- #1
VantagePoint72
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Suppose I have particle in three dimensional space whose position space wavefunction in spherical coordinates is ##\psi(r,\theta,\phi)##. The spherical harmonics ##Y_{\ell,m}## are a complete set of functions on the 2-sphere and so any function ##f(\theta,\phi)## can be expanded as ##f(\theta,\phi) = \sum_{\ell=0}^\infty \sum_{m=-\ell}^{\ell} f_{\ell,m} Y_{\ell,m}(\theta,\phi)##. Since ##\psi## may, in general, have radial dependence, the coefficients ##f_{\ell,m}## will too, i.e., ##f_{\ell,m}(r)##.
I'm curious, then, what the probability is that the particle will be measured have total orbital angular momentum ##\sqrt{\ell_0(\ell_0+1)}\hbar##. I'm used to seeing expansions of quantum states in which the coefficients are just constants and so I'm a bit thrown by the radial dependence in this case. Intuitively, I'd expect that a total orbital angular momentum measurement would yield ##\sqrt{\ell_0(\ell_0+1)}\hbar## with probability
##
P(\ell_0) = \sum_{m=-\ell_0}^{\ell_0} \int_0^\infty dr |f_{\ell_0,m}(r)|^2
##
Is that right?
I'm curious, then, what the probability is that the particle will be measured have total orbital angular momentum ##\sqrt{\ell_0(\ell_0+1)}\hbar##. I'm used to seeing expansions of quantum states in which the coefficients are just constants and so I'm a bit thrown by the radial dependence in this case. Intuitively, I'd expect that a total orbital angular momentum measurement would yield ##\sqrt{\ell_0(\ell_0+1)}\hbar## with probability
##
P(\ell_0) = \sum_{m=-\ell_0}^{\ell_0} \int_0^\infty dr |f_{\ell_0,m}(r)|^2
##
Is that right?