What properties does Baym use to derive the L commutation relation?

[univox360]

In Baym's Lectures on Quantum Mechanics he derives the following formula

[n.L,L]=ih L x n

(Where n is a unit vector)

I follow everything until this line:

ih(r x (p x n)) + ih((r x n) x p) = ih (r x p) x n

I can't seem to get this to work out. What properties is he using here?

 

[George Jones]

Can you show

0 = n x (r x p) + r x (p x n) + p x (n x r) ?

What you want would follow from this.

 

[univox360]

I understand, but is this identity valid since r and p do not commute? This identity is constructed using B(AC)-C(AB) which seems to change order of operation...

 

[George Jones]

Try using

\left( A \times B \right)_i = \epsilon_{ijk} A_j B_k

and

\epsilon_{ijk}\epsilon_{iab} = \delta_{ja}\delta_{kb} - \delta_{jb}\delta_{ka}

in the three terms in your expression (repeated indices are summed over). Also, n is a triple of numbers, and so commutes with r and p.

 

[George Jones]

I have now done the calculation. The identity can be verified by using

Try using

\left( A \times B \right)_i = \epsilon_{ijk} A_j B_k

and

\epsilon_{ijk}\epsilon_{iab} = \delta_{ja}\delta_{kb} - \delta_{jb}\delta_{ka}

in the three terms in your expression (repeated indices are summed over). Also, n is a triple of numbers, and so commutes with r and p.

 

[univox360]

Yes, using that theorem this works. Thanks so much!