Question about quantum numbers:

In summary, the conversation discusses the relationship between the set of quantum numbers (n, l, ml, ms) and the symmetrization of wavefunctions for two identical fermions. The participants question how close two electrons must be for the effects of Pauli's exclusion principle to become noticeable, and whether the wavefunction must always be symmetrized even when the particles are far apart. They also consider the analogy of a double well system to understand the effects of symmetrization. Ultimately, it is concluded that the wavefunction for a two fermion system must always be symmetrized, even when the particles are far apart, and this is due to the fact that there is only one electron field in the universe.
  • #1
VortexLattice
146
0
So, you got n,l,m_l,m_s. If you have two hydrogen atoms on opposite ends of the universe, it seems to me that the electron in each could have the set of quantum numbers (1,0,0,1/2).

Now, if you have a helium atom, because its two electrons are identical fermions, by the PEP they now "know" about each other, and can't have the same set of quantum numbers... So how close do two electrons have to get before this happens?
 
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  • #2
I have been wondering this same thing! Yet I still haven't gotten a straight answer from anyone.
 
  • #3
My friend thinks it's that the "same quantum state" includes the spatial part of the function as well, which kind of makes sense. But I'm still not convinced.
 
  • #4
The quantum numbers n, l, ml, ms are origin-dependent, and so the states labeled by these numbers for atom 1 are not the same states as the ones labelled by the same numbers for atom2.

The antisymmetry is completely unimportant as long as the atoms are far enough apart that the wavefunctions barely overlap. Bring them closely together (an atomic diameter, say) and the fact that the total wavefunction must be antisymmetric starts to matter.
 
  • #5
Bill_K said:
The antisymmetry is completely unimportant as long as the atoms are far enough apart that the wavefunctions barely overlap. Bring them closely together (an atomic diameter, say) and the fact that the total wavefunction must be antisymmetric starts to matter.

This has always bothered me though. How does Pauli "know" when the critical overlap is!? There is always finite overlap, however minuscule. How does more overlap work it's way into symmetrization?
 
  • #6
In place of the 3-D atoms, consider a simpler case: the double well, a pair of identical 1-D wells a distance R apart. This is a standard example covered in QM courses. When the wells are far apart the energy levels are essentially the same as for a single well, and the states are essentially independent of each other. Each well appears to be occupied as if the other did not exist. There's a twofold degeneracy.

But in reality there's a single wavefunction for both wells, and the true eigenstates are symmetric and antisymmetric. And each level that appeared to be twofold degenerate is slightly split: the symmetric function has one fewer node, and is therefore slightly lower in energy than the antisymmetric one. As the wells are moved closer together this difference becomes more and more noticeable. Very crudely, the increase in energy for the antisymmetric state can be thought of as a 'repulsion' caused by the Pauli principle.

Now if the particle occupying the wells is a fermion, the antisymmetric states are the only ones, and the wavefunction is always antisymmetric. For large R the effect of this (the energy level shift) is small and the wells appear to be independently occupied. But as R is decreased the Pauli effect becomes more and more noticeable.
 
  • #7
So while it might seem pedantic, the wavefunction for a two fermion system, even when the fermions are far apart, must still be symmetrerized? If I had the technological power to build a machine, I could measure a difference in the quantum numbers of two fermions a meter apart because of spin-statistics and quantum number splitting?
 
  • #8
Yes in principle, there is only one electron field, and every electron in the universe is an excitation of that field and is antisymmetric with every other one. :cool:
 
  • #9
Thank you :approve:
 
  • #10
Bill_K said:
Yes in principle, there is only one electron field, and every electron in the universe is an excitation of that field and is antisymmetric with every other one. :cool:

http://mlkshk.com/r/DBLT
 
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FAQ: Question about quantum numbers:

What are quantum numbers?

Quantum numbers are a set of four values that describe the energy state of an electron in an atom. These values include the principal quantum number, azimuthal quantum number, magnetic quantum number, and spin quantum number.

What is the principal quantum number?

The principal quantum number, denoted by the symbol "n", describes the energy level or shell of an electron in an atom. It can have integer values ranging from 1 to infinity, with higher values indicating higher energy levels.

What is the azimuthal quantum number?

The azimuthal quantum number, denoted by the symbol "l", describes the shape of the electron's orbital. It can have values ranging from 0 to n-1, with different values corresponding to different subshells (s, p, d, f).

What is the magnetic quantum number?

The magnetic quantum number, denoted by the symbol "ml", describes the orientation of the electron's orbital in space. It can have values ranging from -l to +l, with 2l+1 possible values for each subshell.

What is the spin quantum number?

The spin quantum number, denoted by the symbol "ms", describes the spin of the electron. It can have two possible values: +1/2 (spin up) or -1/2 (spin down).

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