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Problem Setup:

There is a nice proof of the heisenberg uncertianty principle using cauchy-schwarz given here

Stated in variance form it is:

$$ \sigma_x^2 \sigma_p^2 \ge \frac{\hbar^2}{4} $$

Now the original Cauchy schwarz inequality can be generalized into an equality which includes an angle term as:

$$ \left< f \middle| f \right> \left< g \middle| g \right> \cos(\theta_{f,g}) = |\left< f|g \right>|^2 $$

And this is often used to DEFINE what an angle means between two elements of an arbitrary inner product space.

So that suggests there should be a dimensionless quantity $\cos(\theta_{x-\mu_x,p-\mu_p})$ where $\mu_m$ denotes the expected value of the operator $m$.

Such that

$$\sigma_x^2 \sigma_p^2 \cos(\theta_{x-\mu_x,p-\mu_p}) = \frac{\hbar^2}{4} $$

So my question:

What is the physical significance of $\theta_{x-\mu_x,p-\mu_p}$? If one quantum observer measures the value to be quite close to 1 and another measures it to be quite close to 0 what does that say about the observers?

Some Attempted Pondering:

Intuitively this is suggesting the the position and momentum basis are very close together (cosine of the angle is 1), and when they are perpendicular the cosine of the angle is 0.

In some sense this problem is asking what does a large value for $\sigma_x \sigma_p$ mean? This product has units of action.

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