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QM-1: Quantum Mechanics in CPP — The Schrödinger Equation from Phase-Carrying DI-Bit Hopping

Thomas Lee Abshier

Published Jan 1, 2026• 0 citations
Publisher

Abstract

Derivation of the time-dependent Schrödinger equation from phase-carrying DI-bit hopping on the 600-cell lattice. Each DI bit carries a complex amplitude at each Grid Point; the equation emerges from the hopping dynamics. The time-dependent Schr\"odinger equation is derived in Conscious Point Physics from the hopping of phase-carrying Displacement Increment (DI) bits on the 600-cell lattice. Each DI bit carries a complex amplitude _i = \,e^i_i at each Grid Point, where _i is the DI-bit number density and _i is the geometric phase accumulated at velocity c = /. The discrete evolution over one Absolute Moment tick involves a complex hopping Hamiltonian with off-diagonal amplitude T = ^2/(4m s^2), derived from the D

DOI
10.17605/osf.io/fbnjc

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This talk is about a method for solving the numerical initial value problem of the time-dependent Schrodinger equation with a Hamiltonian which may depend on time. It is only for time independent Hamiltonians that I was able to describe the accuracy and stability properties in the form of proven theorems. All practical experience shows, that the time-dependent case behaves equally well. In this method, the Hamiltonian comes in only as a hermitian linear operator and not as a mathematical expression that can be reordered or otherwise transformed in order to speed up the necessary calculations. In all cases where it is a feasible computational task to apply the Hamilton operator to a generic state vector, my suggestion is to try this method as a workhorse. It can be considered a ‘Simpson’s rule’ for quantum dynamics. There is no way for me to explain the matter from a technical point of view in 10 minutes, so I’ll try to convey the basic idea of the method. This idea is to treat quantum dynamics in closer analogy to classical dynamics as one usually does. One may argue, that there is a fundamental difference between classical and quantum dynamics since the Schrodinger equation is first order with respect to the time derivative, whereas classical mechanics is inherently second order, since forces determine accelerations and not velocities. This is not so clear a distinction as one might think, since also classical mechanics can be given the first-order form of canonical equations. I propose the opposite of such an approach, namely to give quantum dynamics a second-order form by simply differentiating the time-dependent Schrodinger equation with respect to time. To be sure, this is only a heuristic device for defining a computationally feasible approximation and not a proposal to change quantum mechanics. For the resulting equation for the second time derivative of the wave function, I then carry over and use the integration methods from the field of granular matter dynamics which I have been using in an industrial project concerning simulation of the toning process in electro-photographic copiers/printers over some years [4]. Such methods come in

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Capstone synthesis: all of quantum mechanics emerges in CPP from four primitives (CPs, DI bits, the 600-cell lattice, the Nexus). The complex DI-bit amplitude evolves by deterministic hopping; the emergent quantum framework is recovered. All of quantum mechanics emerges in Conscious Point Physics from four primitives: Conscious Points (CPs), phase-carrying Displacement Increment (DI) bits, the 600-cell lattice, and the Nexus. The complex DI-bit amplitude _i = \,e^i_i evolves by deterministic hopping, yielding the Schr\"odinger equation in the continuum limit (Paper~2~). Simultaneous propagation along all geodesics gives the discrete Feynman path integral and quantum superposition (Paper~3~). The Nexus maintains non-separable joi

