Some Partial Differential Operators of Quantum Mechanics
In suitable units, the Hamiltonian H for a free particle is the kinetic energy -½∇2 in ℝ3. For a system of N identical particles, it is -½∇2 in ℝ n , with n = 3N. The Schrödinger Hamiltonian is obtained by adding a potential energy term. Self-adjoint versions of these operators are discussed in this chapter. In the case of just one particle (electron) in a Coulomb field (hydrogen-like atom), the relativistic (Dirac) Hamiltonian is also discussed.
KeywordsSchrödinger and Dirac Hamiltonian for a free particle and for hydrogen-like atoms Schrödinger Hamiltonian of n-electron atoms self-adjointness and properties of the spectra resolvent and resolution of the identity for the Laplacian relatively bounded perturbation of a self-adjoint operator essential spectrum absolutely continuous and singular continuous spectra continuous spectrum in the sense of Hilbert absolutely continuous and singular-continuous subspaces. Problems of the relativistic hydrogen atom for different values of Z self-adjointness and spectrum of the Laplacian in a bounded region of space
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