The dual Schrödinger equation is defined as replacing the imaginary number i by —1 in the original one. This paper shows that the dual equation shares the same stationary states as the original one. Different from the original one, it explicitly defines a dynamic process for a system to evolve from any state to lower energy states and eventually to the lowest one. Its power as a global optimization algorithm might be used by nature for constructing atoms and molecules. It shall be interesting to verify its existence in nature.The dual Schrödinger equation is defined as replacing the imaginary number i by —1 in the original one. This paper shows that the dual equation shares the same stationary states as the original one. Different from the original one, it explicitly defines a dynamic process for a system to evolve from any state to lower energy states and eventually to the lowest one. Its power as a global optimization algorithm might be used by nature for constructing atoms and molecules. It shall be interesting to verify its existence in nature.

AIP conference proceedingsJan 1, 20110 citations

A Global Optimization Approach to Quantum Mechanics

Xiaofei Huang

This paper presents a global optimization approach to quantum mechanics, which describes the most fundamental dynamics of the universe. It suggests that the wave-like behavior of (sub)atomic particles could be the critical characteristic of a global optimization method deployed by nature so that (sub)atomic systems can find their ground states corresponding to the global minimum of some energy function associated with the system. The classic time-independent Schrodinger equation is shown to be derivable from the global optimization method to support this argument.This paper presents a global optimization approach to quantum mechanics, which describes the most fundamental dynamics of the universe. It suggests that the wave-like behavior of (sub)atomic particles could be the critical characteristic of a global optimization method deployed by nature so that (sub)atomic systems can find their ground states corresponding to the global minimum of some energy function associated with the system. The classic time-independent Schrodinger equation is shown to be derivable from the global optimization method to support this argument.

arXiv:quant-ph/0605220May 25, 20060 citations

Schrodinger Equation As a General Optimization Algorithm

Xiaofei Huang

One of the greatest scientific achievements of physics in the 20th century is the discovery of quantum mechanics. The Schrodinger equation is the most fundamental equation in quantum mechanics describing the time-based evolution of the quantum state of a physical system. It has been found that the time-independent version of the equation can be derived from a general optimization algorithm. Instead of arguing for a new interpretation and possible deeper principle for quantum mechanics, this paper elaborates a few points of the equation as a general global optimization algorithm. Benchmarked against randomly generated hard optimization problems, this paper shows that the algorithm significantly outperformed a classic local optimization algorithm. The former found a solution in one second with a single trial better than the best one found by the latter around one hour after one hundred thousand trials.One of the greatest scientific achievements of physics in the 20th century is the discovery of quantum mechanics. The Schrodinger equation is the most fundamental equation in quantum mechanics describing the time-based evolution of the quantum state of a physical system. It has been found that the time-independent version of the equation can be derived from a general optimization algorithm. Instead of arguing for a new interpretation and possible deeper principle for quantum mechanics, this paper elaborates a few points of the equation as a general global optimization algorithm. Benchmarked against randomly generated hard optimization problems, this paper shows that the algorithm significantly outperformed a classic local optimization algorithm. The former found a solution in one second with a single trial better than the best one found by the latter around one hour after one hundred thousand trials.

