what's the deal with Lz?

[mattlorig]

In the hydrogen atom, I believe most people are familiar with the following three equations:
L^2 (psi) = l*(l+1)*hbar^2*(psi)
Lz (psi) = ml*hbar*(psi)
H (psi) = -1/n^2*junk*(psi)
where L^2, Lz, and H are the linear operators for total angular momentum squared, angular momentum about the z-axis, and energy. I'm comfortable with eigenvalues, eigenvectors, etc. The thing I don't understand, however, is what Lz really is. Since our choice of axes is completely arbitrary, I could have just as easily chosen Lx to be Lz. But of course, if I know Lz, I can not possibly know anything about Lx (other than, perhaps, its maximum value).

I gues what I'm asking is the following: is Lz just a way to say, that if we were to measure the angular momentum of an electron about a certain axis (which we'll call z) it can only have values of ml*hbar, and once we know what Lz is, we can't possibly know anything about Lx and Ly?

I'd really appreciate it if somebody could straighten this out for me.

[Galileo]

Yes, our choice of axes is of course completely arbitrary.

L_x,L_y and L_z do not commute, but they DO commute with L^2. So we can find a complete basis of eigenvectors common to L^2 and an axis. So we have to make a choice, L_z is used for convenience.

If you measured L_z, then the electron will be in an eigenstate of L_z. Now try to see what the probabilities are of getting m_l\hbar when measuring L_x.
(Use L_x=\frac{1}{2}(L_++L_-))

[dextercioby]

Yes, our choice of axes is of course completely arbitrary.
L_x,L_y and L_z do not commute, but they DO commute with L^2. So we can find a complete basis of eigenvectors common to L^2 and an axis. So we have to make a choice, L_z is used for convenience.
If you measured L_z, then the electron will be in an eigenstate of L_z. Now try to see what the probabilities are of getting m_l\hbar when measuring L_x.
(Use L_x=\frac{1}{2}(L_++L_-))


Sorry,Galileo,but i just couldn't help myself. :tongue2:
So,this convention is one of the many more encountered in physics.Think about the old famous conventions regarding the magnetic field (induction) \vec{B} .Both in electrodynamics (charged particle in magnetic/electromagnetic field) and in QM (Zeeman effect (normal/anomal)) it's always chosen along "Oz(=Ox_{3})" axis.I don't know why,i never met the guys who did that. :wink: You'll have to accept it,the same way you accepted those wicked conventions in geometrical optics,that convention for the sign of work in thermodynamics and many more.

A physicist's mind is twisted in uncountable ways... :cool:

Daniel.