Why is the spin quantum number given a value 1/2? Why not any other value, say +1 for clockwise spin and -1 for anticlockwise spin instead of +1/2 for clockwise spin and -1/2 for anticlockwise spin ?
This prescription takes [tex]\hbar[/tex] as the fundamental unit of angular momentum. So, what's really happening is that spin up has angular momentum [tex]\frac{\hbar}{2}[/tex] and spin down has [tex]\frac{-\hbar}{2}[/tex]. In general this is a matter of convenience, as it is simply defined to be consistent with the use of [tex]\hbar[/tex] as the fundamental unit of angular momentum elsewhere.
This is not just a convention chosen for convenience. The [itex]\pm \hbar/2 [/itex] comes from the eigenvalues equations for a spin-1/2 particle (like the electron). A spin-1 particle has eigenvalues, -1, 0 and 1, and so on...
Are there particles with spin-1 having 3 eigen values -1,0,1? Does 3 eigen values mean that they can have angular momentum due to 3 different spin orientations of the particle?
The word eigenstate is descriptive of the measured state of some object that possesses quantifiable characteristics such as position, momentum, etc. The state being measured and described must be an "observable" (i.e. something that can be experimentally measured either directly or indirectly like position or momentum), and must have a definite value. It is imposible to know two coordinates of the spin at the same time because of the restriction of the Uncertainty principle. So the number of eigenvalues give the number of observable states of a particle which is also subject to cetain selection rules.
All spin-1 particles have 3 eigenstates, often represented as something like |+>, |0> and |->, of say, the S_z operator. The probability of observing the particle in any of these 3 states can be any number [itex]0 \leq p \leq 1 [/itex], subject to the condition that the total probability is 1. Loosely speaking, yes. But specifically, it means that if you measure say the z-component of the spin, you will observe no more than 3 different values.
Well, the possible values for the spin quantum number come from group theory and are basically determined by symmetry. Symmetry means that the QM equations (like the excpectation value) that describe nature must be invariant under certain transformations like rotations. An electron has a spin (intrinsic angular momentum), but it does NOT actually rotate around its axis, guys. The 'rotational nature' of spin comes from the behavior of the Dirac wavefunction (this is a matrix that represents a physical state and arises when solving the Dirac equation. This equation describes a fermion : a particle with non-integer spin) under coordinate-transformations (which are called the rotations). With behavior i mean : how does the physics change if we interchange the components of this Dirac spinor, if we change the parity, if we apply coordinate transformations to the wavefunction and so on....For example, if we rotate the wavefunction 360°, do we still get the same physical laws...You see the pattern ??? It is this specific behavior that yields the name SPINOR because if you rotate it 360°, you get the opposite value. Now, changing coordinates (represented by rotations) and looking how the physics changes or not, is NOT THE SAME as actually rotating. So, spin arises thanks to symmetries involved but there is no actual rotation. marlon