Limitations on radial wavefunction for electron in an atom

[yxgao]

What are the limitations on the radial wavefunction for electron in an atom?

For instance, of the following, which cannot be the radial wave function, and why?

1.) $$e^{-r}$$
2.) $$\sin(br)$$
3.) $$\frac{1}{r}$$

Thanks!

 

[dextercioby]

What are the limitations on the radial wavefunction for electron in an atom?

For instance, of the following, which cannot be the radial wave function, and why?

1.) $$e^{-r}$$
2.) $$\sin(br)$$
3.) $$\frac{1}{r}$$

Thanks!

For an eigenfunction of the hamiltonian to describe physically acceptable states,it needs some requirements:
a)Continuous oner the entire domain (0,+oO)
b)Annulation in asymptotic limit (condition for bound states)
c)Lebesgue square integrabilty.

The only function of your three to have them all is 1),but 2) describes scattering states for the electrons of the atom.It doesn't annulate in asymptotic limit (r->+oO) and is not Labesgue square integrable.But it is an eigenfunction of the Hamiltonian.Because it's not normalizable,it does not fulfill the requirement expressed in the first postulate,therefore it cannot describe quantum states with discrete spectrum for the Hamiltonian.These functions can be normalized in Dirac generalized sens,implying distributions.

Daniel.

 

[R0man]

The radial wavefunction for an electron in an atom is limited by the principles of quantum mechanics. One of the main limitations is that the wavefunction must be continuous and finite at all points in space. This means that it cannot have any sudden changes or discontinuities, and it must approach zero as the distance from the nucleus increases.

Based on this limitation, the first option, e^{-r}, cannot be the radial wavefunction because it does not satisfy the requirement of being finite at all points. As r approaches infinity, e^{-r} approaches zero, but it never actually reaches zero. This means that the wavefunction would not be continuous and would not satisfy the principles of quantum mechanics.

The second option, \sin(br), could potentially be a valid radial wavefunction. However, it would only be valid for certain values of b. If b is too large, the wavefunction would not approach zero at large distances and would violate the continuity requirement. So while \sin(br) could be a valid wavefunction, it is limited by the value of b.

The third option, \frac{1}{r}, is not a valid radial wavefunction because it does not satisfy the requirement of being finite at all points. As r approaches zero, \frac{1}{r} approaches infinity, which is not physically possible. This also violates the continuity requirement and therefore cannot be a valid wavefunction.

Overall, the limitations on the radial wavefunction for an electron in an atom ensure that it accurately describes the behavior of the electron and follows the principles of quantum mechanics. These limitations play a crucial role in understanding the electronic structure of atoms and molecules.