## Melbourne Cup Challenge Full Version 16

June 10, 2022

Melbourne Cup Challenge Full Version 16

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Incorrect expectation value for scalar product with one unit vector?

I am trying to compute the expectation value of
$$\langle a \mid \mathbf{r} \mid b \rangle = a^T \mathbf{r} b = \langle \mathbf{r} \mid a \otimes b \rangle$$
where $\mathbf{r}$ is a vector of unit length, $a$ and $b$ are unit vectors.
So far I have the following expectation value:
$$\langle a \mid \mathbf{r} \mid b \rangle = \langle \mathbf{r} \mid a \otimes b \rangle = \int \int r_\lambda a_\mu b_ u \delta_{\lambda\mu} \delta_{\sigma u} \text{d}\lambda \text{d}\sigma = \int r_\sigma a_\sigma b_\sigma \text{d}\sigma = r_\sigma a_\sigma b_\sigma$$
However, this result does not agree with my expectation. My expectation would be that the integral is zero because both $a$ and $b$ are unit vectors. However, my expectation is that
$$\langle a \mid \mathbf{r} \mid b \rangle = – \langle \mathbf{r} \mid a \otimes b \rangle = \int \int r_\lambda a_\mu b_ u \delta_{\lambda\mu} \delta_{\sigma u} \text{d}\lambda \text{d}\sigma = \int -r_\lambda a_\lambda b_\sigma \text{d}\lambda = 0$$
as $\lambda = \sigma$, and again $a$ and $b$ are unit vectors http://www.theoldgeneralstorehwy27.com/textmilling-4-winmac-2022/

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Oct 23, 2016
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