arXiv:0902.1590Feb 10, 20090 citations

Splitting Analysis for Yukawa Potential

Di Fang, Jiaqi Zhang

Splitting methods are among the most classical and fundamental tools for the simulation of quantum dynamics, and their importance has grown further with the rise of quantum computing. In this work, we analyze the Schrödinger equation with Yukawa potential, a physically relevant and widely used model potential. It may be viewed as a Coulomb interaction with exponential decay at spatial infinity, preserving the Coulomb singularity at the origin while removing the long-range Coulomb tail. We prove that the operator splitting for this unbounded Hamiltonian achieves a global $1/4$ -order convergence rate in the time step for many-body Yukawa interactions, with explicit polynomial dependence on the number of particles. The result holds for all initial wavefunctions in $H^2(\mathbb R^{3N})$ , the natural domain of the Hamiltonian, and our numerical experiments are consistent with the theoretical estimates. To identify the sharp obstruction behind this rate, we prove a short-time lower bound in the one-body setting of order $t^{5/4}$ for the one-step error, which rules out any uniform global estimate of order better than $1/4$ in general. This agreement with the optimal $1/4$ rate in the Coulomb case is particularly interesting, as Yukawa potential is short-ranged compared to Coulomb potential. For the many-body upper bound, one of the new technical ingredients is the explicit polynomial-in-system-size Sobolev estimates of many-body Yukawa systems. These estimates are crucial for obtaining fully a priori bounds that depend only on the norms of the initial states, rather than on the solution at time $t$ . For the one-body lower bound, we leverage a new analysis argument based on Fourier analysis and Kato smoothing.Splitting methods are among the most classical and fundamental tools for the simulation of quantum dynamics, and their importance has grown further with the rise of quantum computing. In this work, we analyze the Schrödinger equation with Yukawa potential, a physically relevant and widely used model potential. It may be viewed as a Coulomb interaction with exponential decay at spatial infinity, preserving the Coulomb singularity at the origin while removing the long-range Coulomb tail. We prove that the operator splitting for this unbounded Hamiltonian achieves a global $1/4$-order convergence rate in the time step for many-body Yukawa interactions, with explicit polynomial dependence on the number of particles. The result holds for all initial wavefunctions in $H^2(\mathbb R^{3N})$, the natural domain of the Hamiltonian, and our numerical experiments are consistent with the theoretical estimates. To identify the sharp obstruction behind this rate, we prove a short-time lower bound in the one-body setting of order $t^{5/4}$ for the one-step error, which rules out any uniform global estimate of order better than $1/4$ in general. This agreement with the optimal $1/4$ rate in the Coulomb case is particularly interesting, as Yukawa potential is short-ranged compared to Coulomb potential. For the many-body upper bound, one of the new technical ingredients is the explicit polynomial-in-system-size Sobolev estimates of many-body Yukawa systems. These estimates are crucial for obtaining fully a priori bounds that depend only on the norms of the initial states, rather than on the solution at time $t$. For the one-body lower bound, we leverage a new analysis argument based on Fourier analysis and Kato smoothing.

