Re: orbitals, flowers, quantum puzzlement;

Cyberkatru wrote:
> > For isolated atoms, rho is spherically symmetric, giving symmetric
> > shapes. For molecules, rho is in fact a function of the coordinates of
> > all nuclei involved, and there is no longer any reason to have more
> > symmetry than the symmetry of the configuration of nuclei has, which is
> > very little and often none.
>
> Hmm. OK. But what determines the relative positions of the nuclei which are
> each in and of themselves, basically spherically symmetric?

To clarify. An isolated atom/molecule/nucleus/anything will be
perfectly spherically symmetric *only if* it is in a state of definite
angular momentum with zero total spin (mathematically, in an eigenstate
of the total angular momentum operator with eigenvalue 0). Many nuclei
have some definite nonzero total spin, hence cannot be spherically
symmetric. In combination with orbiting electrons, a nucleus with
nonzero spin may still from an atom of zero total spin (like the
singlet state of the hydrogen atom), but it may also not.

An interesting puzzle can be formulated here for the experts. If a
system is rotationally invariant (such as any system in isotropic space
and without external fields) will have energy states that are also
eigenstates of the total angular momentum operator. But will there
always be an energy eigenstate that is an eigenstate totale angular
momentum with eigenvalue 0 (i.e. a spherically symmetric state)? It's
clear that any fermion (fundamental or composite) can never be in a
zero spin state, since its spin has to be half integral. But in what
other situations can we put a lower bound on the total spin of the
system?

Physically speaking, given say a long polymer molecule, if the molecule
is completely isolated it's Hilbert space of states splits into finite
dimensional subspaces with definite total spin. But I would imagine
that it'd be very hard to find such a molecule with zero total spin. Is
it because 0 is excluded from the spectrum of the total angular
momentum operator or is it because such a state would simply have much
higher energy than other ones?

Igor

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