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The Hamiltonian describes the total energy of the system. Where i is the imaginary unit, t is time, is the partial derivative with respect to t, is the reduced Planck's constant (Planck's constant divided by 2π), ψ( t) is the wave function, and is the Hamiltonian (a self-adjoint operator acting on the state space). Using Dirac's bra-ket notation, the definition of energy results in the time derivative operator: at time t by. The Schrödinger equation provides a quantitative description of the rate of change of the state vector. As the state of a system generally changes over time, the state vector is a function of time. This state vector encodes the probabilities for the outcomes of all possible measurements applied to the system. In the mathematical formulation of quantum mechanics, each system is associated with a complex Hilbert space such that each instantaneous state of the system is described by a unit vector in that space. It is of central importance to the theory of quantum mechanics, playing a role analogous to Newton's second law in classical mechanics. In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1925, describes the space- and time-dependence of quantum mechanical systems. This singularity also obscures the properties of the system under parity transformations, since the behavior of V ( x ) = γ | x | on ℝ\ will necessarily be singular, and it will not be unique.2007 Schools Wikipedia Selection. In particular, the singularity at the origin makes it unclear what are the boundary conditions (b.c.) to be used there and, consequently, whether the potential is impenetrable or not at the singularity (see, e.g., ). This is due to the fact that the 1D Coulomb potential has a non-integrable singularity at the origin, rendering its mathematical treatment non-trivial. ĭespite its many applications and deceptive simplicity, the 1D hydrogen atom has been a source of great controversy in the literature.
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The 1D Coulomb potential finds applications in several fields, such as the quasi one-dimensional hydrogen atom in astrophysical systems with very strong magnetic fields, in quantum wires and carbon nanotubes (see, e.g., and references therein), etc.-for a recent review of the applications of the 1D Coulomb interaction see Loudon. Finally, we present a systematic study of the bound states within this subfamily, addressing the frequently debated issues of the multiplicity and parity of the bound states, and the boundedness of the ground state energy.Īnd it has been a source of considerable interest since the paper by Loudon, who investigated this problem due to its relevance for the physics of excitons in strong magnetic fields. By requiring that the interaction distribution be invariant under parity, we further restrict the 4-parameter family of interactions to the subfamily of all the parity invariant Coulomb interactions. We replace this ill-defined product by a well-defined interaction distribution, S( x), and by imposing that it should satisfy some fundamental mathematical and physical requirements, we show that this distribution is defined up to a 4-parameter family of contact interactions, in agreement with the method of self-adjoint extensions. This singularity renders the interaction term V( x)ψ( x) in the Schrödinger's equation, where ψ( x) is the wave function, an ill-defined product in the ordinary sense. We consider the one-dimensional Hydrogen atom, with the Coulomb interaction V ( x ) = γ | x | (γ < 0), and use Schwartz's theory of distributions to address the non-integrable singularity at the origin.