Revisiting Schrödinger’s fourth-order, real-valued wave equation and the implication from the resulting energy levels

In his seminal part IV, Annalen der Physik vol. 81, 1926 paper, Schrödinger has developed a clear understanding about the wave equation that produces the correct quadratic dispersion relation for matter-waves and he first presents a real-valued wave equation that is fourth-order in space and second-order in time. In the view of the mathematical difficulties associated with the eigenvalue analysis of a fourth-order, differential equation in association with the structure of the Hamilton–Jacobi equation, Schrödinger splits the fourth-order real operator into the product of two, second-order, conjugate complex operators and retains only one of the two complex operators to construct his iconic second-order, complex-valued wave equation. In this paper, we show that Schrödinger’s original fourth-order, real-valued wave equation is a stiffer equation that produces higher energy levels than his second-order, complex-valued wave equation that predicts with remarkable accuracy the energy levels observed in the atomic line spectra of the chemical elements. Accordingly, the fourth-order, real-valued wave equation is too stiff to predict the emitted energy levels from the electrons of the chemical elements; therefore, the paper concludes that quantum mechanics can only be described with the less stiff, second-order, complex-valued wave equation.


INTRODUCTION
During his effort to construct a matter-wave equation that satisfies the quadratic dispersion relation between the angular frequency  and the wave number  ( = ℏ "#  " with ℏ = $ "% where ℎ = 1 Professor, Dept. of Civil and Environmental Engineering, Southern Methodist Univ, Dallas, TX 75275 (corresponding author).Email: nmakris@smu.edu6.62607 × 10 &'( # !)* + = Planck's constant), Schrödinger in his part IV, 1926 paper [1,2] reaches a real-valued, fourth-order in space and second-order in time differential equation where  is the mass of the elementary, non-relativistic particle and () is its potential energy that is only a function of the position .In his 1926 paper [1], Schrödinger explains in his own words: "Eq.( 1) is thus evidently the uniform and general wave equation for the field scalar ".
He further recognizes that his fourth-order, Eq. ( 1) resembles the fourth-order, equations of motion that emerge from the theory of elasticity and references the governing equation of a vibrating plate.
More precisely, because of the 3-dimensional geometry of atoms, the description of an electron orbiting the atom with Eq. ( 1) resembles to the equation of motion of a vibrating shell [3][4][5][6] which had not been developed at that time.
For standing waves, the spatial and temporal dependence of the matter-wave can be separated so that (, )  = ±  ℏ (, )   !(, )  != −  !ℏ !(, ) In the interest of simplifying the calculations in the eigenvalue analysis of Eq. (1); in association that () does not contain the time, Schrödinger [1,2] substitutes the second of Eq. (3) into Eq.
(1) and recasts it in a factored form ℏ (, )  = − ℏ !2 ∇ !ψ(, ) + ()(, ) At the end of section §1 of his part IV, 1926 paper [1,2] Schrödinger indicates that for "a conservative system, Eq. ( 5) is essentially equivalent to Eq. ( 1), as the real operator may be split up into the product of the two conjugate complex operators if  does not contain the time".
The above equivalence statement is not true, since the fourth-order, real-valued wave equation (1) is a "stiffer" equation than the second-order, complex-valued equation ( 5), yielding higher eigenvalues and therefore higher energy levels.
The higher energy levels predicted by the stiffer fourth-order, real-valued wave equation ( 1) than these predicted by the classical second-order, complex-valued, Schrödinger equation ( 5) are shown in this paper by computing the energy levels of a one-dimensional elementary particle, (, ), trapped in a square well with finite potential .The paper shows that the one-dimensional version of Schrödinger's original fourth-order, real-valued equation is equivalent to the governing equation of a vibrating flexural-shear beam [13,14].By splitting the fourth-order, real-valued operator into the product of two conjugate second-order, complex-valued operators and upon retaining only one of the complex operators, Schrödinger [1,2] essentially removed from his original fourth-order, Eq. (1) its "flexural stiffness" and left it only with its "shear stiffness".
In view of the many predictions with remarkable precision of Schrödinger's second-order, complex-valued Eq. ( 5) for the atomic orbitals of the chemical elements [15][16][17][18][19] in association with the higher energy levels predicted from his original fourth-order, real-valued Eq. ( (therefore, apparently incorrect), this paper offers a straight forward explanation why Quantum Mechanics can only be described with complex-valued functions-a finding that is in agreement with more elaborate recent studies that hinge upon symmetry conditions of real number pairs [20] or involve entangled qubits [21][22][23].These recent studies on entangled qubits [21][22][23] offer the opposite conclusion than the work of McKague et al. [24] which suggests that a real-valued quantum theory can describe a broad range of quantum systems.
This paper shows in a simple, straight-forward manner that Schrödinger's original fourth-order, real-valued wave equation (1), which is the simplest possible real-valued wave equation that satisfies the quadratic dispersion relation,  = ℏ "#  " , is too stiff to predict the visible energy levels that correspond to the visible atomic line spectra of the chemical elements.By splitting the fourth-order, real-valued operator of Eq. ( 1) into the product of two conjugate second order, complexvalued operators, Schrödinger [1,2] extracts a more flexible equation than his original 4 th -order, real-valued Eq. ( 1) at the expense of being complex-valued−that is his iconic Eq. ( 5) which predicted correctly the energy levels of the hydrogen atom; and subsequently made a wealth of fundamental predictions with remarkable precision at the atomic and molecular scale in the century to come [15][16][17][18][19]25].
The question that deserves an answer is how Schrödinger developed the remarkable intuition to proceed from the onset of his efforts with a complex-valued equation for matter-waves-that is only the one factor of the split 4 th order, real-valued equation; which while complex-valued, is flexible enough to predict the correct frequencies manifested in the visible atomic line spectra of the chemical elements in the years to come and abandoned his original fourth-order, real-valued equation that its predictions were apparently never explored.

