S.M. Blinder

Mathematical Methods in Quantum Mechanics

The mathematical methods used in quantum mechanics are developed, with emphasis on linear algebra and complex variables. Dirac notation for vectors in Hilbert space is introduced. The representation of coordinates and momenta in quantum mechanics is analyzed and applied to the Heisenberg uncertainty principle.

Twentieth-Century Quantum Mechanics

The historical development and fundamental principles of quantum mechanics are reviewed. Fundamental differences with classical statistical mechanics are emphasized. The representation of quantum phenomena in Hilbert space is introduced.

Digital and Quantum Computers

The fundamentals of digital and quantum computers are presented. Binary numbers, modular arithmetic, and Boolean algebra are reviewed. Logic gates for both classic and quantum computers are introduced and combined in some simple circuits for carrying out useful computations. The possibility for efficient factoring of large numbers using quantum computers is discussed in detail. This requires a lengthy detour into number theory. The most obvious application is to cryptography and secure communication.

The Schrödinger Equation

The Schrodinger equation is introduced and applied to some elementary problems: particle-in-a box, harmonic oscillator, angular momentum, and the hydrogen atom. The representation of eigenfunctions and eigenvalues is discussed, employing linear operators and matrices. The quantum theory of spin, as well as the Pauli exclusion principle, are described. These enable a theoretical understanding of atomic structure and the periodic table. The two disparate modes of time-dependence of a quantum system–unitary evolution and collapse of the wavefunction–are contrasted.

New Adventures: Isotropic Vectors, Rotations, Spinors, and Groups

Some more specialized mathematical topics are introduced, including isotropic vectors, rotations, spinors, and Lie groups. The concept of invariance in the objective world is discussed. The stereographic projection is introduced to describe the behavior of spinors. The Lie groups SO(3) and SU(2) are studied in detail.

Quantum Entanglement and Bell’s Theorem

A description of quantum entanglement and its applications. Bell’s inequalities and Bell’s theorem are described, along with their implications for local reality and hidden variables. Other topics: applications using electron spin and photon polarization, Aspect’s experiments, decoherence of quantum states.