In this paper the Levy-Leblond procedure for linearizing the Schrödinger equation to obtain the Pauli equation for one particle is generalized to obtain an
N-particle equation with spin. This is achieved by using the more universal matrix factorization,
GG~=∣G∣I=(−K)lI. Here the square matrix
G is linear in the total energy E and all momenta,
G~ is the matrix adjoint of
G,
I is the identity matrix,
∣G∣ is the determinant of
G,
l is a positive integer and
K=H−E is Lanczos' extended Hamiltonian where
H is the classical Hamiltonian of the electro-mechanical system.
K is identically zero for all such systems, so that matrix
G is singular. As a consequence there always exists a vector function
θ with the property
Gθ=0. This factorization to obtain the matrix
G and vector function
θ is illustrated first for a one-dimensional particle in a simple potential well. This same technique, when applied to the classical nonrelativistic Hamiltonian for
N interacting particles in an electromagnetic field, is shown to yield for N=1 the Pauli wave equation with spin and its generalization to
N particles. Finally this nonrelativistic generalization of the Pauli equation is used to treat the simple Zeeman effect of a hydrogen-like atom as a two-particle problem with spin.
The goal of Quantum Mechanics I, A Problem Text, is to enable students to solve problems appropriate to their first course in quantum mechanics. Significantly detailed solutions to traditionally posed problems dominate. Theory and rationale are addressed in prescript and postscript narratives and/or within the solved problems to guide students; and clarify vocabulary, symbology, and terminology. Applications of the postulates are presented initially using arguments from linear algebra appropriate to discrete systems with low-dimensional vectors and matrix operators, introducing orthogonality, orthonormality, Hermiticity, eigenvectors, eigenvalues, probability, expectation value, and degeneracy. These arguments blend into infinite-dimensional vectors and operators, and calculus-based arguments for continuous systems where delta functions, theta functions, and Fourier transforms are initially discussed. Dirac notation is fully developed within these first five chapters. Chapter six discusses Ehrenfest’s theorem, the Heisenberg uncertainty relations, and Gaussian wave functions and wave packets. Increasingly sophisticated potential energy functions are then treated to include the free particle, the infinite square well, the one-dimensional scattering state, the simple harmonic oscillator, the finite square well, orbital angular momentum, and the hydrogen atom. Ladder operators are featured in the developments of the simple harmonic oscillator and orbital angular momentum. The diversity of narratives and detailed solutions intend