I've been struggling to understand this: Let's say I have a gas of one million electrons. Does every single one of those electrons have a different energy (up to the degeneracy from the different momentum components)?
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3 Answers
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They have to do differ in some quantum number or another, but their energies may well be the same. To take a much simpler example, the two electrons in a helium atom have the same energy, unless a magnetic field is present.
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Avogadro constant, which is a good estimate of the order of magnitude for a macroscopic number of particles, is much bigger than a million ($N_A\approx 10^{23}$). Yet, there is nothing difficult in giving every pair of electrons (with different spins) their own momentum state, for momentum is continuous - i.e., the number of available states is infinite.
If we confine the electrons (e.g., in a box) then the spectrum becomes discrete and, for sufficiently small sizes of the confining potential the spacing between levels start become noticeable and limit the number of electrons that can be put isnide of a container - this is routinely observed in nanoscale devices (quantum wires and quantum dots), although at such small scales one cannot consider the electrons as non-interacting - this leads to Coulomb blockage in quantum dots and Luttinger liquid behavior in quantum wires.
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The Pauli principle was derived from observations of the electrons of atoms. It expresses that two electrons with the same level in the atom are distinguishable. By an external magnetic field the orientations of the spin can be manipulated.
So for an electron gas it is possible to align the electrons a bit in the direction of an external magnetic field. But this alignment of course will be disturbed at any time by the chaotic thermal movement of the gas.
answered Oct 20, 2021 at 9:52