math.NACross-listedarXiv:2607.12153Jul 15, 2026

Chapter 9: Optimization with the adiabatic theorem

Giacomo Nannicini

The adiabatic theorem is a powerful result concerning the evolution of quantum mechanical systems. Intuitively, it states the following: suppose we want to study the evolution of a quantum mechanical system governed by a given Hamiltonian $\mathcal H$ (Def. 6.1), and the initial state of the system is the eigenstate of $\mathcal H$ corresponding to its lowest eigenvalue. The laws of quantum physics tell us that the evolution of the system over time follows the Schrüdinger equation, Eq. (6.1). Suppose now that the Hamiltonian is not fixed, but rather, it changes very slowly over time, so that the evolution of the system is described by the differential equation $i\,\frac{\mathrm{d} | \psi(t) \rangle}{\mathrm{d}t} = \mathcal{H}(t) | \psi(t) \rangle$ , and $\mathcal{H}(t)$ is slowly changing. If, for any value of the time parameter $t$ , there is always a gap between the lowest eigenvalue of $\mathcal{H}(t)$ and all other eigenvalues, then the adiabatic theorem asserts that the state of the system always remains in an eigenstate with the lowest eigenvalue throughout the evolution. One specific case of interest in this chapter occurs when the time-dependent Hamiltonian $\mathcal{H}(t)$ varies from an initial (time-independent) Hamiltonian $\mathcal{H}_{\mathrm{I}}$ to a final Hamiltonian $\mathcal{H}_{\mathrm{F}}$ . If the change happens slowly, and the initial state of the system is an eigenstate of $\mathcal{H}_{\mathrm{I}}$ with minimum eigenvalue, the adiabatic theorem asserts that the evolution eventually leads to the eigenstate with minimum eigenvalue of the final Hamiltonian $\mathcal{H}_{\mathrm{F}}$ . This result can be used for optimization: imagine that the initial Hamiltonian encodes an easy problem for which we can construct the eigenstate with minimum eigenvalue, and the final Hamiltonian encodes a difficult optimization problem whose solution is given by the eigenstate with minimum eigenvalue. By the adiabatic theorem, evolving the eigenstate of the initial Hamiltonian while slowly changing the Hamiltonian into the final one leads to the solution of the difficult optimization problem. Of course we need to know how slowly $\mathcal{H}_{\mathrm{I}}$ should be changed into $\mathcal{H}_{\mathrm{F}}$ . The adiabatic theorem is at the heart of the quantum approximate optimization algorithm (QAOA), a framework that has been widely used to implement optimization algorithms on existing quantum hardware, because it has low requirements of quantum resources–although it does not guarantee improvements over classical algorithms in general. In this chapter we first discuss the adiabatic theorem, and then give an overview of QAOA.The adiabatic theorem is a powerful result concerning the evolution of quantum mechanical systems. Intuitively, it states the following: suppose we want to study the evolution of a quantum mechanical system governed by a given Hamiltonian $\mathcal H$ (Def. 6.1), and the initial state of the system is the eigenstate of $\mathcal H$ corresponding to its lowest eigenvalue. The laws of quantum physics tell us that the evolution of the system over time follows the Schrüdinger equation, Eq. (6.1). Suppose now that the Hamiltonian is not fixed, but rather, it changes very slowly over time, so that the evolution of the system is described by the differential equation $i\,\frac{\mathrm{d} | \psi(t) \rangle}{\mathrm{d}t} = \mathcal{H}(t) | \psi(t) \rangle$, and $\mathcal{H}(t)$ is slowly changing. If, for any value of the time parameter $t$, there is always a gap between the lowest eigenvalue of $\mathcal{H}(t)$ and all other eigenvalues, then the adiabatic theorem asserts that the state of the system always remains in an eigenstate with the lowest eigenvalue throughout the evolution. One specific case of interest in this chapter occurs when the time-dependent Hamiltonian $\mathcal{H}(t)$ varies from an initial (time-independent) Hamiltonian $\mathcal{H}_{\mathrm{I}}$ to a final Hamiltonian $\mathcal{H}_{\mathrm{F}}$. If the change happens slowly, and the initial state of the system is an eigenstate of $\mathcal{H}_{\mathrm{I}}$ with minimum eigenvalue, the adiabatic theorem asserts that the evolution eventually leads to the eigenstate with minimum eigenvalue of the final Hamiltonian $\mathcal{H}_{\mathrm{F}}$. This result can be used for optimization: imagine that the initial Hamiltonian encodes an easy problem for which we can construct the eigenstate with minimum eigenvalue, and the final Hamiltonian encodes a difficult optimization problem whose solution is given by the eigenstate with minimum eigenvalue. By the adiabatic theorem, evolving the eigenstate of the initial Hamiltonian while slowly changing the Hamiltonian into the final one leads to the solution of the difficult optimization problem. Of course we need to know how slowly $\mathcal{H}_{\mathrm{I}}$ should be changed into $\mathcal{H}_{\mathrm{F}}$. The adiabatic theorem is at the heart of the quantum approximate optimization algorithm (QAOA), a framework that has been widely used to implement optimization algorithms on existing quantum hardware, because it has low requirements of quantum resources–although it does not guarantee improvements over classical algorithms in general. In this chapter we first discuss the adiabatic theorem, and then give an overview of QAOA.

