orthogonal states side-step Pauli Exclusion Principle ?

[Widdekind]

in a metal, the many many electrons' wave-functions overlap, at least when you take their modulus-squared = (charge) density distribution...

 

so, why don't such electrons repel, according to PXP ?

 

seemingly, if you propelled pairs of electrons at each other, then their wave functions would slam into each other, and of necessity overlap, at least for some mere moments in their interaction...

 

would PXP apply ?

 

or, is the PXP appropriate, to constant-energy stationary states, which are themselves "orthogonal", i.e. [math]\int d^3x \Psi^*_1(x)\Psi_2(x) = 0[/math] ? And, does that say, that (simplistically), the electrons' wave-functions never actually overlap, because at any given instant of time, the "orthogonality" of the wave-functions implies, that "whilst one electron is 'here', the other is 'there'", so that, instant by instant, the electrons are never actually co-occupying the exact same (set of) space-time event / points ? When you "smear over time", and take the squared modulus, you observe, that the electrons do "cycle through (some of) the same regions", i.e. their densities do overlap... but instant by instant, the actual probability amplitudes are "in different places" moment by moment ?

[swansont]

Who says the electrons don't repel?

[Widdekind]

if the electrons repelled, then they would propagate apart, and not continue to exist, in stationary energy-constant states

 

in stationary states, the wave-functions are unchanging (except for an energy-dependent phase factor)

[swansont]

if the electrons repelled, then they would propagate apart, and not continue to exist, in stationary energy-constant states

 

in stationary states, the wave-functions are unchanging (except for an energy-dependent phase factor)

 

And? Why can't you have a stationary state solution with repulsion? Helium exists, and the electrons repel.

[Enthalpy]

A few approximations that can mislead you...

 

Pauli's exclusion principle is not a physical force that repels particles. It's a mathematical impossibility.

 

In a metal, the accessible states for valence electrons occupy the whole crystal. "Overlap" if you wish, but that's an understatement... Every single state is at the same place as any other, with minor variations in the phase or local nodes and antinodes. Take a 1D crystal of 30 atoms: the wavefunctions with 5 and with 6 nodes over this crystal superimpose but are orthogonal, as their signs add over some places and subtract over others, the sum of the product being zero.

 

Electrons do repel an other by electrostatic repulsion, which is not Pauli's principle. But protons attract electrons, which makes atoms. Then, atoms assemble in molecules - like a piece of metal - because this permits electrons to share their room, and as more room means a longer wavelength, the electrons' kinetic energy decreases. The limit to it is Pauli's principle which prevents putting all electrons in the most favourable state, but two electrons can, which is already a covalent bond.

 

A stationary state is not necessarily the wavefunction of a particle. When an electron is trapped at one nucleus, in a metal... then its wavefunction is a weighed sum of stationary states which are orthogonal. This wavefunction is rarely stationary.

 

When an electron is trapped in a small room like one atom, the stationary states' energies are well spaced, often by many kT. This implies that, at thermal equilibrium (not during the absorption or emission of a photon for instance), all electrons occupy the lowest possible states, within the limit of Pauli's exlusion.

 

Though, as particles aren't discernable, we can only tell "these states are occupied, and these are free".

 

In a metal, states' energies are extremely close to an other - very much closer than kT. (The size limit to separated energies is near "quantum dots", which are just few atoms wide). At thermal equilibrium, states clearly lower than the occupation limit (=Fermi's level) are full, those clearly above are empty - and only those few kT away from Fermi's level have an occupation probability perceptibly different from zero or one.

 

At this small energy range, some states are occupied and others are free, with some probability. Though, we can't easily tell if one particular electron combines these states or others, because of lack of discernability. What we can tell is that if one electron has been spotted around one place with about that speed (remember Heisenberg) then we have this many chances of seen it again at that other place with that speed after that time; the few kT width of the uncertain occupation energy range tell that such electrons are broadly delocalized, because confinement in one atom size means an electron kinetic energy of several eV but kT is only 26meV at 300K. And soon, we can't tell where the spotted electron has gone, as its wavefunction is fuzzy, and we mistake it with others.

 

--------------------------------

 

Our representations of electrons in metals claim they're indepedent, and this is grossly false, obviously - I do agree. Though, better representations are hard to use...

 

Metals have about one (a bit less or few more) valence electron per atom, delocalized to the whole crystal, so the mean distance between electrons is one atom size. This implies that the mean eletrostatic repulsion between two electrons is the same as the attraction by hydrogen nucleus, like 10eV - hugely more than kT. This energy is about isotropic because every electron has neighbours in all directions, but even the variation of this repulsion energy over distance (= the force) is very steep.

 

It does not confine electrons irremediably, because tunnel effect permits them to pass by; in fact, tunnel effect by electrons is what defines an atomic radius (electrons tunnel to the "edge" of the orbital), so at similar distance and energy, electrons can tunnel by an other.

 

But because of this strong repulsion, the proper way would be to write one single psi wavefunction resulting from the positions of all the 10²³ electrons, NOT one wavefunction for individual electrons as we do.

 

This is presently done (sometimes) with the (most external) electrons of one atom or a small molecule, where electrostatic repulsion is computed for all the positionS of the electronS. Though, the result is a computer drawing instead of a usable mental representation, and isn't generalized to 10²³ electrons.

