Final answer:
The Stern-Gerlach experiment illustrates the quantization of spin angular momentum, with electrons having non-degenerate eigenvalues of ±ħ/2 as they can only be 'spin up' or 'spin down'. The non-degeneracy is due to the lack of symmetries that could result in more than one eigenvector corresponding to one eigenvalue for the electron's spin.
Step-by-step explanation:
The Stern-Gerlach experiment is a key piece of evidence for quantum mechanics and specifically for the quantization of spin angular momentum. In this experiment, particles such as electrons are passed through a nonuniform magnetic field, which causes their spins to align along the direction of the field.
This results in the particles being deflected into distinct paths, which are detected as discrete spots on a photographic plate. According to quantum mechanics, the intrinsic angular momentum (or spin) of an electron is quantized, exhibiting only certain allowed values. Electrons have a spin quantum number (s) of 1/2, leading to two possible spin projection quantum numbers (ms), +1/2 and -1/2, corresponding to the 'spin up' and 'spin down' states, respectively.
The eigenvalues of the spin operator ^Sz reflect these discrete possible outcomes of a spin measurement. The question posed concerns the assumption of non-degeneracy, referring to each eigenvalue having a unique eigenvector. In the case of the Stern-Gerlach experiment for electrons, since there are only two possible spin states for a single electron, the eigenvalues ±ħ/2 are non-degenerate.
It is the inherent symmetries or lack thereof in the physical system being measured that determine the degeneracy of eigenvalues. In a theoretical scenario where a Stern-Gerlach-like experiment yielded four distinct bands instead of two, it would suggest the existence of a particle with a different spin quantum number allowing for more spin projections and hence degenerate eigenvalues.