Society for Industrial and Applied Mathematics eBooksDec 17, 20250 citations

Nonlinear Phase Modification of the Schroedinger Equation

Waldemar Puszkarz

A nonlinear modification of the Schrödinger equation is proposed in which the Lagrangian density for the Schrödinger equation is extended by terms polynomial in $Δ^{m}\ln (Ψ^{*}/Ψ)$ multiplied by $Ψ^{*}Ψ$ . This introduces a homogeneous nonlinearity in a Galilean invariant manner through the phase $S$ rather than the amplitude $R$ of the wave function $Ψ=R\exp (iS)$ . From this general scheme we choose the simplest minimal model defined in some reasonable way. The model in question offers the simplest way to modify the Bohm formulation of quantum mechanics so as to allow a leading phase contribution to the quantum potential and a leading quantum contribution to the probability current removing asymmetries present in Bohm's original formulation. It preserves most of physically relevant properties of the Schrödinger equation including stationary states of quantum-mechanical systems. It can be thought of as the simplest model of nonlinear quantum mechanics of extended objects among other such models that also emerge within the general scheme proposed. The extensions of this model to $n$ particles and the question of separability of compound systems are studied. It is noted that there exists a weakly separable extension in addition to a strongly separable one. The place of the general modification scheme in a broader spectrum of nonlinear modifications of the Schrödinger equation is discussed. It is pointed out that the models it gives rise to have a unique definition of energy in that the field-theoretical energy functional coincides with the quantum-mechanical one. It is found that the Lagrangian for its simplest variant represents the Lagrangian for a restricted version of the Doebner-Goldin modification of this equation.A nonlinear modification of the Schrödinger equation is proposed in which the Lagrangian density for the Schrödinger equation is extended by terms polynomial in $Δ^{m}\ln (Ψ^{*}/Ψ)$ multiplied by $Ψ^{*}Ψ$. This introduces a homogeneous nonlinearity in a Galilean invariant manner through the phase $S$ rather than the amplitude $R$ of the wave function $Ψ=R\exp (iS)$. From this general scheme we choose the simplest minimal model defined in some reasonable way. The model in question offers the simplest way to modify the Bohm formulation of quantum mechanics so as to allow a leading phase contribution to the quantum potential and a leading quantum contribution to the probability current removing asymmetries present in Bohm's original formulation. It preserves most of physically relevant properties of the Schrödinger equation including stationary states of quantum-mechanical systems. It can be thought of as the simplest model of nonlinear quantum mechanics of extended objects among other such models that also emerge within the general scheme proposed. The extensions of this model to $n$ particles and the question of separability of compound systems are studied. It is noted that there exists a weakly separable extension in addition to a strongly separable one. The place of the general modification scheme in a broader spectrum of nonlinear modifications of the Schrödinger equation is discussed. It is pointed out that the models it gives rise to have a unique definition of energy in that the field-theoretical energy functional coincides with the quantum-mechanical one. It is found that the Lagrangian for its simplest variant represents the Lagrangian for a restricted version of the Doebner-Goldin modification of this equation.

arXiv:quant-ph/9710010Oct 2, 19971 citation

1/4

-order convergence rate in the time step for many-body Yukawa interactions, with explicit polynomial dependence on the number of particles. The result holds for all initial wavefunctions in

H^2(\mathbb R^{3N})

, the natural domain of the Hamiltonian, and our numerical experiments are consistent with the theoretical estimates. To identify the sharp obstruction behind this rate, we prove a short-time lower bound in the one-body setting of order

t^{5/4}

for the one-step error, which rules out any uniform global estimate of order better than

1/4

in general. This agreement with the optimal

1/4

rate in the Coulomb case is particularly interesting, as Yukawa potential is short-ranged compared to Coulomb potential. For the many-body upper bound, one of the new technical ingredients is the explicit polynomial-in-system-size Sobolev estimates of many-body Yukawa systems. These estimates are crucial for obtaining fully a priori bounds that depend only on the norms of the initial states, rather than on the solution at time

t

. For the one-body lower bound, we leverage a new analysis argument based on Fourier analysis and Kato smoothing.

