Standard Basis and Ladder operators

[quasar987]

Cohen-Tanoudji defines a "standard basis" of the state space as an orthonormal basis {|k,j,m>} composed of eigenvectors common to J² and J_z such that the action of J_± on the basis vectors is given by

$$J_{\pm}|k,j,m>=\hbar\sqrt{j(j+1)-m(m\pm 1)}|k,j,m\pm 1>$$

But isn't is automatic that such are the effects of the ladder operators as soon as {|k,j,m>} is an orthonormal basis of simultaneous eigenvectors of J² and J_z?

At least this is the distinct impression I got out of the text preceeding this definition.

 

[dextercioby]

Yes, C-Tannoudji is a little inaccurate. A standard basis in the irreducible space is a a basis formed by eigenvectors of L_{z} and L^{2}. Period.