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In simple quantum mechanical problems such as the infinite square well, we solve the Time Independent Schrodinger's equation by separation of variable, effectively getting the energy eigenstates of the problem, and then express any general solution as a superposition of the stationary states.

This can also be done for the case of say the Hydrogen atom with one electron and coulomb potential in 3D. However, upon solving for the stationary states, we stumble upon the orbitals of your basic chemistry class.

So in quantum mechanics, the electron is in a superposition of all those orbitals, but in chemistry, an electron only occupies ONE orbital

So my question is why do we not talk about the wavefunctions that may be the linear combination of two or more than two stationary states in atoms or are there conditions in chemistry under which electrons occupy only the stationary states?

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  • $\begingroup$ There are certainly places in physics (Rydberg atoms) and chemistry (hybridized orbitals) where superpositions of orbitals are directly considered. I'll also point out that the stationary states you reference are for one-electron atoms - once you add another electron the problem is more complex, yet amazingly remains fairly close to the one-electron solutions. $\endgroup$
    – Jon Custer
    Nov 10, 2021 at 22:30
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    $\begingroup$ Answered at physics.stackexchange.com/questions/671299/… $\endgroup$ Nov 10, 2021 at 22:40
  • $\begingroup$ I'm used to read that superpositions of eigenstates varies with time (but I don't know why, given that at my level time component were alway took apart, right because looking for stationary states). In all cases, it seems to that even in the case you describe, then chemistry relies on the collapse of those eventual superposition. Ie, reacting molecules measure each others. $\endgroup$
    – Alchimista
    Nov 11, 2021 at 8:50

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There is an edit at the end trying to see why the answer is not satisfying the readers.Because I do not think what I am writing below is wrong, so I must not be answering the question.

The solution of the Schrodinger equation for the hydrogen atom, has been validated with data, because the energy levels come out of the solutions of the equation. Those energy level spectra had already been fitted with the Bohr model before the discovery that the Schrodinger equation solutions, could do an equally good job.

There is no superposition when modelling the hydrogen atom, the orbital solutions come out because of the postulates of quantum mechanics., the wavefunction being one of them. :

5.For a system described by a given wavefunction, the expectation value of any property q can be found by performing the expectation value integral with respect to that wavefunction.

So it is a probability distribution for the particular Bohr orbits, called orbitals. The individual particle modeled is a point in the probability distribution for the particular variable. For the Hydrogen atoms this gives rise to orbitals that can accept electrons.

orbital

There have been experiments where orbitals have been seen.

The energy levels themselves have a width, due to higher level interactions than just the Coulomb potential , in that sense in an experiment there is a superposition of slightly different wavefunction probabilities , that give the spectral widths, but I do not think this is the superposition you are talking about.

So in quantum mechanics, the electron is in a superposition of all those orbitals, but in chemistry, an electron only occupies ONE orbital

In quantum mechanics an electron occupies one orbital also , given by the solutions of the QM equation.

Edit

Quantum mechanics is a general theory using mathematics. To have relevance to observations,the relevant wave equation has to be solved with the specific potentials and the specific boundary conditions of a given problem. Your presumptive statement in italics:

we solve the Time Independent Schrodinger's equation by separation of variable, effectively getting the energy eigenstates of the problem, and then express any general solution as a superposition of the stationary states.

is not correct, for a particular solution the boundary conditions of the system need to be applied.

So my question is why do we not talk about the wavefunctions that may be the linear combination of two or more than two stationary states in atoms or are there conditions in chemistry under which electrons occupy only the stationary states?

Because there are very strict boundary condition that have to be applied to the general solutions , including conservation laws of quantum numbers, that result in a specific spectrum of eigenstates of the particular system, with a possible width in energy as explained above . A random superposition will not model the system of the atoms under study.

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