Disclaimer: I've asked a very similar question before (which I can provide if desired, but it shouldn't contain anything that is not stated here), but it was downvoted and eventually deleted for reasons unknown to me (hence, if you're going to downvote this question, please tell me why!). However, while I did not get an answer to my question, the comments did help to clarify the issue a bit, so I am now trying to phrase my question a bit more clearly. Also, I'm sorry that this turned out to be such a long question, but I felt the need to be as accurate as possible about my question, so that the same thing will not happen again.

When solving the Schrödinger equation for the coulomb potential, one usually starts by splitting the wave function into separate temporal, angular, and radial parts, which is justified by the assertion that all possible solutions to the Schrodinger equation are obtainable as linear combinations of the special ones. The angular part is then split further into two more parts, and discrete sets of solutions are obtained for both, such that, in the end, the (splittable) solutions to the angular equation are exactly the spherical harmonics, which can be uniquely parametrized by two integer numbers $l$ and $m$.

What had bothered me about this before was that, in splitting the angular equation into two parts, one has to choose an axis along which to construct the polar coordinate system, which then results in the spherical harmonics being oriented about that axis, such that in the end the choice of this axis influences the obtained set of solutions. Really, the set of solutions to the angular equation should be closed under rotation, since the equation itself is also rotationally symmetric.

The comments to my previous question have confirmed my guesses that these rotated variants of the Schrödinger equation then come up as exactly those linear combinations of the fundamental solutions obtained by splitting the angular function, such that, in the end, the solutions to the angular equation are in fact closed under rotation. However, this then means that the set of solutions is not in fact a discrete one and cannot be described only by the two integers $l$ and $m$.

This, then, is exactly the part which confuses me, since everywhere in literature it is claimed that the state of an electron inside an atom (if we approximate the potential as a coulomb potential and disregard interactions between electrons, and we assume the state to be one of definite energy (or, equivalently, time-independent) can be uniquely described by giving the four integers $n$ (which gives the energy level, or, equivalently, specifies a solution of the radial equation), $l$, $m$ (which specify the angular part), and $s$ (the spin). However, if what I was told is correct, even if we assume our electron to occupy one specific shell, and hence to factor into radial and angular parts, the state of a general electron can still not be described by these four numbers, since these apparently disregard the possible linear combinations, or equivalently, rotated versions of the discrete set of spherical harmonics.

Then again, this claim of an electron's state being uniquely described by $n$, $l$, $m$, and $s$ is then used to apply Pauli's exclusion principle in order to derive the fact that certain electron shells of an atom can only be occupied by a certain number of electrons, since there only are that many possible distinct states for the electron to be in.

This does not agree with my previous impression that there ought to be infinitely many possible states for an electron in a given shell, due to the continuous rotational symmetry. For example, according to my understanding, one could take one possible state of an electron in the second shell (whose angular part should not be constant), and rotate its angular part by small angles, each time obtaining a different wave function. According to the way that I have always heard the Pauli Exclusion Principle being stated before (two indistinguishable fermions cannot have the EXACT same wave function and spin (i.e. the same state)) it should not forbid this configuration.

This seems like a contradiction to me, so I would like to know where the error in my reasoning lies. My personal conjecture is that my idea of the Pauli Inclusion Principle for more than two particles is incorrect, and that it actually says something along the lines of the set of states of indistinguishable fermions always being linearly independent or mutually orthogonal or something similar. That would be sort of intuitive (and the 'linearly independent' version does reduce to the classical statement for two particles) and I feel like it should resolve this problem (since, if I recall correctly, the specific solutions to the spherical harmonics do form an orthogonal basis), but I haven't been able to find any such formulation of the principle anywhere (in fact, it has always been the same statement which I have given above) and I have not gotten very far trying to prove it, since I am not yet sure how one would represent the Hilbert space corresponding to a joint system of indistinguishable fermions algebraically. (In fact, I do not yet know how to even rigourously formulate indistinguishability given the Hilbert spaces, so if someone could help me with that I would be very grateful as well.)

An attempt at the final formulation of my question: If the set of possible states of electrons in a given shell of an atom is indeed infinite, how does one use the Pauli Exclusion Principle (and: what is its precise statement for more than two particles) to derive that only a limited number of electrons can be in this shell at a time?


1 Answer 1


"Really, the set of solutions to the angular equation should be closed under rotation, since the equation itself is also rotationally symmetric." this not correct. Solutions should transform as a representation of the rotation group. They need to be covariant not necessarily invariant. Only a full shell is invariant.

" there ought to be infinitely many possible states for an electron in a given shell, due to the continuous rotational symmetry" there are indeed infinitely many, but only $2(2{\cal l} +1)$ are linearly independent. You are free to choose which ones by choosing the so called quantization axis.

Once the set of linearly independent orbitals is selected, they can be filled following the exclusion principle.

Note that an accurate calculation of orbitals requires so called self consistent field theory.

  • $\begingroup$ To your first point, that might have been inaccurate wording on my part, I did not mean that every solution should be rotationally symmetric, just that a rotated version of a solution she again be a solution. I think that is what you are saying there as well. $\endgroup$ Commented Jan 22, 2019 at 21:02
  • $\begingroup$ To your second point: Does that then mean that my interpretation of the Pauli Exclusion Principle (that a collection of indistinguishable fermions cannot have linearly independent states) is (approximately) correct? At least that's what I think you are saying with choosing the quantization axis. $\endgroup$ Commented Jan 22, 2019 at 21:06
  • 1
    $\begingroup$ Linearly dependent. Probably a typo. $\endgroup$
    – my2cts
    Commented Jan 22, 2019 at 21:10
  • $\begingroup$ Has my answer helped to clarify your questions? Do you have any remaining questions? $\endgroup$
    – my2cts
    Commented Jan 22, 2019 at 21:31
  • $\begingroup$ I think it has, since the contradiction is now resolved. I also can't think of any further questions, so thanks again :) $\endgroup$ Commented Jan 22, 2019 at 21:37

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