For almost 75 years, the general solution for the Schrödinger equation was assumed to be generated by an exponential or a time-ordered exponential known as the Dyson series. We study the unitarity of a solution in the case of a singular Hamiltonian and provide a new methodology that is not based on the assumption that the underlying space is L2(R). Then, an alternative operator for generating the time evolution that is manifestly unitary is suggested, regardless of the choice of Hamiltonian. The new construction involves an additional positive operator that normalizes the wave function locally and allows us to preserve unitarity, even when dealing with infinite dimensional or non-normed spaces. Our considerations show that Schrödinger and Liouville equations are, in fact, two sides of the same coin and together they provide a unified description for unbounded quantum systems.
For almost 75 years, the general solution for the Schrödinger equation was assumed to be generated by an exponential or a time-ordered exponential known as the Dyson series. We study the unitarity of a solution in the case of a singular Hamiltonian and provide a new methodology that is not based on the assumption that the underlying space is L2(R). Then, an alternative operator for generating the time evolution that is manifestly unitary is suggested, regardless of the choice of Hamiltonian. The new construction involves an additional positive operator that normalizes the wave function locally and allows us to preserve unitarity, even when dealing with infinite dimensional or non-normed spaces. Our considerations show that Schrödinger and Liouville equations are, in fact, two sides of the same coin and together they provide a unified description for unbounded quantum systems.
In this chapter, we finally introduce time evolution. The development is carried out for finite, enumerable systems (i.e., systems whose states live in a finite dimensional, inner product space) which, of course, allows a straight forward extension to the continuum and infinite dimensional state spaces. The necessary mathematical background of unitary operators, and their properties relevant to quantum mechanics are first introduced. In particular, the fact that preservation of norm under the action of an operator implies unitarity, and that every unitary operator can be expressed as the exponential of i timesa Hermitian operator are introduced. Time evolution is then introduced in its integral form through the time evolution operator, and the Hamiltonian operator is introduced in this context. This approach, however, necessitates the restriction to conservative systems. Next, the differential form of the law of time evolution, i.e., Schrodinger equation, is derived from the time evolution operator, and this is postulated to be the general law of time evolution. We then demonstrate how, for conservative systems, the solution of the Schrodinger equation essentially reduces to solving the eigenvalue problem of the Hamiltonian. Constants of motion and Heisenberg picture have been relegated to the problem section.
In this paper, the Dyson series corresponding to the time-varying Hamiltonian of a finite dimensional quantum mechanical system is expanded in terms of products of exponentials of a complete basis of commutator superoperators in the corresponding Liouville space. The Cayley-Hamilton theorem and the Wei-Norman formula allow to express explicitly the functional relation between the Dyson series and the product of exponentials via a set of first order differential equations. Since the method is structure preserving, it can be used for the exact unitary integration of the driven Liouville-von Neumann equation.
This paper derives the non-relativistic time-dependent Schrödinger equation as the logically unavoidable dynamical law for de Broglie matter waves, extending the reconstruction begun in the companion work “The Logical Origin of the de Broglie Matter Wave and its Physical Meaning (the Born Rule)” (Chan, 2026). Starting from the same minimal premises—continuous position in real space, Fourier conjugacy (and the logical uncertainty principle), translation invariance via Noether’s theorem, and probability conservation via statistical energy flux—the Schrödinger equation follows deductively once norm preservation and classical non-relativistic correspondence are enforced. The derivation proceeds in clear stages: Plane-wave solutions for free particles and constant potentials, Quadratic dispersion relation from classical kinematics, Exact local plane-wave structure (defined via phase gradients) that must hold at every point for any smooth wave function, The unique linear differential operator of order ≤ 2 that consistently “glues” these local structures across smooth external potentials V(x,t) while preserving the L²-norm (unitarity). No canonical quantization, operator promotion, commutation relations, or postulated wave equations are used. The Born rule is taken as already established from the prior paper and serves only to constrain unitary evolution. The Schrödinger equation thus emerges not as an empirical postulate or convenient ansatz, but as the mathematically and symmetrically forced description of indivisible particles propagating in continuous, translationally invariant (or smoothly perturbed) space. Domain of validity and natural limitations (non-relativistic kinematics, smooth potentials, single-particle/mean-field regime, closed-system unitarity) are explicitly delineated.
Starting from our idea of combining the Feynman path integral spirit and the Dyson series kernel, we find an explicit and general form of time evolution operator that is a c-number function and a power series of perturbation including all order approximations in the unperturbed Hamiltonian representation. Based on it, we obtain an exact solution of the Schrödinger equation in general quantum systems independent of time. Comparison of our exact solution with the existed perturbation theory makes some features and significance of our exact solution clear. The conclusions expressly indicate that our exact solution is obviously consistent with the usual time-independent perturbation theory at any order approximation, it explicitly calculates out the expanding coefficients of the unperturbed state in the non-perturbation method, and it fully solves the recurrence equation of the expansion coefficients of final state in the unperturbed Hamiltonian representation from a view of time-dependent perturbation theory. At the same time, the exact solution of the von Neumann equation is also given. Our results can be thought of as theoretical developments of quantum dynamics, and are helpful for understanding the dynamical behavior and related subjects of general quantum systems in both theory and application. Our exact solution, together with its sequence studies on perturbation theory [An Min Wang, quant-ph/0611217] and open system dynamics [An Min Wang, quant-ph/0601051] can be used to establish the foundation of theoretical formulism of quantum mechanics in general quantum systems. Further applications of our exact solution to quantum theory can be expected.
The Magnus expansion offers a powerful geometric method to express a time-ordered exponential operator solution, typical of time-dependent linear differential equations, as an ordinary operatorial exponential. This representation possesses advantageous theoretical and qualitative features, such as preserving the underlying Lie algebraic structure of the system while remaining an exact solution to the original differential equation. For any finite-dimensional Hermitian Hamiltonian governing a quantum system, the standard Magnus expansion guarantees a manifestly unitary representation at every finite order of truncation. However, this foundational property is no longer preserved if the underlying Hamiltonian is non-Hermitian. In this work, we derive a generalized, unitarized expansion framework that systematically restores and maintains the property of geometric unitarity for all bounded finite-dimensional non-Hermitian Hamiltonians.
Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.