Chapter 6 - Spin

A few new operators (or new names for the same ones!) The three axes, x, y, z spin components can be measured with $SA_x$, $SA_y$, and $SA_z$ devices.

We’ll use $\hbar=1$ for numerical results, this is fairly standard practice, but can be tricky to remember.

from numpy import sin,cos,pi,sqrt
from qutip import *

pz = Qobj([[1],[0]])
mz = Qobj([[0],[1]])
px = Qobj([[1/sqrt(2)],[1/sqrt(2)]])
mx = Qobj([[1/sqrt(2)],[-1/sqrt(2)]])
py = Qobj([[1/sqrt(2)],[1j/sqrt(2)]])
my = Qobj([[1/sqrt(2)],[-1j/sqrt(2)]])
Sx = 1/2.0*sigmax()
Sy = 1/2.0*sigmay()
Sz = 1/2.0*sigmaz()

py

Example: determine $P(S_x = \frac{\hbar}{2} ||-y\rangle)$¶

abs(px.dag() * my) ** 2

Example: verify the commutation relation: $\left[\hat{S}_x,\hat{S}_z\right] = -i\hbar\hat{S}_y$¶

Sx*Sz - Sz*Sx == -1j*Sy  # remember, h = 1

Ex: find $\langle \hat{S}_x\rangle$ for the state $|\psi\rangle=|+Z\rangle$.¶

pz.dag()*Sx*pz

This makes sense given that $S_x$ can be either $\frac{+\hbar}{2}$ or $\frac{-\hbar}{2}$ with equal probability. Similarly, if the state is $|\psi\rangle=|+x\rangle$.

px.dag()*Sx*px

Again, in units of $\hbar$.