Do all spin matrices always have same eigenvalues? this is the case for \(S_{x},S_{y},S_{z}\) for electron.
NO. depends spin number.
How do we get the probability of measuring \(S_{y}=-\frac {h}{2}\) or \(S_{y}=\frac {h}{2}\) to be \(\frac {1}{2}\)? is it because there are two
eigenvalues, and it is 50% each? see class notes lecture 5b. page 9. Answer: Current
state vector is \(|S_{x}=\frac {h}{2}\rangle \).
Does the order matter? In page 5, lecture 5B, could we do \(C_{+}=\langle S_{x}=\frac {h}{2}|S_{z}=\frac {h}{2}\rangle \) or \(C_{+}=\langle S_{z}=\frac {h}{2}|S_{x}=\frac {h}{2}\rangle \) ? Resolved.
Why is \(\langle V|S_{x}|V\rangle \) gives the The statistical average of measuring \(S_{x}\) given current state vector is \(|V\rangle \)
? Resolved.
Can we just move the \(H\) operator to RHS, as in \(x^{\prime \prime }+Mx=0\) instead of \(x^{\prime \prime }=-Mx\). This way no need to work
with negative eigenvalues? Yes.
HW 5, last problem, I do not see how \(M,N\,\) share all the 3 eigenvectors. I get only one
common eigenvector. I also do not understand the comment in my solution to refer to
set of vectors as basis? What does this mean? Also, we know \(M,N\) commute, and so they
share a common basis, but the question is asking which ones they share? Resolved.
For Pauli matrices, \(\left [ \sigma _{i},\sigma _{j}\right ] =2i\sum \epsilon _{ijk}\sigma _{k}\). and for spin \(\frac {1}{2}\) it is \(\left [ S_{i},S_{j}\right ] =i\hbar \sum _{k}\epsilon _{ijk}S_{k}\). So what is it for spin \(1\)? is it still \(\left [ S_{i},S_{j}\right ] =i\hbar \sum _{k}\epsilon _{ijk}S_{k}\) ? Yes.
I think \(\Psi \left (x,t\right ) \) is just the eigenfunction corresponding to the eigenvalue just measured. So
if the operator used is the position operator \(X\), then it is called \(\Psi \relax (x) \). If the operator used
is momentum operator \(\mathrm {P}\), we call it \(\phi _{p}(x)\), but should it be really be \(\Psi _{p}\relax (x) \)? If the operator is
Hamiltonian \(\hat {H}\), then the eigenvalue is the energy level \(E\) and the \(\Psi \) is called \(\Psi _{E}\relax (x) \). Any of these
are also called the wave function \(\Psi \relax (x) \). Is this correct? I think so.