Energy Bands of Solids: Exploring Electrons & Intuitive Reasons

[gareth]

Hi guys,

I'm researching excited electrons in metals at the moment.

I have a question which has probably being asked somewhere before but I can't find it using the search.

I understand that the Pauli excl. princ. staes that you can't have two electrons with the same quantum numbers in the same atom. I understand this falls out of the maths somewhere but is there an intuitive reason for this?

My second question is about energy bands in solids;

when atoms combine in a solid, they basically act as one big atom, so there has to be lots of states available to the electrons, hence energy bands, right? But if this is so, and the amount of energy levels is dependant on the number of atoms, if you have a huge piece of metal, will you have loads of high energy bands, perhaps high enough to escape the metal itself?

Thanks as always
gareth

 

[genneth]

As with so much of physics, the Pauli exclusion principle is something postulated based on observation. There isn't really any good theoretical reason for it having to be the case (though there's plenty to show that it is possible).

When atoms combine... you get *molecules*. The electron structure of molecules is all based around creating bonding and antibonding molecular orbitals. Wiki or google those terms --- a diagram would help a lot rather than dry text. The essence is that you get 2 bands for every atomic orbital, separated in energy. The fact that bands have a continuum of energies rather than the discrete energy levels of (small) molecules is because of the macroscopic size of lattices in solids.

 

[Modey3]

gareth,

To answer your first question. The Pauli exclusion principal states that in a quantum system (such as an atom) you cannot have two electrons in the same quantum state ( same n,l,m and spin quantum states). What this really means is that two electrons with the same spin cannot be described by the same probability distribution. The primary reason is because of coulombic repulsion of the electrons. If the electrons have different spins the coulumbic repulsion is offset by the spin coupling. As my graduate adviser always says: "systems always try to go to the lowest state of energy". Think about the particle in the box, it has an infinite number of solutions, but each of those solutions will be a stationary state (i.e dE/d(psi) = 0). You might ask, in terms of energy, what would happen if two electrons (disregarding their spin component) in the particle in the box model were in the same quantum state say n1 = n2 = m. The result would be that the system would be in a higher state of energy than if the electrons were in different states say n1=m and n2=n. The reason being that the coulumbic potential energy component will be higher in this state when n1 = n2 = m than when n1=m and n2=n. Think of quantum systems in terms of energy and the Pauli exclusion principal will become more intuitive. I hope this helps.

Modey3

 

[gareth]

Thanks for the helpful replies folks,

So certain energy states are more favorable than others because the system is trying to settle to its lowest energy value.

But why would there be more coulombic potential between two electrons in the same state than two electrons different states, it it because they will be have a higher probality of being closer together when they are in the same state?

Thanks again
gareth

 

[genneth]

What this really means is that two electrons with the same spin cannot be described by the same probability distribution. The primary reason is because of coulombic repulsion of the electrons.

Woah, stop just a minute there. The fermionic statistics of electrons have nothing to do with the mutual coulomb repulsion. Two electrons can't be in the same quantum state, period. It wouldn't change if for some reason electrons attracted instead of repelled. There is no *reason* why this is so --- since things like photons can be in the same state --- it's just an observed phenomenon, and incorporated into the mathematics in various ways.

I repeat: electrons cannot be in the same state, because *we observe them to not*. There is no further explanation at this current time.

 

[Gokul43201]

The Exclusion Principle is not an artifact of Coulomb repulsion. Neither is it a solely empirical observation (see Spin-Statistics Theorem).

What is true however, is that the Heisenberg Hamiltonian for spin-spin interactions is an artifact of Exclusion and Coulomb repulsion.

 

[gareth]

OK,

So if we just take it for granted that the uncertainty principle holds true, what actually defines a quantum system?

I assume there are two electrons somewhere in the universe with the same quantum numbers, but under what specific circumstances can these two electrons be subject to this principle and have to change their quantum description (EG spin)?

Is it when they are confined in the same solid?

Or can they be just in close proximity?

Thanks

 

[Modey3]

Ok. If the exclusion principal is purely a law of quantum mechanical behavior it doesn't take away from the fact that there must be a energetic explanation for why the exclusion principal exists. Consider the particle in the box model. There must be some sort of repulsion that keeps two electrons from being in the same state. After all, what is really an electronic state? It's a probability distribution for an electron nothing more nothing less. If I do recall the Exchange-Energy in the Hartree-Fock model takes into account the exclusion principal indirectly.

 

[genneth]

Ok. If the exclusion principal is purely a law of quantum mechanical behavior it doesn't take away from the fact that there must be a energetic explanation for why the exclusion principal exists. Consider the particle in the box model. There must be some sort of repulsion that keeps two electrons from being in the same state. After all, what is really an electronic state? It's a probability distribution for an electron nothing more nothing less. If I do recall the Exchange-Energy in the Hartree-Fock model takes into account the exclusion principal indirectly.

