Checkpoint: Evolve a State

This checkpoint pulls together everything from the module: building $U(t)$, evolving a superposition, tracking expectation values, and using conservation. You will compute a single number that secretly encodes three separate facts.

The scenario

A spin starts in $|+\rangle = \tfrac{1}{\sqrt2}(|0\rangle + |1\rangle)$ and evolves under

for a time such that $\omega t = \tfrac{2\pi}{3}$. From the module you can write down the Bloch components without re-deriving them:

• Precession (expectation-value dynamics): $\langle\sigma_x\rangle(t) = \cos\omega t$ and $\langle\sigma_y\rangle(t) = \sin\omega t$.
• Conservation: $\sigma_z$ commutes with $\hat H$, so $\langle\sigma_z\rangle(t) = 0$ for all time.
• Unitarity / purity: the Bloch vector keeps unit length, $\langle\sigma_x\rangle^2 + \langle\sigma_y\rangle^2 + \langle\sigma_z\rangle^2 = 1$, because $U(t)$ is unitary and a pure state stays pure.

What to compute

1. Form the squared length $s = \langle\sigma_x\rangle^2 + \langle\sigma_y\rangle^2 + \langle\sigma_z\rangle^2$. It must equal $1$ — your built-in sanity check that the dynamics conserve probability.
2. Form $\langle\sigma_x\rangle(t) = \cos\omega t$.
3. Return the product $s \cdot \langle\sigma_x\rangle(t)$. Because $s = 1$, this is just $\langle\sigma_x\rangle(t)$.

At $\omega t = \tfrac{2\pi}{3}$ we have $\cos\tfrac{2\pi}{3} = -\tfrac12$, so the answer is $-\tfrac12$.

Try it

This is a numerical exercise — your code should return a number. Compute $s\cdot\langle\sigma_x\rangle(t)$ at $\omega t = 2\pi/3$. The correct value is $-0.5$.

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