Final answer:
The spin quantum number of an electron can only be +1/2 or -1/2 because intrinsic spin angular momentum is quantized. The Pauli exclusion principle requires these two unique values to differentiate electrons occupying the same orbital. The Stern-Gerlach experiment showing more than two distinct bands would indicate a particle with more possible spin states.
Step-by-step explanation:
The electron spin quantum number, denoted as ms, can only take on two possible values, +1/2 or -1/2. This is because electrons have an intrinsic spin angular momentum that is quantized, meaning it can only exist in certain fixed values. In quantum mechanics, this intrinsic spin is described by the electron's spin quantum number. If the Stern-Gerlach experiment showed four distinct bands instead of the expected two, it would suggest that the particle being tested has a different spin quantum number, allowing for more possible spin states.
The existence of only two possible values for electron spin states, also known as the a and ß states, is due to the rules of quantum mechanics and is fundamental to the Pauli exclusion principle. This principle states that no two electrons in an atom can have the exact same set of four quantum numbers. Because the spin quantum number can only be +1/2 or -1/2, each orbital in an atom can be occupied by at most two electrons. For helium (He) atoms, which have two electrons, this means that both can occupy the 1s orbital, but they must have opposite spin quantum numbers to comply with the exclusion principle, leading to the sets {1, 0, 0, +1/2} and {1, 0, 0, -1/2} for the two electrons.