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arXiv:1812.03056 · Journal of Physics A Mathematical and Theoretical

Merits of using density matrices instead of wave functions in the stationary Schrödinger equation for systems with symmetries

E. V. Shpagina, Filipp Gennadevich Uskov, N. Il’in, Oleg Valentinovich Lychkovskiy

Published Dec 20, 2019• 3 citations
arXivHTMLPDF

Abstract

Abstract The stationary Schrödinger equation can be cast in the form , where H is the system’s Hamiltonian and is the system’s density matrix. We explore the merits of this form of the stationary Schrödinger equation, which we refer to as SSE , applied to many-body systems with symmetries. For a nondegenerate energy level, the solution of the SSE is merely a projection on the corresponding eigenvector. However, in the case of degeneracy is non-unique and not necessarily pure. In fact, it can be an arbitrary mixture of the degenerate pure eigenstates. Importantly, can always be chosen to respect all symmetries of the Hamiltonian, even if each pure eigenstate in the corresponding degenerate multiplet spontaneously breaks the symmetries. This and other features of the solutions of the SSE can prove helpful by easing the notations and providing an unobscured insight into the structure of the eigenstates. We work out the SSE for a general system of spins 1/2 with Heisenberg interactions, and address simple systems of spins 1. Eigenvalue problem for quantum observables other than Hamiltonian can also be formulated in terms of density matrices. As an illustration, we provide an analytical solution to the eigenproblem , where is the total spin of N spins 1/2, and is chosen to be invariant under permutations of spins. This way we find an explicit form of projections to the invariant subspaces of .

