Hi guys! Many time reader, first time poster... I've struggled big time with the following. Any advice at all would be great. I'm so muddled, it's just not funny any more... (plus I'm not really familiar with who to write the mathematic script so please be patient)
I have an operator, say A representing observable A with two normalised eigenstates with eigenvalues a1, a2 respectively (and the same with B in place of A)...
and if |A1> = 1/5(3|B1> +4|B2>) and |A2> = 1/5(4|B1> - 3|B2>). My question is how do i find the commutator of this, and if I do find the commutator of this, how can I then use this to determine the generalised uncertainty relation for operators Aand B?? So desperate!
<(A)^2><(B)^2> > 1/4|<[A,B]>|^2
The Attempt at a Solution
I'm not sure why, but I felt that I could represent:
A: 1/5[3b1 4b1; 4b2 -3b2] ; and
B: 1/5[3a1 4a1; 4a2 -3a2]
for which I can then do matrix multiplication and subtraction (i.e. [A,B] = AB - BA (all operators))
and I get:
4/25 [4(b1a2-a1b2) -3(b1a2+a1b2); 3(b2a1-a2b1) 4(b2a1-a2b1)]
then not sure if i have anything remotely ok, although I do notice that the matrix looks kinda Hermitian or whatever?. really stuck from here. Basically have no idea how to get the 'expectation' value for this or whatever... any help would be great (and asap cos I've tried to work it out all week and now left my time seriously short - not that anyone HAS to help me of course!...) Cheers very much! humfri
(p.s. sorry if this post was annoying and confusing... still learning)