Rotations in Quantum Mechanics Question

[AdamBourke]

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 η_ij according to the equation:

l² = η_ij xⁱx^j​

where Einstein Summation Convention is used

The action of a geometrical transformation R acting on the vectors can be written:

x' ⁱ = R ⁱ_j 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 = η_ij xⁱx^j = (R ^Tη_ij [SIZE="5"]R) [SIZE="5"]R ⁱ_a x ^a[SIZE="5"]R ^j_b x ^b​

But then I wasn't really sure a) if the η_ij should be encased by the [SIZE="5"]Rs, and b) what do do after that. I tried to write ([I][B][SIZE="5"]R[/B] [SUP]T[/SUP][/I]η[SUB]ij[/SUB] [I][B][SIZE="5"]R[/B][/I]) as [SIZE="5"][FONT="Times New Roman"][I][B]R[/B][/I][SUP]p[/SUP][SUB]i[/SUB]η[SUB]pq[/SUB][I][B]R[/B][/I][SUP]j[/SUP][SUB]q[/SUB], but that gives an awful lot of [FONT="Times New Roman"][I][B][SIZE="5"]R[/B][/I]s and indices which I just can't see how to get rid of:

[SIZE="5"][FONT="Times New Roman"][CENTER]η[SUB]ij[/SUB] [I]x[/I][SUP]i[/SUP][I]x[/I][SUP]j[/SUP] = [I][B]R[/B][/I][SUP]p[/SUP][SUB]i[/SUB]η[SUB]pq[/SUB][I][B]R[/B][/I][SUP]j[/SUP][SUB]q[/SUB] [I][B][SIZE="5"]R[/B][/I] [SUP]i[/SUP][SUB]a[/SUB] [I]x[/I] [SUP]a[/SUP][I][B][SIZE="5"]R[/B][/I] [SUP]j[/SUP][SUB]b[/SUB] [I]x[/I] [SUP]b[/SUP][/CENTER]

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

 

[fzero]

If distances are preserved, then the length of $${x'}^i = {R^i}_j x^j$$ must be equal to the length of $$x^i$$. So it must be that

$$\eta_{ij}{x'}^i{x'}^j = \eta_{ij}x^i x^j.$$

You therefore want to determine the condition on $${R^i}_j$$ so that this is true.

The approach you took would have worked too, but you would have had to be much more careful about all of the indices in the expression, as well as knowing a property of the inverse of the matrix $$R$$.