Chapter 5: Spin One – Stern-Gerlach Experiment

Unlike classical mechanics, the quantum treatment relies entirely on probability amplitudes rather than classical concepts like angular momentum, offering a universally applicable framework. The apparatus demonstrates that atoms entering the magnetic field emerge in one of three clearly defined base states, designated as (+), (0), or (−). By strategically using quantum filtering (blocking masks) to select only a specific beam, atoms can be prepared into a pure state or polarized beam. Complex experiments involving multiple, successive Stern-Gerlach devices (like S and T filters, where T may be rotated by an angle alpha) reveal that subsequent measurements reorient the atomic state. A fundamental rule of quantum mechanics is established concerning orthogonal base states: the amplitude of transition between non-identical base states is zero, but between identical states, it is one. This essential relationship is formally summarized by the Kronecker delta relation. When all beams are permitted to pass through an apparatus—creating a wide-open filter—the total quantum mechanical amplitude for reaching a final state is determined by the superposition (sum) of the amplitudes for all possible intervening paths, illustrating the principle of amplitude interference. The operational machinery of quantum mechanics is formalized by showing that any filtering apparatus or complex experimental setup can be described by a nine-component matrix of complex numbers. These numbers represent all possible transition amplitudes between the three base states. Finally, the chapter demonstrates the transformation of a quantum state vector from one coordinate basis (like S) to another (like T, which is rotated), showing how the matrix provides the necessary coefficients for conversion, drawing a clear analogy to vector transformations in classical physics.