Editing Atomic orbital (section)

=== Qualitative understanding of shapes ===
The shapes of atomic orbitals can be qualitatively understood by considering the analogous case of [[Vibrations of a circular drum|standing waves on a circular drum]].<ref>{{cite journal| last1=Cazenave, Lions|first1=T., P.|title=Orbital stability of standing waves for some nonlinear Schrödinger equations| journal=Communications in Mathematical Physics|year=1982| volume=85|issue=4 | pages= 549–561|doi = 10.1007/BF01403504 |bibcode = 1982CMaPh..85..549C| last2=Lions| first2=P. L.|s2cid=120472894}}</ref> To see the analogy, the mean vibrational displacement of each bit of drum membrane from the equilibrium point over many cycles (a measure of average drum membrane velocity and momentum at that point) must be considered relative to that point's distance from the center of the drum head. If this displacement is taken as being analogous to the probability of finding an electron at a given distance from the nucleus, then it will be seen that the many modes of the vibrating disk form patterns that trace the various shapes of atomic orbitals. The basic reason for this correspondence lies in the fact that the distribution of kinetic energy and momentum in a matter-wave is predictive of where the particle associated with the wave will be. That is, the probability of finding an electron at a given place is also a function of the electron's average momentum at that point, since high electron momentum at a given position tends to "localize" the electron in that position, via the properties of electron wave-packets (see the [[Uncertainty principle|Heisenberg uncertainty principle]] for details of the mechanism).

This relationship means that certain key features can be observed in both drum membrane modes and atomic orbitals. For example, in all of the modes analogous to '''s'''&nbsp;orbitals (the top row in the animated illustration below), it can be seen that the very center of the drum membrane vibrates most strongly, corresponding to the [[antinode]] in all '''s'''&nbsp;orbitals in an atom. This antinode means the electron is most likely to be at the physical position of the nucleus (which it passes straight through without scattering or striking it), since it is moving (on average) most rapidly at that point, giving it maximal momentum.

A mental "planetary orbit" picture closest to the behavior of electrons in '''s'''&nbsp;orbitals, all of which have no angular momentum, might perhaps be that of a [[Keplerian orbit]] with the [[orbital eccentricity]] of 1 but a finite major axis, not physically possible (because [[particle]]s were to collide), but can be imagined as a [[limit (mathematics)|limit]] of orbits with equal major axes but increasing eccentricity.<!-- could somebody make an illustration? -->

Below, a number of drum membrane vibration modes and the respective wave functions of the hydrogen atom are shown. A correspondence can be considered where the wave functions of a vibrating drum head are for a two-coordinate system {{math|ψ(''r'', ''θ'')}} and the wave functions for a vibrating sphere are three-coordinate {{math|ψ(''r'', ''θ'', ''φ'')}}.

<gallery mode="nolines" perrow="3" widths="200px" caption="s-type drum modes and wave functions">
File:Drum vibration mode01.gif|Drum mode <math>u_{01}</math>
File:Drum vibration mode02.gif|Drum mode <math>u_{02}</math>
File:Drum vibration mode03.gif|Drum mode <math>u_{03}</math>
File:Phi 1s.gif|Wave function of 1s orbital (real part, 2D-cut, <math>r_\mathrm{max}=2 a_0</math>)
File:Phi 2s.gif|Wave function of 2s orbital (real part, 2D-cut, <math>r_\mathrm{max}=10 a_0</math>)
File:Phi 3s.gif|Wave function of 3s orbital (real part, 2D-cut, <math>r_\mathrm{max}=20 a_0</math>)
</gallery>

None of the other sets of modes in a drum membrane have a central antinode, and in all of them the center of the drum does not move. These correspond to a node at the nucleus for all non-'''s''' orbitals in an atom. These orbitals all have some angular momentum, and in the planetary model, they correspond to particles in orbit with eccentricity less than 1.0, so that they do not pass straight through the center of the primary body, but keep somewhat away from it.

In addition, the drum modes analogous to '''p''' and '''d''' modes in an atom show spatial irregularity along the different radial directions from the center of the drum, whereas all of the modes analogous to '''s'''&nbsp;modes are perfectly symmetrical in radial direction. The non-radial-symmetry properties of non-'''s''' orbitals are necessary to localize a particle with angular momentum and a wave nature in an orbital where it must tend to stay away from the central attraction force, since any particle localized at the point of central attraction could have no angular momentum. For these modes, waves in the drum head tend to avoid the central point. Such features again emphasize that the shapes of atomic orbitals are a direct consequence of the wave nature of electrons.

<gallery mode="nolines" perrow="3" widths="200px" caption="p-type drum modes and wave functions">
File:Drum vibration mode11.gif|Drum mode <math>u_{11}</math>
File:Drum vibration mode12.gif|Drum mode <math>u_{12}</math>
File:Drum vibration mode13.gif|Drum mode <math>u_{13}</math>
File:Phi 2p.gif|Wave function of 2p orbital (real part, 2D-cut, <math>r_\mathrm{max}=10 a_0</math>)
File:Phi 3p.gif|Wave function of 3p orbital (real part, 2D-cut, <math>r_\mathrm{max}=20 a_0</math>)
File:Phi 4p.gif|Wave function of 4p orbital (real part, 2D-cut, <math>r_\mathrm{max}=25 a_0</math>)
</gallery>

<gallery mode="nolines" perrow="3" widths="200px" caption="d-type drum modes">
File:Drum vibration mode21.gif|Drum mode <math>u_{21}</math> 
File:Drum vibration mode22.gif|Drum mode <math>u_{22}</math>
File:Drum vibration mode23.gif|Drum mode <math>u_{23}</math>
</gallery>