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We now may bring the time-dependent state functions ( )tx,Ψ into the picture, taking into account, as we already did in Eq.(29.9), that only the state function but not its position coordinate x varies with time, so that ( ) ( ) t tx dt txd ∂ Ψ∂≡Ψ ,, (30.3) Upon insertion of the expression (29.24) for the expectation value, Eq.(30.2) becomes ˆˆ ˆ ˆ ˆd d fi f i f dx i f f dx i dt dt t t t ⎛ ⎞∂Ψ ∂Ψ ∂= Ψ Ψ = Ψ +Ψ + Ψ Ψ⎜ ⎟∂ ∂ ∂⎝ ⎠∫ ∫ ∫= = = = dx∗ (30.4) On the other hand, as the Hamiltonian Hˆ is a Hermitian operator, we have ( )ˆ ˆˆ ˆ ˆˆ ˆ ˆ, f ff H i Hf f H dx i dt t ∞ ∞ ∂ ∂⎡ ⎤ + = −Ψ Ψ +Ψ Ψ + Ψ Ψ⎣ ⎦ ∂ ∂∫ ∫= = x (30.5) ( ) dx t fidxHffH Ψ∂ ∂Ψ+⎥⎦ ⎤⎢⎣ ⎡ ΨΨ+ΨΨ−= ∫∫ ∗∗ ˆˆˆˆˆ = From Eqs.(30.4) and (30.5) it follows that the evolution equation (30.2) is true only if Ψ=∂ Ψ∂ H t i ˆ= (30.6) and this is a basic result called the time-dependent Schrödinger equation. This equation describes the evolution of quantum systems in terms of time-dependent state functions that obey Eq.(30.6), but neglect the time dependence of the quantum operators. Such a description is known as the Schrödinger picture. The fifth postulate states that the timedependent Schrödinger equation must always be satisfied. Postulate 5. The evolution of the state function of a quantum system is given by the time-dependent Schrödinger equation (30.6), where Hˆ is the Hamiltonian of the system.