DOI
10.1088/1751-8121/ab64a1

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Abstract The stationary Schrödinger equation can be cast in the form , where H is the system’s Hamiltonian and is the system’s density matrix. We explore the merits of this form of the stationary Schrödinger equation, which we refer to as SSE , applied to many-body systems with symmetries. For a nondegenerate energy level, the solution of the SSE is merely a projection on the corresponding eigenvector. However, in the case of degeneracy is non-unique and not necessarily pure. In fact, it can be an arbitrary mixture of the degenerate pure eigenstates. Importantly, can always be chosen to respect all symmetries of the Hamiltonian, even if each pure eigenstate in the corresponding degenerate multiplet spontaneously breaks the symmetries. This and other features of the solutions of the SSE can prove helpful by easing the notations and providing an unobscured insight into the structure of the eigenstates. We work out the SSE for a general system of spins 1/2 with Heisenberg interactions, and address simple systems of spins 1. Eigenvalue problem for quantum observables other than Hamiltonian can also be formulated in terms of density matrices. As an illustration, we provide an analytical solution to the eigenproblem , where is the total spin of N spins 1/2, and is chosen to be invariant under permutations of spins. This way we find an explicit form of projections to the invariant subspaces of .
For systems which obey quantum mechanics, the formulation of the problem of treating large numbers of particles is, of course, somewhat different than it is for classical systems. The microscopic description of the system is provided by a wave function which (in the absence of spin) is a function of the classical coordinates { q i }. The mathematical model is provided by a Hamiltonian operator H which is often obtained from the corresponding classical Hamiltonian by the replacement p i → ( ħ /i)(∂/∂ q i ). In other cases the form of the Hamiltonian operator is simply postulated. The microscopic dynamics are provided by the Schrödinger equation i ħ (∂Ψ/∂ t ) = H Ψ which requires as initial condition the knowledge of the wave function Ψ({ q i }, t ) at some initial time t 0 . (Boundary conditions on Ψ({ q i }, t ) must be specified as part of the description of the model as well.) The results of experiments in quantum mechanics are characterized by operators, usually obtained, like the Hamiltonian, from their classical forms and termed observables. Operators associated with observables must be Hermitian. In general, the various operators corresponding to observables do not commute with one another. It is possible to find sets of commuting operators whose mutual eigenstates span the Hilbert space in which the wave function is confined by the Schrödinger equation and the boundary conditions. A set of such (time independent) eigenstates, termed ψ ν ( q ), is a basis for the Hilbert space.
The main differential equations of quantum theory are the eigenequations based on the energy operator; they have the energy as eigenvalues and the wave functions as eigenfunctions. A usual complexity of these equations makes their accurate solutions accessible easily only for very few physical cases. One of the methods giving the approximate solutions is the Schrödinger perturbation theory in which both the energies and wave functions of a more complicated eigenproblem are approached with the aid of similar parameters characteristic for a less complicated eigenproblem. No time parameter is necessary to be involved in these calculations. The present paper shows that the Schrödinger perturbation method for non-degenerate stationary quantum states, i.e. the states being independent of time, can be substantially simplified by applying a circular scale of time separately for each order of the perturbation theory. The arrangement of the time points on the scale, combined with the points contractions, gives almost immediately the series of terms necessary to express the stationary perturbation energy of a given eigenproblem. The Schrödinger’s method is compared with the Born-Heisenberg-Jordan perturbation approach.
The proof of Smith, which gives the solution to the N-representability problem of the first-order density matrix for the case of N even and a nondegenerate energy eigenstate, is also shown to be valid for certain linear combinations of degenerate states. A connection is shown between these proper linear combinations and the eigenfuctions of a perturbed Hamiltonian for which all of the degeneracies have been lifted. For the case of S=0, all the eigenfunctions of the pth-order density matrix (natural p functions) can be taken to be eigenfunctions of total spin (Σ lim i=1pSi)2,and if orbital angular momentum is a constant of the motion, these natural p functions are eigenfunctions of the total orbital angular momentum for L=0. Simple wavefunctions are presented for which the natural 2 functions are not all eigenfunctions of the total pair spin. It is then shown that for MS=0 (any integral spin) or for L=1, ML=0, the natural 2 functions are eigenfunctions of (S1+S2)2 and (L1+L2)2, respectively.
An algebraic procedure to find extremal density matrices for the expectation value of a finite Hamiltonian matrix is established. The extremal density matrices for pure states provide a complete description of the system, that is, its corresponding energy spectrum and projectors. For density matrices representing mixed states, one gets the most probable eigenstates that yield extremal mean values of the energy. The procedure uses mainly the stationary solutions of the von Neumann equation of motion, the orbits of the Hamiltonian, and the positivity conditions of the density matrix. The method is illustrated for matrix Hamiltonians of dimensions d = 2 and d = 3.
This paper presents a comprehensive reformulation and extension of the Scretching–Schrödinger Equation (SSE) across three complementary quantum-mechanical formalisms. First, the SSE is developed in its wave-mechanical operator form by connecting the spectroscopically constrained Hamiltonian structure to the Tao–Hamilton operator framework and the Zhang–Svidzinsky probability-amplitude derivation of the Schrödinger equation. Second, the SSE is recast in Heisenberg matrix-mechanical form, where its constraints appear as operator and matrix equations governed by explicit spectroscopic closure conditions. Third, rigorous relativistic extensions are constructed through the Klein–Gordon–SSE and Dirac–SSE equations, including spinor structure, gauge covariance, and compatibility with standard relativistic quantum mechanics. The paper further integrates hyperfine-structure, Zeeman-effect, and Lamb-shift literature; the Andrews pure-wave-system derivation; and Kocher et al.’s nonlinear Schrödinger numerical methods to show that the SSE closure operator, V^SQC\hat{V}_{\mathrm{SQC}}V^SQC, can be expressed within every major formalism of quantum theory. New derived results include the SSE Heisenberg equation of motion, SSE density matrix, SSE path-integral action, SSE propagator, and a dimensionally correct SI optical susceptibility equation. All numerical results are triple-verified using CODATA 2022 constants and benchmarked against NIST reference standards where applicable. Hydrogen is used throughout as the zero-correction benchmark, with V^SQC(H)≈0\hat{V}_{\mathrm{SQC}}(\mathrm{H}) \approx 0V^SQC(H)≈0 confirmed by the Lyman-α\alphaα dipole comparison, giving δμ=0.027%\delta_{\mu} = 0.027\%δμ=0.027%. This establishes hydrogen as the reference case in which the ordinary Schrödinger, Heisenberg, Klein–Gordon, and Dirac descriptions remain fully consistent with the SSE spectroscopic closure, while also providing the foundation for applying SSE corrections to more complex atoms, molecules, and quantum systems.
Solving for the lowest energy eigenstate of the many-body Schrodinger equation is a cornerstone problem that hinders understanding of a variety of quantum phenomena. The difficulty arises from the exponential nature of the Hilbert space which casts the governing equations as an eigenvalue problem of exponentially large, structured matrices. Variational methods approach this problem by searching for the best approximation within a lower-dimensional variational manifold. In this work we use graph neural networks to define a structured variational manifold and optimize its parameters to find high quality approximations of the lowest energy solutions on a diverse set of Heisenberg Hamiltonians. Using graph networks we learn distributed representations that by construction respect underlying physical symmetries of the problem and generalize to problems of larger size. Our approach achieves state-of-the-art results on a set of quantum many-body benchmark problems and works well on problems whose solutions are not positive-definite. The discussed techniques hold promise of being a useful tool for studying quantum many-body systems and providing insights into optimization and implicit modeling of exponentially-sized objects.