THE "FLEXURAL-SHEAR BEAM" EQUATION FOR MATTER-WAVES
In the interest of illustrating that the fourth-order, real-valued wave equation ( 1) is a stiffer equation than Schrödinger's second-order, complex-valued Eq. ( 5), we consider for simplicity a single elementary, non-relativistic practice with mass  > 0 in one-dimension moving along the positive direction, , within an energy potential ().The total energy of the elementary particle, , is described with its Hamiltonian, where  = / is the momentum of the elementary particle and .0 = " represents its kinetic energy.Using Einstein's [26] quantized energy expression,  = ℎ = ℏ and de Broglie's [27] momentum− wavelength relation,  = ℎ/ = ℏ, where  = 2/ is the wave number, the Hamiltonian of the elementary particle given by Eq. ( 6) assumes the form For a particle moving freely in the absence of a potential (() = 0), Eq. ( 7) leads to a quadratic dispersion relation  = ℏ "#  " for matter-waves as opposed to the linear dissipation relation,  = , of electromagnetic waves of shear waves in a solid continuum.
The simplest expression for a matter-wave travelling along the positive  − direction is (, ) =  1  2()/&40) and upon using that  = /ℏ and  = /ℏ (, ) =  ( The time-derivative of Eq. (8) gives Substitution of the expression for the energy, , given by Eq. ( 6) into Eq.( 9) gives The 2 nd space-derivative of Eq. ( 8) gives and substitution of the quantity  " (, ) from Eq. (11) into Eq.( 10) yields the one-dimensional version of the time-dependent Schrödinger equation given by Eq. ( 5) We now proceed by taking higher derivatives.The time-derivative of Eq. ( 9) in association with Eq. ( 8) gives whereas by raising the Hamiltonian given by Eq. ( 6) to the second power gives Substitution of the expression for  " given by Eq. ( 14) into Eq.( 13) yields Upon differentiating of Eq. ( 11) in space two more times, The substitution of the quantity  ( (, ) from Eq. ( 16) and of the quantity  " (, ) from Eq.
(11) into Eq.(15) gives Equation ( 17) is the one-dimensional version of the real-valued Eq. ( 1) originally presented by Schrödinger [1,2] which satisfies the quadratic dispersion relation of matter-waves as dictated by Eq. (7).We coin this time-dependent equation: the "flexural-shear beam wave equation" because of the striking similarities with an approximate beam equation that was proposed by Heidebrecht and Stafford Smith [13] to model the dynamics of tall buildings which consist of a strong corewall that offers flexural resistance acting in parallel with the surrounding framing system of the building that offers shear resistance to lateral loads.

WAVES
The corresponding time-independent equation for standing waves (mode shapes) of Eq. ( 17) is derived with the standard method of separation of variables where (, ) = ()(). Accordingly, and Substitution of the expressions for the partial derivatives given by Eqs. ( 18) and ( 19) into Eq.( 17) and upon dividing with ()() gives The left hand-side of Eq. ( 20) is a function of time alone; whereas, the right-hand side is a function of space alone.Accordingly, where  is a spring constant with units Accordingly, Eq. ( 21) is the equation of motion of a harmonic oscillator with a real-valued solution () =  sin   +  cos  (22) where  = J/ is the natural frequency of the harmonic oscillator.
Returning to Eq. ( 20), its right-hand side is also equal to the spring constant  =  " .

