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3.5.2. Normalizing the time-dependent wave function Normalizing the time-dependent wave function is similar to that of its time-independent counter part, with the exception that the derivation of N by (19) causes N to become a time-dependen quantity N(t) in its own right due to the normalization integral being only over space. However when the definition Vporm(r,t) = N(t)U(r,t) of N is substituted into (1) as was done in (19), on is forced to again conclude that N must be constant. This apparent contradiction is resolved by observing three primary facts about the nature of (1). The first is that (1) assumes a normalize: solution from the beginning, while the second is that the act of solving (1) does not automaticalh guarantee normalization. Rather, one must solve (1) and then proceed with normalization. Th third fact is that the time-dependent potential term, together with boundary conditions if transient is the force that causes the wave function to evolve through time. To guarantee that the systen remains normalized at all times, the normalization parameter must evolve likewise through time Intuitively speaking, a change in the wave function implies a change in the correction factor usec to ensure normalization, and this reality is implicit in (1) but must be made explicit in its solution Thus, the evidence falls in favor of a time-dependent N. Performing the same procedure as wit! (19) to derive N produces a time-dependent N, and multiplying this N(t) by W (r,t) yields th normalized time-dependent wave functions (22) for both bounded and unbounded domains.

Figure 3 3.5.2. Normalizing the time-dependent wave function Normalizing the time-dependent wave function is similar to that of its time-independent counter part, with the exception that the derivation of N by (19) causes N to become a time-dependen quantity N(t) in its own right due to the normalization integral being only over space. However when the definition Vporm(r,t) = N(t)U(r,t) of N is substituted into (1) as was done in (19), on is forced to again conclude that N must be constant. This apparent contradiction is resolved by observing three primary facts about the nature of (1). The first is that (1) assumes a normalize: solution from the beginning, while the second is that the act of solving (1) does not automaticalh guarantee normalization. Rather, one must solve (1) and then proceed with normalization. Th third fact is that the time-dependent potential term, together with boundary conditions if transient is the force that causes the wave function to evolve through time. To guarantee that the systen remains normalized at all times, the normalization parameter must evolve likewise through time Intuitively speaking, a change in the wave function implies a change in the correction factor usec to ensure normalization, and this reality is implicit in (1) but must be made explicit in its solution Thus, the evidence falls in favor of a time-dependent N. Performing the same procedure as wit! (19) to derive N produces a time-dependent N, and multiplying this N(t) by W (r,t) yields th normalized time-dependent wave functions (22) for both bounded and unbounded domains.

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Substituting (14) into (12) and the result into (13) fully specifies the general wave function (15) in terms of its boundary and initial conditions. (15) is the general solution of (1) in any number n of spatial dimensions on a bounded domain 2 C (R” x ¢ € R) or on an unbounded domain R” xt € R. The time-independent Schrédinger equation (16) is used to model systems whose wave functions need not evolve through time. It is clear that the Riccati-Sturm-Liouville transform (1-3) and the Email: alan.longfellow.118@gmail.com
Substituting (14) into (12) and the result into (13) fully specifies the general wave function (15) in terms of its boundary and initial conditions. (15) is the general solution of (1) in any number n of spatial dimensions on a bounded domain 2 C (R” x ¢ € R) or on an unbounded domain R” xt € R. The time-independent Schrédinger equation (16) is used to model systems whose wave functions need not evolve through time. It is clear that the Riccati-Sturm-Liouville transform (1-3) and the Email: alan.longfellow.118@gmail.com
Email: alan.longfellow.118@gmail.com The Schrédinger equation for the helium atom [13] is given by (24) where ri and rg are the dis- tances between the nucleus and the two electrons respectively, e is the elementary charge, €o is the permittivity of free space, u = te is the reduced mass of either electron me with respect to the nucleus my, and FE is the total energy of the system. The boundary conditions are the same as those of the hydrogen atom, namely lim,_,.. V(r) = 0 and lim,_,9 U(r) € C.
Email: alan.longfellow.118@gmail.com The Schrédinger equation for the helium atom [13] is given by (24) where ri and rg are the dis- tances between the nucleus and the two electrons respectively, e is the elementary charge, €o is the permittivity of free space, u = te is the reduced mass of either electron me with respect to the nucleus my, and FE is the total energy of the system. The boundary conditions are the same as those of the hydrogen atom, namely lim,_,.. V(r) = 0 and lim,_,9 U(r) € C.
