Lab 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.



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



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

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



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









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0.49999999999999978

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



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









    Out[5]:





True

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



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pz.dag()*Sx*pz









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Quantum object: dims = [[1], [1]], shape = [1, 1], type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}0.0\\\end{array}\right)\end{equation*}

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 $|\psi\rangle=|+x\rangle$.



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px.dag()*Sx*px









    Out[7]:





Quantum object: dims = [[1], [1]], shape = [1, 1], type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}0.500\\\end{array}\right)\end{equation*}

Again, in units of $\hbar$.

Lab instructions:

Calculate the expectation value $\langle S_z \rangle$ for the states $|+x\rangle, |+y\rangle, |+z\rangle$
Calculate the expectation value $\langle S_z \rangle$ for the state $|\psi\rangle = \frac{1}{\sqrt{2}}|+x\rangle + \frac{1}{\sqrt{2}}|-x\rangle$. Interpret this result, does it make sense? Is there a simpler way to write this state?
Represent the spin-squared operator $\hat S^2$ as a matrix.
Verify that the $\hat S^2$ commutes with each spin component: $\hat S_x, \hat S_y, \hat S_z$
Find the expectation value of $S^2$ for the three postive spins: $|+x\rangle, |+y\rangle, and |+z\rangle$. Interpret your results. Do the values differ? Should they?
Problem 6.9, 6.11, 6.12 from Chapter 6



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