ELEMENTARY PARTICLE TRAPPED IN A FINITE POTENTIAL SQUARE WELL WITH STRENGTH 𝑽 > 𝟎
Given that both the 4 th -order, real-valued flexural-shear beam Eq. ( 17) and the 2 nd -order, complexvalued Schrödinger Eq. ( 12) satisfy the quadratic dispersion relation offered by Eq. ( 7) as dictated by the Hamiltonian; we proceed by comparing the prediction of these two equations in an effort to show that Schrödinger's original, fourth-order, real-valued Eq. ( 1) is a stiffer differential equation than his second-order, complex-valued Eq. ( 5) or Eq. ( 12) in one dimension.The quadratic Hamiltonian operator appearing in the flexural-shear beam Eq. ( 27) leads to elaborate calculations even for simple cases; therefore, we select as a test case the response analysis of an elementary, particle with mass  trapped in a square potential well with finite potential  and width 2.
Accordingly, the potential at the bottom of the well is zero as shown in Fig. 1.This simple, one dimensional idealization has been employed to determine the wavelengths for color-center absorption [28].For the case where the elementary particle happens to be outside the well (|| ≥ ), () =  > 0, and Eq. ( 24) gives The solutions of the homogeneous Eq. ( 28) are expected to be of the form () =  8/ and Eq. ( 28) yields the following characteristic equation where  >  > 0. The four roots of the characteristic Eq. ( 29) are Accordingly, for the case || ≥  where () =  >  > 0 the solution for () is For the case where the elementary particle is within the potential well (|| ≤ ), () = 0 and Eq. ( 24) gives By setting (4 " /ℏ ( ) " =  ( , Eq. (33) assumes the form Eq. (34) has a real-valued solution [29,30].
It is worth nothing that Eq. ( 34 For this case where  ≤ −, the solution () given by Eq. (32) remains finite when  " =  ( = 0. Consequently, for this case in which  -and  ' are real-valued and given by Eqs. ( 30) and (31).
For this case () is given by Eq. (35).
Consequently, for this case in which  -and  ' are real-valued and given by Eqs.(30) and (31).
The solution of the wave equation () has to be continuous over the entire domain −∞ <  < ∞.Accordingly, at  = −, Eq. (36) from the left and Eq. ( 35) from the right need to satisfy the following continuity equations: Similarly, at  = , Eq. ( 35) from the left and Eq.(37) from the right need to satisfy the following continuity equations.
The homogeneous system of eight equations that is generated by the eight continuity Eqs. ( 38) and (39) can be decomposed in four equations that produce the even eigenfunctions  9 : () and four equations produce the odd eigenfunctions  9 ; ().The homogeneous system that produces the even eigenfunctions is where  = (/ℏ)√2 is a dimensionless positive real number that expresses the strength of the potential well and  =  = (/ℏ)√2 are the eigenvalues of the even eigenfunctions to be determined.The eigenvalues  9 depend on the dimensionless product  rather than on the individual values of  and  and they are calculated by setting the determinant of the 4 × 4 matrix appearing on the left of Eq. ( 42) equal to zero.As an example, for  = 10 the characteristic equation of the homogeneous system given by Eq. ( 42) yields four real roots (eigenvalues,  ∈ {1, 2, 3, 4}) for  9 = (/ℏ)J2 9 = 1.9747, 4.6204, 7.2901, and 9.7999.For larger value of  (deeper and wider potential well) the number of real eigenvalues increases given that the unknown eigenvalue  needs to remain smaller than  for the radical √ "  " −  " of the last column of the matrix appearing in Eq. ( 42) to remain positive.
Similarly, the homogenous system as results from the continuity equations that produces the odd eigenfunctions is The finite eigenvalues  9 = (/ℏ)J2 9 that corresponds to the odd eigenfunctions are computed by setting the determinant of the 4 × 4 matrix appearing on the left of Eq. ( 43) equal to zero.As an example, for  = 10 the characteristic equation of the homogeneous system given by Eq. ( 43) yields three real roots (eigenvalues,  ∈ {1, 2, 3}) for  9 = (/ℏ)J2 9 = 3.2887, 5.9574, and 8.5976.For larger value of  (deeper and wider potential well) the number of real roots of the characteristic equation (eigenvalues) increases as long as  <  so that the radical √ "  " −  " appearing in the last column of the 4 × 4 matrix Eq. ( 43) remains real.