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•Jan 1, 2026•0 citations
Derivation of the time-dependent Schrödinger equation from phase-carrying DI-bit hopping on the 600-cell lattice. Each DI bit carries a complex amplitude at each Grid Point; the equation emerges from the hopping dynamics. The time-dependent Schr"odinger equation is derived in Conscious Point Physics from the hopping of phase-carrying Displacement Increment (DI) bits on the 600-cell lattice. Each DI bit carries a complex amplitude _i = ,e^i_i at each Grid Point, where _i is the DI-bit number density and _i is the geometric phase accumulated at velocity c = /. The discrete evolution over one Absolute Moment tick involves a complex hopping Hamiltonian with off-diagonal amplitude T = ^2/(4m s^2), derived from the D
This talk is about a method for solving the numerical initial value problem of the time-dependent Schrodinger equation with a Hamiltonian which may depend on time. It is only for time independent Hamiltonians that I was able to describe the accuracy and stability properties in the form of proven theorems. All practical experience shows, that the time-dependent case behaves equally well. In this method, the Hamiltonian comes in only as a hermitian linear operator and not as a mathematical expression that can be reordered or otherwise transformed in order to speed up the necessary calculations. In all cases where it is a feasible computational task to apply the Hamilton operator to a generic state vector, my suggestion is to try this method as a workhorse. It can be considered a ‘Simpson’s rule’ for quantum dynamics. There is no way for me to explain the matter from a technical point of view in 10 minutes, so I’ll try to convey the basic idea of the method. This idea is to treat quantum dynamics in closer analogy to classical dynamics as one usually does. One may argue, that there is a fundamental difference between classical and quantum dynamics since the Schrodinger equation is first order with respect to the time derivative, whereas classical mechanics is inherently second order, since forces determine accelerations and not velocities. This is not so clear a distinction as one might think, since also classical mechanics can be given the first-order form of canonical equations. I propose the opposite of such an approach, namely to give quantum dynamics a second-order form by simply differentiating the time-dependent Schrodinger equation with respect to time. To be sure, this is only a heuristic device for defining a computationally feasible approximation and not a proposal to change quantum mechanics. For the resulting equation for the second time derivative of the wave function, I then carry over and use the integration methods from the field of granular matter dynamics which I have been using in an industrial project concerning simulation of the toning process in electro-photographic copiers/printers over some years [4]. Such methods come in
Capstone synthesis: all of quantum mechanics emerges in CPP from four primitives (CPs, DI bits, the 600-cell lattice, the Nexus). The complex DI-bit amplitude evolves by deterministic hopping; the emergent quantum framework is recovered. All of quantum mechanics emerges in Conscious Point Physics from four primitives: Conscious Points (CPs), phase-carrying Displacement Increment (DI) bits, the 600-cell lattice, and the Nexus. The complex DI-bit amplitude _i = \,e^i_i evolves by deterministic hopping, yielding the Schr\"odinger equation in the continuum limit (Paper~2~). Simultaneous propagation along all geodesics gives the discrete Feynman path integral and quantum superposition (Paper~3~). The Nexus maintains non-separable joi
We now may bring the time-dependent state functions ( )tx,Ψ into the picture, taking into account, as we already did in Eq.(29.9), that only the state function but not its position coordinate x varies with time, so that ( ) ( ) t tx dt txd ∂ Ψ∂≡Ψ ,, (30.3) Upon insertion of the expression (29.24) for the expectation value, Eq.(30.2) becomes ˆˆ ˆ ˆ ˆd d fi f i f dx i f f dx i dt dt t t t ⎛ ⎞∂Ψ ∂Ψ ∂= Ψ Ψ = Ψ +Ψ + Ψ Ψ⎜ ⎟∂ ∂ ∂⎝ ⎠∫ ∫ ∫= = = = dx∗ (30.4) On the other hand, as the Hamiltonian Hˆ is a Hermitian operator, we have ( )ˆ ˆˆ ˆ ˆˆ ˆ ˆ, f ff H i Hf f H dx i dt t ∞ ∞ ∂ ∂⎡ ⎤ + = −Ψ Ψ +Ψ Ψ + Ψ Ψ⎣ ⎦ ∂ ∂∫ ∫= = x (30.5) ( ) dx t fidxHffH Ψ∂ ∂Ψ+⎥⎦ ⎤⎢⎣ ⎡ ΨΨ+ΨΨ−= ∫∫ ∗∗ ˆˆˆˆˆ = From Eqs.(30.4) and (30.5) it follows that the evolution equation (30.2) is true only if Ψ=∂ Ψ∂ H t i ˆ= (30.6) and this is a basic result called the time-dependent Schrödinger equation. This equation describes the evolution of quantum systems in terms of time-dependent state functions that obey Eq.(30.6), but neglect the time dependence of the quantum operators. Such a description is known as the Schrödinger picture. The fifth postulate states that the timedependent Schrödinger equation must always be satisfied. Postulate 5. The evolution of the state function of a quantum system is given by the time-dependent Schrödinger equation (30.6), where Hˆ is the Hamiltonian of the system.
The time-dependent Schrödinger equation is a cornerstone of quantum physics and governs all phenomena of the microscopic world. However, despite its importance, its origin is still not widely appreciated and properly understood. We obtain the Schrödinger equation from a mathematical identity by a slight generalization of the formulation of classical statistical mechanics based on the Hamilton-Jacobi equation. This approach brings out most clearly the fact that the linearity of quantum mechanics is intimately connected to the strong coupling between the amplitude and phase of a quantum wave.
Superposition and interference emerge from coherent summation of phase-carrying DI bits propagating along all available geodesics of the 600-cell lattice — the discrete Feynman path integral in CPP. Superposition and interference in Conscious Point Physics emerge from the coherent summation of phase-carrying Displacement Increment (DI) bits propagating simultaneously along all available geodesics of the 600-cell lattice --- the discrete Feynman path integral. Each DI bit emitted from source vertex s accumulates a deterministic geometric phase _k along path k, determined by the DI-bit velocity c = / and the local SSV potential. The total amplitude at detector vertex d is (d) = _k A_0 e^i_k,
Derivation of Bell inequality violation in CPP via non-separable Nexus states between spin-1/2 CP aggregates in singlet states; phase encoding via ZBW helix structure. We derive quantum entanglement and Bell inequality violation in Conscious Point Physics from first principles. A spin-12 CP aggregate carries a two-component DI-bit state whose phase encodes the spin direction via the ZBW helix (companion~C4~). When two such aggregates are created in a total-spin-zero state, their joint DI-bit state is non-separable: it cannot be written as a product of independent single-particle states (Theorem~). The Nexus maintains this joint state globally at each Absolute