This is the solution to the fifth (and sixth) problem of the first section of Quantum Mechanics for the work professional. You can find the details and notes here:
https://pasayten.org/quantum-mechanic...
We define and practice with an alternative basis for the vector space of quantum states for our two-level system. We compute probabilities and eventually show how to make a generic choice of basis vectors by a rotation through some angle theta.
The course itself has a website here:
https://pasayten.org/quantum-mechanic...