I am trying to solve an exercise, but I can't seem to get it to work.

  I get given this rule,

$$\langle \sigma_x \rangle^2 + \langle \sigma_y \rangle^2+\langle \sigma_z \rangle^2 =1 $$

and I am asked to verify this for $|\psi\rangle = \cos\theta|0\rangle + \sin\theta|1\rangle$.

  I first expand upon the rule, by actually computing the probabilities for each base.

This then leads me to this:

This leads me to this:

I assume my mistake lies here.

When inserting $a_0 = \cos\theta$ and $a_1 = \sin\theta$ I get this:

This is incorrect though, because the relationship at the beginning should, according to the exercise, hold for all $\theta$ and not just when $\theta = n\pi$ $\forall n \in \mathbb N$, because it asks me to verify it for that specific state ($|\psi\rangle= \cos\theta|0\rangle + \sin\theta|1\rangle$). Can anyone help me out here? Thanks, Tom

I am trying to solve an exercise, but I can't seem to get it to work. I have to show $$\langle \sigma_x \rangle^2 + \langle \sigma_y \rangle^2+\langle \sigma_z \rangle^2 =1 $$

for $|\psi\rangle = \cos\theta|0\rangle + \sin\theta|1\rangle$. I first expand upon the rule, by actually computing the probabilities for each base.

This then leads me to this:

This leads me to this:

I assume my mistake lies here.

When inserting $a_0 = \cos\theta$ and $a_1 = \sin\theta$ I get this:

This is incorrect though, because the relationship at the beginning should, according to the exercise, hold for all $\theta$ and not just when $\theta = n\pi$ $\forall n \in \mathbb N$, because it asks me to verify it for that specific state ($|\psi\rangle= \cos\theta|0\rangle + \sin\theta|1\rangle$). Can anyone help me out here? Thanks, Tom

I am trying to solve an exercise, but I can't seem to get it to work.

I get given this rule and I am asked to verify this for $|\psi> = \cos\theta|0> + \sin\theta|1>$.

I first expand upon the rule, by actually computing the probabilities for each base.

This then leads me to this:

This leads me to this:

I assume my mistake lies here.

When inserting $a_0 = \cos\theta$ and $a_1 = \sin\theta$ I get this:

This is incorrect though, because the relationship at the beginning should, according to the exercise, hold for all $\theta$ and not just when $\theta = n\pi$ $\forall n \in \mathbb N$, because it asks me to verify it for that specific state ($|\psi> = \cos\theta|0> + \sin\theta|1>$). Can anyone help me out here? Thanks, Tom

I am trying to solve an exercise, but I can't seem to get it to work.

I get given this rule,

$$\langle \sigma_x \rangle^2 + \langle \sigma_y \rangle^2+\langle \sigma_z \rangle^2 =1 $$

and I am asked to verify this for $|\psi\rangle = \cos\theta|0\rangle + \sin\theta|1\rangle$.

I first expand upon the rule, by actually computing the probabilities for each base.

This then leads me to this:

This leads me to this:

I assume my mistake lies here.

When inserting $a_0 = \cos\theta$ and $a_1 = \sin\theta$ I get this:

This is incorrect though, because the relationship at the beginning should, according to the exercise, hold for all $\theta$ and not just when $\theta = n\pi$ $\forall n \in \mathbb N$, because it asks me to verify it for that specific state ($|\psi\rangle= \cos\theta|0\rangle + \sin\theta|1\rangle$). Can anyone help me out here? Thanks, Tom