- #1
AdamBourke
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Homework Statement
This is more maths than QM I think, but it's at the beginning of my Quantum Questions. Basically, it's about rotations preserving length:
xi is the ith component of a vector, and the length of a vector is determined by the metric ηij according to the equation:
l2 = ηij xixj
where Einstein Summation Convention is used
The action of a geometrical transformation R acting on the vectors can be written:
x' i = R ij x j
Show that R is an isometry (i.e distances are preserved by rotations) if and only if:
R Tη R = η
Homework Equations
None, other than the ones in the question. Unless I'm wrong, which would explain why I can't do the question...
The Attempt at a Solution
Well, I'm not entirely sure where to begin.
I started with the condition:
l2 = ηij xixj = (R Tηij R) R ia x aR jb x b
But then I wasn't really sure a) if the ηij should be encased by the Rs, and b) what do do after that. I tried to write (R Tηij R) as RpiηpqRjq, but that gives an awful lot of Rs and indices which I just can't see how to get rid of:
ηij xixj = RpiηpqRjq R ia x aR jb x b
Am I looking at this completely wrong, or just not seeing something simple? It's an assignment, so I don't want the answer, but I need some help I think.
Thanks,
Adam