It is often assumed that bound states of quantum mechanical systems are intrinsically non-perturbative in nature and therefore any power series expansion methods should be inapplicable to predict the energies for attractive potentials. However, if the spatial domain of the Schrödinger Hamiltonian for attractive one-dimensional potentials is confined to a finite length L, the usual Rayleigh-Schrödinger perturbation theory can converge rapidly and is perfectly accurate in the weak-binding region where the ground state's spatial extension is larger than L. Once the binding strength is so strong that the ground state's extension is less than L, the power expansion becomes divergent, consistent with the expectation that bound states are non-perturbative. However, we propose a new truncated Borel-like summation technique that can recover the correct bound state energy from the diverging sum. We also show that perturbation theory becomes divergent in the vicinity of an avoided-level crossing. Here the same numerical summation technique can be applied to reproduce the correct finite energies from the diverging perturbative sums.

We propose a non-perturbative approach to calculate bound state energies and wave functions for quantum field theoretical models. It is based on the direct diagonalization of the corresponding quantum field theoretical Hamiltonian in an effectively truncated and discretized Hilbert space. We illustrate this approach for a Yukawa-like interaction between fermions and bosons in one spatial dimension and show where it agrees with the traditional method based on the potential picture and where it deviates due to recoil and radiative corrections. This method permits us also to obtain some first insight into the spatial characteristics of the distribution of the fermions in the ground state, such as the bremsstrahlung-induced widening. † (x) γ 0 Ψ d (x)] φ(x), where the parameter λ is the coupling strength, Ψ b and Ψ d are the two-component Dirac field operators for the fermions and φ denotes the scalar boson operator. For the special case of m=0 this model could also be used to study simplified QED interactions, where the "photon" has spin zero. The three field operators can be expanded in terms of annihilation and creation operators that fulfill the usual anti-commutator and commutator relationships [b(p), b † (p')] + = [d(p), d † (p')] + = [a(p), a † (p')]-= 3 8/15/2016 δ(p-p'). For couplings λ that are not exceedingly large, fermionic pair-creation processes are not important. The terms in the Hamiltonian that would couple the first fermion to its own antiparticle are proportional to b † (p+k) B † (p) [a † (-2p-k)+a(2p+k)] and b(p+k) B(p)[a † (2p+k)+a(-2p-k)]. Here the anti-particle operators B and B † fulfill the anticommutator relationships [b(p), B † (p')] + = 0 and [B(p), B † (p')] + = δ(p-p'). Similar terms characterize also the second fermion. As very energetic bosons are required in these interactions and the corresponding coupling function decreases rapidly with the boson momentum, we therefore neglect anti-fermions. This leads to the Hamiltonian of

Crowded environments modify the diffusion of macromolecules, generally slowing their movement and inducing transient anomalous subdiffusion. The presence of obstacles also modifies the kinetics and equilibrium behavior of tracers. While previous theoretical studies of particle diffusion have typically assumed either impenetrable obstacles or binding interactions that immobilize the particle, in many cellular contexts bound particles remain mobile. Examples include membrane proteins or lipids with some entry and diffusion within lipid domains and proteins that can enter into membraneless organelles or compartments such as the nucleolus. Using a lattice model, we studied the diffusive movement of tracer particles which bind to soft obstacles, allowing tracers and obstacles to occupy the same lattice site. For sticky obstacles, bound tracer particles are immobile, while for slippery obstacles, bound tracers can hop without penalty to adjacent obstacles. In both models, binding significantly alters tracer motion. The type and degree of motion while bound is a key determinant of the tracer mobility: slippery obstacles can allow nearly unhindered diffusion, even at high obstacle filling fraction. To mimic compartmentalization in a cell, we examined how obstacle size and a range of bound diffusion coefficients affect tracer dynamics. The behavior of the model is similar in two and three spatial dimensions. Our work has implications for protein movement and interactions within cells.

We solve the Maxwell-Dirac equations to study the dynamics of a spatially localized charged particle in one spatial dimension. While the coupling of the Maxwell equations to the Dirac equation predicts correctly the attractive or repulsive interaction between different particles, it also reveals an unphysical interaction of a single electron or positron with itself leading to an enhanced spatial spreading of the wave packet. Using a comparison with a relativistic ensemble of mutually interacting classical quasiparticles, we suggest that this quantum mechanical self-repulsion can be understood in terms of relativistic classical mechanics. We show that due to the simple form of the Coulomb law in one spatial dimension it is possible to find analytical expressions of the time-dependent spatial width for the interacting classical ensemble. A better understanding of the dynamical impact of this unavoidable self-repulsion effect is relevant for recent studies of the field-induced pair creation process from the vacuum.

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