Principles of Advanced Mathematical Physics pp 222-240 | Cite as

# Some Partial Differential Operators of Quantum Mechanics

Chapter

## Abstract

In suitable units, the Hamiltonian *H* for a free particle is the kinetic energy -½∇^{2} in ℝ^{3}. For a system of *N* identical particles, it is -½∇^{2} in *ℝ*^{ n }, with *n* = 3*N*. 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.

## Keywords

Schrödinger 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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## Copyright information

© Springer-Verlag New York Inc. 1978