COMPARISON OF THE EIGENVALUES PREDICTED FROM THE 4 TH -ORDER FLEXURAL-SHEAR BEAM EQUATION AND FROM THE CLASSICAL 2 ND -ORDER SCHRÖDINGER EQUATION
For any given value of the strength of the square potential well,  the resulting eigenvalues of the fourth-order, flexural-shear beam Eq. ( 24) or Eq. ( 27),  9 = (/ℏ)J2 9 , yield the admissible energy levels of the elementary particle in the finite square potential well,  9 = ( 9 " ℏ " )/(2 " ).
Clearly, the predicted energy levels,  9 , are different than the corresponding energy levels,  9 , predicted from the solution of the second-order, time-independent Schrödinger equation.
The predicted eigenvalues,  9 = (/ℏ)J2 9 of an elementary particle in a finite square potential well with the second-order, Schrödinger equation are the roots of the transcendental Eqs.
Table 1 compares the predicted eigenvalues for a non-relativistic particle in a finite square potential well with potential  from the fourth-order, flexural-shear beam wave equation and the secondorder, Schrödinger wave equation for  = 10 and 30.Table 1 also shows the limiting eigenvalues for a particle trapped in an infinitely deep potential well ( = as they result from the secondorder, Schrödinger equation,  9 = (/ℏ)J2 9 = /2 [31] and from the fourth-order, flexuralshear beam equation which are the solution of the characteristic equation cos(2) cosh(2) = 1 as shown in the sequel.

Table 1:
The seven eigenvalues (energy levels)  9 = < ℏ J2 9 for a particle in a finite potential well with strength  = < ℏ √2 = 10, when described with the fourth-order, flexural-shear beam wave equation and with the classical second-order, Schrödinger wave equation, together with the first 9 corresponding eigenvalues when  = 30 and ∞.
Table 1 reveals that when  = 10 all seven eigenvalues that result from the fourth-order, flexuralshear beam equation are larger than the corresponding seven eigenvalues that result from the classical second-order, Schrödinger equation.The same is true for the case when  = 30.
Consequently, this analysis shows that the 4 th -order, real-valued flexural-shear beam equation for matter-waves given by Eq. ( 17) is a stiffer equation than the classical 2 nd -order, complex-valued Schrödinger equation given by Eq. ( 13).Therefore, Schrödinger's equivalence statement that Eq.
Furthermore, Table 1 reveals that when  = 10, the first two eigenvalues  -= 1.9747 and  " = 3.2887 that result from the fourth-order, flexural-shear beam equation are even larger than the first two eigenvalues  -= /2, and  " =  that result from the classical second-order, Schrödinger equation at the limiting case when the strength of the potential well is infinite ( = 0 ℏ √2 = ∞) [31].This pattern where the eigenvalues predicted from the fourth-order, flexuralshear beam equation when trapped in a finite potential well exceed the eigenvalues predicted by the second-order, Schrödinger equation when the particle is trapped in an infinite potential well becomes more dominant as the strength,  of the finite potential well increases.For instance, when  = 30, the first seven eigenvalues that result from the fourth-order, flexural-shear beam equation are larger than the first seven eigenvalues that result from the classical second-order, Schrödinger equation at the limiting case of an infinitely strong potential well.Accordingly, there is a need to calculate the energy levels of an elementary particle trapped in an infinitely strong potential well ( = ∞) when described with the fourth-order, flexural-shear beam wave equation (24) or (27).
The wavefunctions (eigenmodes) associated with the energy levels (eigenvalues) appearing in Table 1 for the situation where the elementary particle is described with the fourth-order, flexuralshear beam wavefunction are offered by Eq. (35) for 0 ≤ || ≤  and by Eq. (37) for  ≥ .

