Angular momentum and coordinates

[M. next]

let us denote the joint eigenstate of \hat{L^{2}}and \hat{L_{z}} by ll,m> and we know that if we are in spherical coordinates,
\hat{L^{2}} and \hat{L_{z}} depend on θ and ∅, so we denote the joint eigenstate by: <θ∅l l,m>.. why?

 

[dextercioby]

The bra-ket is a complex number, the value of the wavefunction (singled out by a choice of l and a choice of m) in terms of the spherical angles, i.e. coordinate variables.

 

[andrien]

are you looking for simultaneous eigenstate.then L² and L_z commute so that it is possible to find simultaneous eigenstates.

 

[jmcelve]

let us denote the joint eigenstate of \hat{L^{2}}and \hat{L_{z}} by ll,m> and we know that if we are in spherical coordinates,
\hat{L^{2}} and \hat{L_{z}} depend on θ and ∅, so we denote the joint eigenstate by: <θ∅l l,m>.. why?

We denote it that way because we've chosen to project the eigenstates and the operators in terms of \theta and \phi. We could equally express the eigenstates |l, m \rangle and the operators \hat{L^{2}} and \hat{L_{z}} in terms of Cartesian coordinates. We express the eigenstates in the standard Cartesian basis as \langle x, y, z|l, m \rangle. They aren't as pretty to work with, which is why we choose spherical coordinates to express spherical harmonics.

 

[M. next]

Thanks jmcelve. It is clear now.