Applying Theorem 2 by performing the transformation outlined in (16-19) maps (25) to a six- dimensional spherically coordinated Poisson equation (26). Approximating the system barycenter as located at the center of the nucleus requires that Mn — CO and consequently pp» — me. All first-order partial derivatives of the wave function thus cancel. The Schrodinger equation for the helium atom (24) is now written in a form (25) compatible with Theorem 2.
Applying Theorem 2 by performing the transformation outlined in (16-19) maps (25) to a six- dimensional spherically coordinated Poisson equation (26). Approximating the system barycenter as located at the center of the nucleus requires that Mn — CO and consequently pp» — me. All first-order partial derivatives of the wave function thus cancel. The Schrodinger equation for the helium atom (24) is now written in a form (25) compatible with Theorem 2.
Reversing the change of variables in Theorem 2 which culminates in (21) allows the wave function to be recovered (32) in terms of the Poisson equation sum solution (31) where = = 1,79, l1, 02,1, My. While the wave function appears complete sans a deterministic expression for the electronic position vectors, it remains necessary to choose coefficients C; and C such that the wave function is quantized according to the principal quantum numbers n, and nz. The physical interpretation of ’ is uncertain, however it may play a part in providing assurance of quantization.
Reversing the change of variables in Theorem 2 which culminates in (21) allows the wave function to be recovered (32) in terms of the Poisson equation sum solution (31) where = = 1,79, l1, 02,1, My. While the wave function appears complete sans a deterministic expression for the electronic position vectors, it remains necessary to choose coefficients C; and C such that the wave function is quantized according to the principal quantum numbers n, and nz. The physical interpretation of ’ is uncertain, however it may play a part in providing assurance of quantization.
4.1.3. Extension to an arbitrary atom and general multiparticulate systems Since the transformation (16-18) applied to (25) yielding (26) renders this diparticulate system sep- arable, and separability is not limited by the number of space dimensions needed to describe the sys- tem, this pathway to a solution for two particles may be extended to multiparticulate systems. For an atom with n, electrons and Z protons whose barycenter is approximated as the center of its nu- cleus, the reasoning (25-32) for helium is applied to the multielectronic Schrédinger equation to yield the general framework of the wave function (33) where © = n1,...,1%n.,41,---;4ne,M1,---;Mn, and n = (f1,---In,). The solution to the time-independent Schrédinger equation is therefore shown to be adaptable to multiparticulate systems by reason of the separability of Poisson equations with constant forcing terms. Since constant-forced heat equations are likewise separable, this argument for the possibility of multiparticulate solutions is further extensible to time-dependent cases.
4.1.3. Extension to an arbitrary atom and general multiparticulate systems Since the transformation (16-18) applied to (25) yielding (26) renders this diparticulate system sep- arable, and separability is not limited by the number of space dimensions needed to describe the sys- tem, this pathway to a solution for two particles may be extended to multiparticulate systems. For an atom with n, electrons and Z protons whose barycenter is approximated as the center of its nu- cleus, the reasoning (25-32) for helium is applied to the multielectronic Schrédinger equation to yield the general framework of the wave function (33) where © = n1,...,1%n.,41,---;4ne,M1,---;Mn, and n = (f1,---In,). The solution to the time-independent Schrédinger equation is therefore shown to be adaptable to multiparticulate systems by reason of the separability of Poisson equations with constant forcing terms. Since constant-forced heat equations are likewise separable, this argument for the possibility of multiparticulate solutions is further extensible to time-dependent cases.
While the Cole-Hopf transform was famously applied to solve the viscous Burgers equation without a forcing term, a similar solution to the forced Burgers equation (34) remains a mystery in fluid mechanics. Exact solutions have been found for simplified cases such as f(x, t) = fi (x) + fo(t), but perhaps the closest the literature comes to revealing u(x,t) for arbitrary f(x,t) has been by Lie group symmetry analyses, one of which was held back from finding a tractable formula for u(z,t) by an encounter with the likewise unsolved Riccati equation [15] while another analysis remarked that the form of the solution is heavily dependent on the form of f(x, t) [16]. Despite the difficulty of the orced Burgers and Riccati equations, there exists a well-known sequence of two changes of variables 17] [18] which converts (34) into a real-valued version of the Schrédinger equation. Since we have already solved the Schrédinger equation, the solution to (34) is derived likewise in this subsection. Since this fluid mechanical application does not directly deal with probability distributions, we use the unnormalized solution (15) to the time-dependent Schrédinger equation.