\mathcal H

(Def. 6.1), and the initial state of the system is the eigenstate of

\mathcal H

corresponding to its lowest eigenvalue. The laws of quantum physics tell us that the evolution of the system over time follows the Schrüdinger equation, Eq. (6.1). Suppose now that the Hamiltonian is not fixed, but rather, it changes very slowly over time, so that the evolution of the system is described by the differential equation

i\,\frac{\mathrm{d} | \psi(t) \rangle}{\mathrm{d}t} = \mathcal{H}(t) | \psi(t) \rangle

, and

\mathcal{H}(t)

is slowly changing. If, for any value of the time parameter

t

, there is always a gap between the lowest eigenvalue of

\mathcal{H}(t)

and all other eigenvalues, then the adiabatic theorem asserts that the state of the system always remains in an eigenstate with the lowest eigenvalue throughout the evolution. One specific case of interest in this chapter occurs when the time-dependent Hamiltonian

\mathcal{H}(t)

varies from an initial (time-independent) Hamiltonian

\mathcal{H}_{\mathrm{I}}

to a final Hamiltonian

\mathcal{H}_{\mathrm{F}}

. If the change happens slowly, and the initial state of the system is an eigenstate of

\mathcal{H}_{\mathrm{I}}

with minimum eigenvalue, the adiabatic theorem asserts that the evolution eventually leads to the eigenstate with minimum eigenvalue of the final Hamiltonian

\mathcal{H}_{\mathrm{F}}

. This result can be used for optimization: imagine that the initial Hamiltonian encodes an easy problem for which we can construct the eigenstate with minimum eigenvalue, and the final Hamiltonian encodes a difficult optimization problem whose solution is given by the eigenstate with minimum eigenvalue. By the adiabatic theorem, evolving the eigenstate of the initial Hamiltonian while slowly changing the Hamiltonian into the final one leads to the solution of the difficult optimization problem. Of course we need to know how slowly

\mathcal{H}_{\mathrm{I}}

should be changed into

\mathcal{H}_{\mathrm{F}}

. The adiabatic theorem is at the heart of the quantum approximate optimization algorithm (QAOA), a framework that has been widely used to implement optimization algorithms on existing quantum hardware, because it has low requirements of quantum resources–although it does not guarantee improvements over classical algorithms in general. In this chapter we first discuss the adiabatic theorem, and then give an overview of QAOA.

A nonlinear modification of the Schrödinger equation is proposed in which the Lagrangian density for the Schrödinger equation is extended by terms polynomial in

Δ^{m}\ln (Ψ^{*}/Ψ)

multiplied by

Ψ^{*}Ψ

. This introduces a homogeneous nonlinearity in a Galilean invariant manner through the phase

S

rather than the amplitude

R

of the wave function

Ψ=R\exp (iS)

. From this general scheme we choose the simplest minimal model defined in some reasonable way. The model in question offers the simplest way to modify the Bohm formulation of quantum mechanics so as to allow a leading phase contribution to the quantum potential and a leading quantum contribution to the probability current removing asymmetries present in Bohm's original formulation. It preserves most of physically relevant properties of the Schrödinger equation including stationary states of quantum-mechanical systems. It can be thought of as the simplest model of nonlinear quantum mechanics of extended objects among other such models that also emerge within the general scheme proposed. The extensions of this model to

n

particles and the question of separability of compound systems are studied. It is noted that there exists a weakly separable extension in addition to a strongly separable one. The place of the general modification scheme in a broader spectrum of nonlinear modifications of the Schrödinger equation is discussed. It is pointed out that the models it gives rise to have a unique definition of energy in that the field-theoretical energy functional coincides with the quantum-mechanical one. It is found that the Lagrangian for its simplest variant represents the Lagrangian for a restricted version of the Doebner-Goldin modification of this equation.