Fermi hole
Encyclopedia
The idea of a Fermi hole requires some background in the idea of anti-symmetrized wavefunctions. The Pauli exclusion principle
is the "rule" that no more than two electrons can be in the same orbital. The "rule" traces to a deep algebraic property of nature that has nothing whatsoever to do with the charge on electrons. The essence is that many-electron
wave function
s must change sign when the labels on any two electrons are interchanged. This property is called antisymmetry, and its essential consequence is that electrons either stay out of one another's way, forming what is called a Fermi
hole, or clump together, forming what is called a Fermi
heap. Since electrons repel one another electrically, Fermi holes and Fermi heaps have drastic effects on the energy of many-electron atoms.
It is useful to consider an excited state of the helium atom in which electron 1 is in the 1s orbital and electron 2 has been excited to the 2s orbital. It is not possible, in principle, to distinguish electron 1 from electron 2. In other words, electron 2 might be in the 1s orbital with electron 1 in the 2s orbital. While there are 4 possible spin
states for this system, only the ones in which the spins of both electrons are aligned (pointing in the same direction) will be considered. (This is the triplet state, there exists a singlet state with the spins paired).
Because electrons are fermions, they must be antisymmetric with respect to exchange. This means that if electrons 1 and 2 are switched, exactly the same wavefunction would be obtained, but with a negative sign in front. This antisymmetry can arise either from the spin part (the intrinsic angular momentum of the electron) or the spatial part (the position of the electron as a function of r, theta, and phi) of the wavefunction. If the spatial part of the wavefunction is antisymmetric, the spatial wavefunction will look something like this (for the helium atom described above):
1s(1) 2s(2) – 1s (2) 2s(1)
where we cannot distinguish which electron is in which orbital (so we have separate terms for each case), and if we exchange the electrons, the wavefunction becomes:
1s(2) 2s(1) – 1s(1) 2s(2)
which is equal to:
- [1s(1) 2s(2) – 1s (2) 2s(1)]
Thus this spatial wavefunction is antisymmetric. Because of this result, if electrons 1 and 2 occupy exactly the same point in space, the wavefunction will vanish! Because the wavefunction squared gives the probability density for the electron, this antisymmetry means that the two electrons will never be found directly on top of each other. This gives rise to the phenomenon called the Fermi
hole – the region around an electron in which no other electron with parallel spin will come.
Fermi holes give rise to the Pauli exclusion principle
and are responsible for the space-occupying properties of matter. (This principle does not hold for bosons, which may all occupy a single state as in lasers and Bose-Einstein condensates.)
A related phenomenon, called the Fermi heap, occurs when the wavefunction antisymmetry arises from the spin part of the wavefunction, giving the spatial wavefunction (this is the singlet state described above):
1s(1) 2s(2) + 1s (2) 2s(1)
In this case, for paired spins, there is actually a slightly higher probability of finding the electrons together. Fermi heaps play an important role in chemical bond
ing by allowing both electrons to be localized in the internuclear region and thus shielding the positively charged nuclei from electrostatic repulsion with one another.
Since electrons repel one another electrically, Fermi holes and Fermi heaps have drastic effects on the energy of many-electron atoms. The most profound result is the periodic properties of the elements.
Animations of Fermi holes and Fermi heaps in the carbon atom are here. Details of the origin and significance of Fermi holes and Fermi heaps in the structure of atoms are discussed here.
