the particle's energies at low speeds when γ V ~ 1 [5]. From Equation (38)  1 .
Equation (39) gives 2 3 V γ = when l = 1 and L C = λ C ; evaluating for these values Equation (38) results in K 1 ~ 0.7mc 2 , which is smaller than the value K 1 ~ mc 2 that would be obtained, using the Schrödinger equation, for the state with minimum non-zero angular momentum (l = 1) of a quantum rotor with L C = λ C [5]. Moreover, this result is precise because the calculated energy (K 1 ~ 0. 7mc 2 ) is quasi-relativistic. In contrast, Equation (39) gives 2 9 V γ = when l = 1 and L C = λ C /2; evaluating for these values Equation (38) results in K 1 = 2mc 2 . The number of particles may not be constant at this energy. Consequently, Equation (3) also establishes the following fundamental connection between quantum mechanics and especial theory of relativity: there is a stable orbit with minimum length that a quantum particle of mass m, moving with constant non-zero speed in a sphere, can have. This length is equal to the Compton wavelength associated to the particle's mass. Combining Equations (38) and (39) allow for rewritten Equation , Equation (40) gives the know values of the energies calculated using the Schrödinger equation for a non-relativistic quantum rotor [5]. However, in general, the values of K l calculated using Equation (40) are smaller than the ones calculated using the Schrödinger equation. Moreover, and more significant for experiments, the differences in energies between different energy levels are slightly different when obtained using Equations (1) and (3).

Conclusion
Relativistic quantum mechanics has evolved a lot since 1925, when Erwin , , x y z = r is a 3D spatial vector is rectangular coordinates. Evidently, Equation (A4) is a 3D version of Equation (2). Therefore, a 3D version of Equation (6) is a plane wave solution of Equation (A4), which is a Lorentz-scalar because the wave's phase can be rewritten as the scalar product of two Lorentz-invariant four-components vectors [7] [8] [9]: The equation in the variables x and t that results, after using Equation (A14) for transforming the differential operators of Equation (A12), do not need to be equal to Equation (3). Therefore, the second function of ε(x, t) is to guarantee that ψ(x, t) given by Equations (A13) and (A15) satisfies Equation (3). If there is a function ε(x, t) satisfying these two requirements, then ψ(x, t) and ( ) , x t ψ ′ ′ ′ both satisfy the same equation and both have equal square module values; therefore, both described the same physical reality [6]. i.e., Equation (3) would be Galilean invariant. It can be shown that for the Schrödinger equation, ε Sch (x, t) is given by the following equation [6]: Therefore, ψ Sch (x, t) is a boosted version of ( ) , Sch x t ψ ′ ′ ′ because for a non-relativistic particle: In addition ψ Sch (x, t) satisfies Equation (1) [6]. What follows is the demonstration that Equation (3) is approximately Galileo invariant when o V V  , and K ~ mc 2 . One should find a function ε(x, t) for Equation (A12) that satisfies the two requirements discussed above. First, using the Galilean relations given by Equation (A14), the differential operators in the variables x' and t' in Equation (A12) transform to the following differential operators in the variables x and t [6]: