It is often said in Japan that teaching is learning. I am always convinced that this saying is true everytime I prepare for lectures and discussions with students. Last week it was the partial wave expansion of plane wave that reminded me of the saying again.
Obvioulsly, the Hamiltonian for a free particle has translational and rotational symmetries. So, linear and angular momenta are good quantum numbers. However, the linear and angular momentum operators do not commute with each other, which means we need to choose one of them for labelling the free-particle wave function. In the scattering problem, linear momentum is usually chosen because of the nature of the scattering experiment (you know, the incident beam is produced with a good linear momentum...). This means the wave function far from the target is a plane wave.
In the vicinity of the target, however, angular momentum plays more important role than linear momentum because the interaction between the target and the beam causes the trajectory of the beam bent. So, in many cases, the plane wave is expanded by wave functions labelled by angular momentum, which has a product form of the spherical Bessel function and the spherical Harmonics.
Then, I had a question. When the flux of the free-particle wave function is calculated with the plane wave, the result is well-known to be the constant velocity (in a non-relativistic limit). Even after a conversion into the spherical polar coordinates, the flux has only \theta (orientation) dependence, and there is no dependence of distance (from the origin). However, with the angular momentum representation ( that is, with spherical Bessel and sherical harmonics), the resultant flux seems r-dependent (that is, dependent on the distance). What is wrong with this?
After analysing several mathematical relations, I found that the spherical Bessel functions satisfies a kind of conditions similar to the completeness condition. But this completeness condition is realised only with the weight factor (2L+1), where L represents angular momentum. I have never seen this in the literatures.
I talked to J. on last Friday about this problem. His expertise is the nuclear reaction theory. He said that he was aware of this problem, but could not give the instant answer. "OK, so this problem is not so trivial." is my impression. He continued that he has seen an article in the Am. J. Phys. discussing this problem many years ago. Maybe, the answer can be obtained from the paper.... if J's information is correct and the paper really exists....
