How to obtain theta and phi from Bloch sphere trajectory

How to obtain theta and phi from Bloch sphere trajectory

by Daniel Andrzej Polka -
Number of replies: 2

Hi, I’m not quite sure how we’re supposed to rewrite our state vector psi(t) as cos(theta/2)•up + e^(i•phi)•sin(theta/2)•down for exercises 1b and 1c. I tried taking the coefficients a and b of my final state vector (so let’s say that my final state vector psi(t) = a•up + b•down), and writing a = cos(theta/2) and b = e^(i•phi)•sin(theta/2) and solving the system. However this simply yields theta = 2•arccos(…) with a complex number in the arccos, so the result is pretty disgusting and doesn’t really help interpreting what values theta might actually take (and given what theta looked like, I didn’t bother trying to solve for phi). How can we do this more tidily and in such a way that yields coherent, interpretable results?

In reply to Daniel Andrzej Polka

Re: How to obtain theta and phi from Bloch sphere trajectory

by Maximilian Felix Müller -
Have a look at the „Solution of Schrödinger‘s Equation for spin 1/2“ in the lecture on 12 Oct (https://moodle.epfl.ch/mod/resource/view.php?id=1104975, last page). Note that the coefficient a must be real (cosine of a real number, so precisely in [-1, 1]). To achieve this, we extracted a global phase which can be ignored/removed afterwards. It may be worth trying to find a global phase of the shape e^(i*alpha).
I hope this helps.