Pauli exclusion principle
The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...
is the "rule" that no more than two electrons can be in the same orbital. The "rule" traces to a deep algebraic property of nature that has nothing whatsoever to do with the charge on electrons. The essence is that many-electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
wave function
Wavefunction
Not to be confused with the related concept of the Wave equationA wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of...
s must change sign when the labels on any two electrons are interchanged. This property is called antisymmetry, and its essential consequence is that electrons either stay out of one another's way, forming what is called a Fermi
Enrico Fermi
Enrico Fermi was an Italian-born, naturalized American physicist particularly known for his work on the development of the first nuclear reactor, Chicago Pile-1, and for his contributions to the development of quantum theory, nuclear and particle physics, and statistical mechanics...
hole, or clump together, forming what is called a Fermi
Enrico Fermi
Enrico Fermi was an Italian-born, naturalized American physicist particularly known for his work on the development of the first nuclear reactor, Chicago Pile-1, and for his contributions to the development of quantum theory, nuclear and particle physics, and statistical mechanics...
heap. Since electrons repel one another electrically, Fermi holes and Fermi heaps have drastic effects on the energy of many-electron atoms.
It is useful to consider an excited state of the helium atom in which electron 1 is in the 1s orbital and electron 2 has been excited to the 2s orbital. It is not possible, in principle, to distinguish electron 1 from electron 2. In other words, electron 2 might be in the 1s orbital with electron 1 in the 2s orbital. While there are 4 possible spin
Spin quantum number
In atomic physics, the spin quantum number is a quantum number that parameterizes the intrinsic angular momentum of a given particle...
states for this system, only the ones in which the spins of both electrons are aligned (pointing in the same direction) will be considered. (This is the triplet state, there exists a singlet state with the spins paired).
Because electrons are fermions, they must be antisymmetric with respect to exchange. This means that if electrons 1 and 2 are switched, exactly the same wavefunction would be obtained, but with a negative sign in front. This antisymmetry can arise either from the spin part (the intrinsic angular momentum of the electron) or the spatial part (the position of the electron as a function of r, theta, and phi) of the wavefunction. If the spatial part of the wavefunction is antisymmetric, the spatial wavefunction will look something like this (for the helium atom described above):
1s(1) 2s(2) – 1s (2) 2s(1)
where we cannot distinguish which electron is in which orbital (so we have separate terms for each case), and if we exchange the electrons, the wavefunction becomes:
1s(2) 2s(1) – 1s(1) 2s(2)
which is equal to:
- [1s(1) 2s(2) – 1s (2) 2s(1)]
Thus this spatial wavefunction is antisymmetric. Because of this result, if electrons 1 and 2 occupy exactly the same point in space, the wavefunction will vanish! Because the wavefunction squared gives the probability density for the electron, this antisymmetry means that the two electrons will never be found directly on top of each other. This gives rise to the phenomenon called the Fermi
Enrico Fermi
Enrico Fermi was an Italian-born, naturalized American physicist particularly known for his work on the development of the first nuclear reactor, Chicago Pile-1, and for his contributions to the development of quantum theory, nuclear and particle physics, and statistical mechanics...
hole – the region around an electron in which no other electron with parallel spin will come.
Fermi holes give rise to the Pauli exclusion principle
Pauli exclusion principle
The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...
and are responsible for the space-occupying properties of matter. (This principle does not hold for bosons, which may all occupy a single state as in lasers and Bose-Einstein condensates.)
A related phenomenon, called the Fermi heap, occurs when the wavefunction antisymmetry arises from the spin part of the wavefunction, giving the spatial wavefunction (this is the singlet state described above):
1s(1) 2s(2) + 1s (2) 2s(1)
In this case, for paired spins, there is actually a slightly higher probability of finding the electrons together. Fermi heaps play an important role in chemical bond
Chemical bond
A chemical bond is an attraction between atoms that allows the formation of chemical substances that contain two or more atoms. The bond is caused by the electromagnetic force attraction between opposite charges, either between electrons and nuclei, or as the result of a dipole attraction...
ing by allowing both electrons to be localized in the internuclear region and thus shielding the positively charged nuclei from electrostatic repulsion with one another.
Since electrons repel one another electrically, Fermi holes and Fermi heaps have drastic effects on the energy of many-electron atoms. The most profound result is the periodic properties of the elements.
Animations of Fermi holes and Fermi heaps in the carbon atom are here. Details of the origin and significance of Fermi holes and Fermi heaps in the structure of atoms are discussed here.