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1D is indeed not good one for radial operator (and if one uses it then it corresponds to half line with reflection boundary condition at x=0, and then the boundary term vanishes there so again h/i*d/dr is hermitian). But on higher dimensions, e.g. 3, one need to think of inner product with $r^2$ weight function, so it cancels the the boundary term. Of course this gives another term when doing integration by parts, and that is why $d/dr$ alone if not enough for hermiticity.

Of course this corresponds also to the fact that "plane wave" in radial coordinate has some function of $r$ in the denominator, because of energy conservation, e.g. $e^{ikr}/r$ in 3D.

1D is indeed not good one for radial operator (and if one uses it then it corresponds to half line with reflection boundary condition at x=0, and then the boundary term vanishes there so again h/i*d/dr is hermitian). But on higher dimensions, e.g. 3, one need to think of inner product with $r^2$ weight function, so it cancels the the boundary term. Of course this gives another term when doing integration by parts, and that is why $d/dr$ alone if not enough for hermiticity.

Of course this corresponds also to the fact that "plane wave" in radial coordinate has some function of $r$ in the denominator, because of energy conservation, e.g. $e^{ikr}/r$ in 3D.

Also see this nice paper, which describes the failure of this opproach in 2D.

1D is indeed not good one for radial operator. But on higher dimensions, e.g. 3, one need to think of inner product with $r^2$ weight function, so it cancels the the boundary term. Of course this gives another term when doing integration by parts, and that is why $d/dr$ alone if not enough for hermiticity.

Of course this corresponds also to the fact that "plane wave" in radial coordinate has some function of $r$ in the denominator, because of energy conservation, e.g. $e^{ikr}/r$ in 3D.

1D is indeed not good one for radial operator (and if one uses it then it corresponds to half line with reflection boundary condition at x=0, and then the boundary term vanishes there so again h/i*d/dr is hermitian). But on higher dimensions, e.g. 3, one need to think of inner product with $r^2$ weight function, so it cancels the the boundary term. Of course this gives another term when doing integration by parts, and that is why $d/dr$ alone if not enough for hermiticity.

Of course this corresponds also to the fact that "plane wave" in radial coordinate has some function of $r$ in the denominator, because of energy conservation, e.g. $e^{ikr}/r$ in 3D.

1D is indeed not good one for radial operator. But on higher dimensions, i.e. 3, one need to think of inner product with r2 weight function, so it cancels the the boundary term. If course this gives another term when doing integration by parts, and that is why d/dr alone if not enough for hermiticity.

Of course this corresponds also to the fact that "plane wave" in radial coordinate has some function of r in the denumerator, because of energy conservation, i.e. e(ikr)/r in 3D

1D is indeed not good one for radial operator. But on higher dimensions, e.g. 3, one need to think of inner product with $r^2$ weight function, so it cancels the the boundary term. Of course this gives another term when doing integration by parts, and that is why $d/dr$ alone if not enough for hermiticity.

Of course this corresponds also to the fact that "plane wave" in radial coordinate has some function of $r$ in the denominator, because of energy conservation, e.g. $e^{ikr}/r$ in 3D.