Chapter 6: Spin One-Half & Rotations in Quantum Mechanics

The transformation of a quantum state (psi) from one set of base states (S) to another (T) is achieved through a calculation involving a summation of products, where the transformation coefficients (R sub IJ) relate the amplitudes. These base states are fundamentally orthogonal, meaning the amplitude for a state to be in two different base states is zero. While the core physics must remain invariant under spatial rotations, the transformation coefficients for rotation are only unique up to an arbitrary common phase factor. The discussion utilizes an improved Stern-Gerlach apparatus to experimentally visualize these concepts, showing how the apparatus can prepare pure (plus S) or (minus S) spin states. A key insight developed is the quantum mechanical description of rotations, where the rotation matrix is defined. The text mathematically demonstrates that successive rotations can be combined using matrix multiplication to find the resulting transformation. Specific rotations are analyzed: a 360-degree rotation about the z-axis is shown to change the sign of the quantum amplitudes (meaning the new amplitude C-prime is the negative of the original amplitude C), demonstrating that while the resulting physical state is indistinguishable from the original, the quantum amplitudes themselves are not identical. Transformations for rotations about the y-axis, including 180-degree and 90-degree, are derived by ensuring consistency and invariance requirements are met. Finally, the framework for describing arbitrary rotations is established using three sequential rotations defined by the Euler angles (alpha, beta, gamma). The chapter concludes by presenting the complete formulas and summary tables for the amplitude transformations caused by rotations about the standard coordinate axes.