 

I have the intuition that non-individual electrons (hence opposed to the electron gas mental representation) that repel an other are the key to

- All valence electrons mobile in the Hall effect, but very few mobile in the thermal capacity of metals, which is a paradox to my understanding;

- Superconductivity.

Again, this is only an intuition.

[Enthalpy]

Instead of individual particles that collide sometimes, like molecules in a gas, electrons in a metal could form well-ordered chunks, and these chunks move against the others. This would explain that:

- many electrons move as the Hall effect tells;

- but electrons store little heat, as they are immobile within a chunk;

- conductivity increases at cold, as chunks grow, and losses occur at their boundaries only, since individual electrons have stiff "positions within" (rather: sets of eigenfunctions) a chunk

- the heat capacity increases near the transition, as chunks coalesce.

 

Weaknesses:

- No isotopic effect!

- Chunks should be very stiff to store no heat

- The heat capacity would increase over a rather broad temeprature range

 

Strengths:

- Explains why high carrier concentration eases superconductivity. Silicon becomes a superconductor with difficulty, and only at huge doping.

- Favours superconductivity in 2D atom sheets

- Electrostatic repulsion is much stronger than spin coupling invoked by BCS, so it must have some role.

 

BEWARE this is extremely speculative to say the least. I haven't even checked if this idea was already proposed (it resembles coupling through the magnetic moment) and refuted.

Marc Schaefer, aka Enthalpy

[Widdekind]

the previous posts seem thoroughly thoughtful

 

i'm simply stating, that orthogonal wave-functions "avoid" PXP, i.e. their orthogonality (integrating the product of their phases) permits the particles to co-exist (?)

 

something somewhat similar, to a time-share on a vacation condominium, one electron is "there" whilst the other is "somewhere else", crudely quasi-classically considering the meaning of their mathematical "orthogonality"

 

 

--------------------

 

[math]\Psi_1^*(r) \Psi_2(r) = (a' - \imath b') \times (a + \imath b) = (a a' + b b') + \imath (a' b - b' a)[/math]

 

overall, the only way of getting zero, is for one of the "phasors" to be zero, so that where one wave-function is, the other is not...

orthogonality seems to be a sophisticated sense of the same

[Widdekind]

electrons have two-component wave-functions, [math]\alpha \Psi_u (x) |u>[/math] + [math]\beta \Psi_d (x) |d>[/math]

 

at any place in space (xyz), you need four numbers to describe an electron's probability amplitude (R+I for spin up, R+I for spin down)

 

so, some electron, excited but bound to a proton, could conceivably exist with the following wave-function:

 

[math]\alpha \Psi_{1S} (x) |u>[/math] + [math]\beta \Psi_{2P} (x) |d>[/math]

 

and another with

 

[math]\alpha \Psi_{2S} (x) |u>[/math] + [math]\beta \Psi_{2P} (x) |d>[/math]

 

if i understand correctly, the overlap integral is

 

[math]< \Psi_1 | \Psi_2 > = |\beta|^2[/math]

 

so the two electrons' wave-functions are not orthogonal...

 

does PXP prevent two electrons from partially occupying the same (e.g. 2P-with-spin-down) state ?

@Enthalpy

 

most conduction electrons are available, for thermal conduction in metals, if i understand correctly...

 

TC is proportional to KT x Del(KT) = Cs x Cs x d(KT)/dx = speed for flux density x (dKT/dt) carried by electrons down gradient

 

i may have mis-stated something somewhere, about thermal conduction involving only electrons near the Fermi energy... but the above relations require no such sort of stuff... all electrons are involved... if they are more involved, when their metal is subject to strong EM forces, vs. passive meandering thru the metal in heat conduction, then such seems sensical

[Widdekind]

electrons have two-component wave-functions, [math]\alpha \Psi_u (x) |u>[/math] + [math]\beta \Psi_d (x) |d>[/math]

 

at any place in space (xyz), you need four numbers to describe an electron's probability amplitude (R+I for spin up, R+I for spin down)

 

so, some electron, excited but bound to a proton, could conceivably exist with the following wave-function:

 

[math]\alpha \Psi_{1S} (x) |u>[/math] + [math]\beta \Psi_{2P} (x) |d>[/math]

 

and another, bound to the same proton, existing with

 

[math]\alpha \Psi_{2S} (x) |u>[/math] + [math]\beta \Psi_{2P} (x) |d>[/math]

 

if i understand correctly, the overlap integral is

 

[math]< \Psi_1 | \Psi_2 > = |\beta|^2[/math]

 

so the two electrons' wave-functions are not orthogonal...

 

Q: does PXP prevent two electrons from partially occupying the same (e.g. 2P-with-spin-down) state ?

 

Q: Can two electrons exist, with exactly the same (say) spin-down wave-functions (but w/ differing spin-up components) ?

[swansont]

The PXP keeps two fermions from having the same state. Instead of worrying about a two state wave function, use the basis where it's in one spin state or the other. Two electrons will not both have the same n, l and m quantum numbers.

[Widdekind]

does the PXP require, that two fermions, exist in orthogonal states, [math]< \Psi_1 | \Psi_2 > = 0[/math] ?

 

and, would that requirement only apply, to stationary states, i.e. time-independent solutions ? (In some hypothetical collision, the wave-functions, of two free electrons (say), could conceivably "crash" into each other, resulting, if ephemerally, in a positive overlap integral, between both particles ?