While the Cole-Hopf transform was famously applied to solve the viscous Burgers equation without a forcing term, a similar solution to the forced Burgers equation (34) remains a mystery in fluid mechanics. Exact solutions have been found for simplified cases such as f(x, t) = fi (x) + fo(t), but perhaps the closest the literature comes to revealing u(x,t) for arbitrary f(x,t) has been by Lie group symmetry analyses, one of which was held back from finding a tractable formula for u(z,t) by an encounter with the likewise unsolved Riccati equation [15] while another analysis remarked that the form of the solution is heavily dependent on the form of f(x, t) [16]. Despite the difficulty of the orced Burgers and Riccati equations, there exists a well-known sequence of two changes of variables 17] [18] which converts (34) into a real-valued version of the Schrédinger equation. Since we have already solved the Schrédinger equation, the solution to (34) is derived likewise in this subsection. Since this fluid mechanical application does not directly deal with probability distributions, we use the unnormalized solution (15) to the time-dependent Schrédinger equation.
The initial and boundary conditions of u(a,t) are recovered (39) from those of ® (x, t) by invert ing the changes of variables that led (34) to (37), as is the velocity field (40). Substituting (39) int« (38) and the result into (40) yields the exact and general solution of (34). It is important to not that the function g(t) appearing as the potential term of (36) is not included in (39-40) because i carries no objective meaning in light of the inclusion of all initial and boundary data.
The initial and boundary conditions of u(a,t) are recovered (39) from those of ® (x, t) by invert ing the changes of variables that led (34) to (37), as is the velocity field (40). Substituting (39) int« (38) and the result into (40) yields the exact and general solution of (34). It is important to not that the function g(t) appearing as the potential term of (36) is not included in (39-40) because i carries no objective meaning in light of the inclusion of all initial and boundary data.
A general and analytical solution to the Schrodinger equation was found by adapting known and very broadly applicable methods for solving second-order PDEs to the peculiarities of the Schrodinger equation, especially the Riccati-Sturm-Liouville and modified Cole-Hopf transformations. Unlike its known solutions, which apply only to cases of simplified potential terms, this solution encapsulates cases of any and all potentials, whether they are functions of space, time, or both. Because its derivation process hinged on the linear differential operators of vector calculus, the solution can be extended to any number of space dimensions or particles. This shocking degree of generality allows a nearly complete theory of the helium atom to be derived, which is able to be generalized to describe any atom as well as a large variety of multiparticulate systems. Since the helium atom is the simplest system exhibiting quantum entanglement, the analytical study of quantum entanglement may now begin, as may that of higher orbitals of multielectronic atomic systems. In a second application, the solution to the Schrdédinger equation was adapted to solve the forced Burgers equation of fluid mechanics, enabling progress toward a mathematical theory of the elusive phenomenon of turbulence. These results are expected to make significant progress toward a thorough analytical understanding of quantum systems and linear PDEs with variable coefficients more generally.
A general and analytical solution to the Schrodinger equation was found by adapting known and very broadly applicable methods for solving second-order PDEs to the peculiarities of the Schrodinger equation, especially the Riccati-Sturm-Liouville and modified Cole-Hopf transformations. Unlike its known solutions, which apply only to cases of simplified potential terms, this solution encapsulates cases of any and all potentials, whether they are functions of space, time, or both. Because its derivation process hinged on the linear differential operators of vector calculus, the solution can be extended to any number of space dimensions or particles. This shocking degree of generality allows a nearly complete theory of the helium atom to be derived, which is able to be generalized to describe any atom as well as a large variety of multiparticulate systems. Since the helium atom is the simplest system exhibiting quantum entanglement, the analytical study of quantum entanglement may now begin, as may that of higher orbitals of multielectronic atomic systems. In a second application, the solution to the Schrdédinger equation was adapted to solve the forced Burgers equation of fluid mechanics, enabling progress toward a mathematical theory of the elusive phenomenon of turbulence. These results are expected to make significant progress toward a thorough analytical understanding of quantum systems and linear PDEs with variable coefficients more generally.
Related topics:
Partial Differential EquationsQuantum PhysicsQuantum ChemistryAtomic PhysicsAtomic and Molecular PhysicsSpherical and Ellipsoidal HarmonicsHelium AtomSolution of Schrodinger EquationBurgers equationHeat Equation
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