The (group and spin space) matrix Hamiltonian describing the dynamics of a non-relativistic spin-½ particle moving in a static, but spatially dependent, non-Abelian magnetic field in two spatial dimensions is shown to take the form of an anticommutator of a nilpotent operator and its Hermitian conjugate. Consequently, the (group space) matrix Hamiltonians for the two different spin projections form partners of a supersymmetric quantum mechanical system. The resulting supersymmetry algebra is exploited to explicitly construct the exact zero energy ground state wave function(s) for the system. The remaining eigenstates and eigenvalues of the two partner Hamiltonians form positive energy degenerate pairs.
The (group and spin space) matrix Hamiltonian describing the dynamics of a non-relativistic spin-½ particle moving in a static, but spatially dependent, non-Abelian magnetic field in two spatial dimensions is shown to take the form of an anticommutator of a nilpotent operator and its Hermitian conjugate. Consequently, the (group space) matrix Hamiltonians for the two different spin projections form partners of a supersymmetric quantum mechanical system. The resulting supersymmetry algebra is exploited to explicitly construct the exact zero energy ground state wave function(s) for the system. The remaining eigenstates and eigenvalues of the two partner Hamiltonians form positive energy degenerate pairs.
The (group and spin space) matrix Hamiltonian describing the dynamics of a nonrelativistic spin 1/2 particle moving in a static, but spatially dependent, non-Abelian magnetic field in two spatial dimensions is shown to take the form of an anticommutator of a nilpotent operator and its hermitian conjugate. Consequently, the (group space) matrix Hamiltonians for the two different spin projections form partners of a supersymmetric quantum mechanical system. The resulting supersymmetry algebra is exploited to explicitly construct the exact zero energy ground state wavefunction(s) for the system. The remaining eigenstates and eigenvalues of the two partner Hamiltonians form positive energy degenerate pairs.
Using supersymmetry quantum mechanics, we explain the covariance of the eigenvalue equation for the Jaynes-Cummings (JC) Hamiltonian under the particle-hole transformation. The JC Hamiltonians for the original spin-1/2 ``particle'' and the transformed partner (the ``hole'') are shown to be the fermionic and the bosonic parts of a supersymmetric Hamiltonian. The Hamiltonian, unlike ordinary one-dimensional supersymmetric quantum-mechanical models, can have a degenerate zero-energy ground state despite exact supersymmetry. The degeneracy is shown to arise from the hidden particle-hole exchange symmetry as well as supersymmetry in the Hamiltonian.
We find that in a supersymmetric quantum mechanics (SUSY QM) system, in addition to supersymmetric algebra, an associated SU(2) algebra can be obtained by using semiunitary (SUT) operator and projection operator, and the relevant constants of motion can be constructed. Two typical quantum systems are investigated as examples to demonstrate the above finding. The first example is the quantum system of a nonrelativistic charged particle moving in x-y plane and coupled to a magnetic field along z-axis. The second example is provided with the Dirac particle in a magnetic field. Similarly there exists an SU τ (2) ⊗ SU σ (2) symmetry in the context of the relativistic Pauli Hamiltonian squared. We show that there exists also an SU(2) symmetry associated with the supersymmetry of the Dirac particle.
It is shown that the eigenproblem of any 2×2 matrix Hamiltonian with discrete eigenvalues is involved with a supersymmetric quantum mechanics. The energy dependence of the superalgebra marks the disparity between the deduced supersymmetry and the standard supersymmetric quantum mechanics. The components of an eigenspinor are superpartners\textemdash up to a SU(2) transformation\textemdash which allows to derive two reduced eigenproblems diagonalizing the Hamiltonian in the spin subspace. As a result, each component carries all information encoded in the eigenspinor. We also discuss the generalization of the formalism to a system of a single spin-2p coupled with external fields. The unconventional supersymmetry can be regarded as an extension of the Fulton-Gouterman transformation, which can be established for a two-level system coupled with multi oscillators displaying a mirror symmetry. The transformation is exploited recently to solve Rabi-type models. Correspondingly, we illustrate how the supersymmetric formalism can solve spin-boson models with no need to appeal a symmetry of the model. Furthermore, a pattern of entanglement between the components of an eigenstate of a many-spin system can be unveiled by exploiting the supersymmetric quantum mechanics associated with single spins which also recasts the eigenstate as a matrix product state. Examples of many-spin models are presented and solved by utilizing the formalism.
We study an effective Hamiltonian governing the relative motion of two charged particles confined to a two-dimensional plane under a uniform magnetic field, with interactions described by a generalized central potential that combines harmonic, Coulombic, linear, and inverse-square terms. After formal separation of the center-of-mass coordinates and a unitary transformation to a standard one-dimensional radial Hilbert space, the resulting equation is solved exactly through a non-Hermitian supersymmetric factorization: although the factorization operators are not mutually adjoint, they yield Hermitian partner Hamiltonians that exhibit shape invariance with respect to a weighted inner product, enabling the derivation of closed-form analytical expressions for the energy spectrum and radial wave functions. The constraint relation among potential parameters that emerges from this construction is not a physical law but a mathematical solvability condition–an essential feature of a deliberately idealized model whose purpose is to reveal novel algebraic structures underlying exactly solvable quantum systems. We explicitly discuss the formal nature of the model, the parameter regimes ensuring potential boundedness, and the mathematical framework required for self-adjointness. Мы изучаем эффективный гамильтониан, описывающий относительное движение двух заряженных частиц, ограниченных двумерной плоскостью в однородном магнитном поле, причем взаимодействия описываются обобщенным центральным потенциалом, объединяющим гармонические, кулоновские, линейные и обратноквадратичные члены. После формального разделения координат центра масс и унитарного преобразования в стандартное одномерное радиальное гильбертово пространство полученное уравнение решается точно с помощью неэрмитовой суперсимметричной факторизации: хотя операторы факторизации не являются взаимно сопряженными, они дают эрмитовы гамильтонианы-партнеры, которые демонстрируют инвариантность формы относительно взвешенного скалярного произведения, что позволяет вывести аналитические выражения в замкнутой форме для энергетического спектра и радиальных волновых функций. Связь между потенциальными параметрами, возникающая из этой конструкции, является не физическим законом, а математическим условием разрешимости — существенной особенностью намеренно идеализированной модели, цель которой — выявить новые алгебраические структуры, лежащие в основе точно решаемых квантовых систем. Мы подробно обсуждаем формальную природу модели, режимы параметров, обеспечивающие потенциальную ограниченность, и математическую основу, необходимую для самосопряженности.
It is shown that for a given Hermitian Hamiltonian possessing supersymmetry, there is alwayas a non-hermitian Jaynes-Cummings-type Hamiltonian(JCTH) admitting entirely real spectra. The parent supersymmetric Hamiltonian and the corresponding non-hermitian JCTH are simultaneously diagonalizable. The exact eigenstates of these non-hermitian Hamiltonians are constructed algebraically for certain shape-invariant potentials, including a non-hermitian version of the standard Jaynes-Cummings model for which the parent supersymmetric Hamiltonian is the superoscillator. The positive-definite metric operator in the Hilbert space is constructed explicitly along with the introduction of a new inner product structure, so that the eigenstates form a complete set of orthonormal vectors and the time-evolution is unitary.