JesseM
Science Advisor
- 8,519
- 17
You can only expand a dot product in row and column vectors if you are using matrix multiplication; i.e. (ax, ay, az).(sx, sy, sz) is equal to (ax ay az)x(sx, sy, sz), where I am using x to represent matrix multiplication. So with the matrix multiplication made explicit, your step 5 is something like this:wm said:I am for sure wondering what I have missed? So, from my maths post, with explanations for the moves from (4) to (7). And more to come if I'm still not clear:
(3) <(a.s) (s'.b')>
Since s' = -s we have:
(4) = - <(a.s) (s.b')>
Since we can expand a dot product using row and column vectors http://en.wikipedia.org/wiki/Column_vector
(5) = - <(ax ay az)x(sx, sy, sz)x(sx sy sz)x(bx', by', bz')>
This is valid in itself. But your mistake is here:
Wrong! Again, (sx, sy, sz)x(sx sy sz) is not a dot product with a single value, it is a 3 x 3 matrix whose 9 entries look like this:(6) = - (ax ay az) <(sx, sy, sz) (sx sy sz)> (bx', by', bz')
The ensemble average is now over a dot product between s and s.
sx*sx sx*sy sx*sz
sy*sx sy*sy sy*sz
sz*sx sz*sy sz*sz
Yes. I showed a counterexample to your proof in post #99, and what's more, in post #76 I've given a proof that the correct value for -<(a.s)*(s.b)> would be -(1/2)cos(a - b), using the rule that if the outcome of a given experiment will be some function f(x) of a parameter x whose value is between A and B and whose probability distribution is given by p(x), then the expectation value is \int_{A}^{B} p(x)*f(x) \, dx. In this case, the outcome is a function of the angle \theta, namely cos(\theta - a)*cos(\theta - b), and \theta must be equally likely to take any value from 0 to 2pi, so we must use the probability distribution p(\theta) = 1/2\pi to ensure that if we integrate the probability from 0 to 2pi, the answer is 1. Thus, the expectation value for -<(a.s)*(s.b)> must be - \frac{1}{2\pi} \int_{0}^{2\pi} cos(\theta - a)*cos(\theta - b) \, d\theta, which works out to -(1/2)cos(a - b).wm said:SO: Have I made a mistake that cannot be corrected?
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