I'm trying to understand the concept of spin in Quantum Mechanics. I'm reading "Road to Reality" by Penrose, which despite not being a textbook, is reputed to give one a deep insight into physical processes.
- Let us suppose that we have a spin $\frac{1}{2}$ particle. It has two eigenstates- $|\uparrow\rangle$ and $|\downarrow\rangle$. I would assume that spin $S$ is an operator such that when it acts on the wavefunction $\alpha |\uparrow\rangle+\beta|\downarrow\rangle$, it collapses it to one eigenstate, with an eigenvalue which would be the spin (so $\frac{1}{2}$ here). However, Penrose says that the spin can be thought of as the point $[\alpha:\beta]$ in $\Bbb{C}P^1$.
Let us suppose thatWhy is this? Why do we not have a collapse to an eigenstate?
- Let us suppose that we have a particle with spin $j$. Then $N=2j$. An angular momentum eigenstate of such a particle can be written as $\psi_{AB\dots N}$. I know that there are $N+1$ independent eigenstates, and that for even values of $N$, the eigenstates are spherical harmonic functions while for odd values of $N$ the eigenstates are spin-weighted spherical harmonic functions.
But how do you calculate the spin $\frac{1}{2}$of this particle. It has two eigenstates- $|\uparrow\rangle$ and $|\downarrow\rangle$. I would assume? One might say that the spin $S$ is an operator such that when it acts on the wavefunctionjust $\alpha |\uparrow\rangle+\beta|\downarrow\rangle$, it collapses it to one eigenstate$j$. However, with an eigenvalue which would befrom the example of spin (so $\frac{1}{2}$ here). However, Penrose saysI would figure that the spin canis suppose to be thoughtan element of as the point $[\alpha:\beta]$ in $\Bbb{C}P^1$a projective space.
Why is Is this? Why do we not have a collapse to an eigenstatetrue?