EIGENVALUES OF THE 4 TH -ORDER MATTER-WAVE EQUATION OF AN ELEMENTARY PARTICLE TRAPPED IN AN INFINITE-POTENTIAL SQUARE WELL
Figure 2 reveals that as the strength of the finite potential well increases, the eigenfunctions  9 () that result from the solution of the 4 th -order, wave equation ( 24) or ( 27) meet the walls of the square potential well at a decreasing slope which eventually tends to zero, ( = ) = 0, as the strength of the potential well, , tends to infinity.These zero-slope boundary conditions of the eigenmodes of the trapped particle at the walls of the infinitely strong potential well are drastically different than the finite-slope boundary conditions of the eigenmodes of the trapped particle when described with the 2 nd -order, Schrödinger equation ( 9 () = y " A sin < 9% A = with 0 < x < a = 2L) [31].These fixed-end (zero-slope) boundary conditions (clamped eigenmodes) is another proof that the 4 th -order, real-valued Eq. (1) originally proposed by Schrödinger [1,2] is a stiffer equation than his classical 2 nd -order, complex-valued Eq. ( 5).The eigenfunctions of the particle trapped in an infinitely strong potential well when described with the 4 th -order, flexural-shear beam wave equation (24) are given by Eq. (35), and the integration constants  -,  " ,  ' , and  ( are derived by enforcing the boundary conditions (−) = () = 0 The roots of Eq. (50),  9 =  9  = (/ℏ)J2 9 are the eigenvalues of the fixed-end eigenmodes appearing in Table 1 under  = ∞.

DISCUSSION AND SUMMARY
In this paper we show that Schrödinger's original 4 th -order, real-valued Eq. ( 1) for matter-waves is a stiffer description (higher energy levels) of the behavior of elementary particles than the description offered from his classical, 2 nd -order, complex-valued Eq. ( 5).Given the remarkable predictions of the complex-valued Eq. ( 5) for the visible energy levels of the chemical elements as manifested from their visible atomic line-spectra [15][16][17][18][19]25], in association that his original 4 thorder, real-valued equation predicts invariably higher-energy levels (therefore, apparently incorrect) this paper shows that Quantum Mechanics can only be described with the less stiff, complex-valued wave equation ( 5).This finding is in agreement with more elaborate recent studies that hinge upon symmetry conditions of real number pairs [20] or involve entangled qubits [21][22][23].
At the same time, the paper brings forward that Schrödinger's 2 nd -order, complex-valued equation was extracted from his original 4 th -order, real-valued equation by splitting the 4 th -order, real operator in Eq. ( 1) into the product of two conjugate complex operators and subsequently retaining only one of the two complex 2 nd -order, operators−a rather disruptive mathematical intervention that removed the "flexural stiffness" from his original 4 th -order, real-valued equation.
This disruptive mathematical intervention that alters the physics of his original Eq. ( 1) motivates the conjecture that perhaps the visible energy levels of the chemical elements as manifested from the visible atomic line-spectra is only a fraction of the total emitted energy by the atoms and molecules.The conjecture advanced herein is that perhaps Schrödinger's original 4 th -order, realvalued Eq. ( 1) is the correct equation that predicts the total emitted energy (visible and dark) and the excess of energy above the visible energy predicted with Schrödinger's 2 nd -order complexvalued Eq. ( 1) is merely dark-energy that is not visible on the atomic line spectra of the chemical elements.In this event, @;AB .() ?@C;@ =  D#E#<B; !%* ?@C;@ +  CA@F Fig. 3. Normalized energy levels,  9 " = ( " /ℏ " )2 9 of an elementary particle trapped in a finite potential well with strength  = (/ℏ)√2 = 10 predicted with the 4 th -order, flexural-shear beam equation ( 24) and the 2 nd -order, classical Schrödinger's equation.
The proportion of the dark energy (dark bars) to the visible energy (dotted lines) shown in Fig. 3 for the one-dimensional idealization of an electron trapped in a finite potential square well is much smaller when compared to the current estimate that roughly 68% of the universe is dark energy.
In a realistic 3-dimensional analysis for the energies emitted by the electrons of the chemical elements, the differences between the energy levels predicted from the 3-dimentional "shell" Eq.
(1) and the 3-dimentional "membrane" Eq. ( 5) are expected to be much higher than the difference from their one-dimensional versions; and therefore, the predictions of Schrödinger's original 4 thorder Eq. ( 1) perhaps deserve to be further investigated.

Fig. 1 .
Fig. 1.The finite potential square well with constant strength  outside the well with width 2.
of the four coefficients is assigned an arbitrary value and the other three coefficients are calculated in proportion to the arbitrary assigned value of the first coefficient since the eigenfunctions  9 : () and  9 ; () are eigenmodes of arbitrary amplitude which subsequently can be normalized according to some normalizing rule such as ∫ |()| " dx > &> = ∫  " ()dx = 1 > &>