No. There is only the fact that two electrons just plain aren't allowed to be in the same state. It doesn't even make sense to talk about the energy of two electrons in the same state, because that is not a valid state of the system. There are various ways in which this condition is enforced mathematically, but they are all just mathematical ways to say the same thing.

The exchange energy is only peripherally related. The exchange-energy is taking into account the fact that electrons also have spin, which introduces extra degrees of freedom. Thus if two electrons are spin aligned, then their spatial wavefunctions are anti-symmetric and that a lower energy configuration than if their spins were anti-aligned and their spatial wavefunctions symmetric.

 

[gareth]

So the probabilty distribution describes the chances of finding an electron in a certain place. Is this determined by the quantum numbers? I was under the impression that the quantum numbers just defined the energy state of the electron.

In general the impression I'm getting is that the exclusion principle only comes into play in atoms because of the restricitve nature of the allowed energy bands, whereas in a free gass of electrons the chances of finding two electrons with the same energetic description is low, and the chances of finding those two electrons in close proximity is even lower. So at the end of the day it can be decribed by normal kinetic equations (Boltzmann distribution).

Is this any closer to reality?

 

[genneth]

So the probabilty distribution describes the chances of finding an electron in a certain place. Is this determined by the quantum numbers? I was under the impression that the quantum numbers just defined the energy state of the electron.

In general the impression I'm getting is that the exclusion principle only comes into play in atoms because of the restricitve nature of the allowed energy bands, whereas in a free gass of electrons the chances of finding two electrons with the same energetic description is low, and the chances of finding those two electrons in close proximity is even lower. So at the end of the day it can be decribed by normal kinetic equations (Boltzmann distribution).

Is this any closer to reality?

The basic structure of QM is that states are vectors in a Hilbert space. Thus there are only as many truly distinct states are there are dimensions in the Hilbert space. You can then choose to label these states in any way you wish. Quite often however, there are measurements which are useful, like energy, or position; these correspond to Hermitian operators on the Hilbert space, and have eigenstates which span the entire space. It is then useful to label these states with their eigenvalues. We call these (for historical reasons) "quantum numbers". Note that because eigenvectors can be degenerate, it is useful to pick a set (rather than just one) of commuting operators, so that you can assign unique labels to the distinct states.

When you combine two systems together, you use the tensor product. However, we observe exclusion, so if you combine two identical systems for representing one fermion each together, you have to "knock out" certain dimensions, because of the anti-symmetry requirement.

When considering an atom, we usually neglect the motion of the atom itself, and then we get a fairly small Hilbert space, labelled by things like n, l, m_l, spin, etc. A free electron is labelled by things like momentum and spin. Now momentum is a continuum, so you get a truly vast number of dimensions. But the principle is still the same. You can't have two electrons with the same momentum and spin. We usually don't label free electron states by their energy because for a given energy you still have a huge degeneracy in terms of which direction the momentum is.

In a free electron gas, you still have to worry about the exchange statistic, except in the dilute, high temperature limit. In that (one) case, as you said, the occupation of states tends towards Boltzmann from the strictly correct Fermi-Dirac.

 

[Modey3]

gareth,

So the probabilty distribution describes the chances of finding an electron in a certain place. Is this determined by the quantum numbers? I was under the impression that the quantum numbers just defined the energy state of the electron.

Yes, the quantum number defines both the wave-function (and the hence the probability distribution) and the energy of that system.

In general the impression I'm getting is that the exclusion principle only comes into play in atoms because of the restricitve nature of the allowed energy bands, whereas in a free gass of electrons the chances of finding two electrons with the same energetic description is low, and the chances of finding those two electrons in close proximity is even lower. So at the end of the day it can be decribed by normal kinetic equations (Boltzmann distribution).

Lets say you have 100 isolated quantum systems each at the same temperature. If you sampled the energies of each system simultaneously then the energy distribution would obey Boltzmann statistics. In systems where you have multiples electrons you have to use Fermi-Dirac statistics to take into account the exclusion principal.

 

[gareth]

Genneth, you lost me a bit with the mathematics there but the bottom line as in your post states;

In a free electron gas, you still have to worry about the exchange statistic, except in the dilute, high temperature limit. In that (one) case, as you said, the occupation of states tends towards Boltzmann from the strictly correct Fermi-Dirac.

So you say a dilute, high temperature gas can be described by the classical theory, is their a general marker in science to indicate when you have to change theories?

I.e at what temperature and what diluteness?

So...

What the exclusion principle is really telling us is that you can't have two electrons in the same place at the same time, but they can have the same energies, either at different times or different places.

 

[genneth]

The heuristic (i.e. not rigorous) rule is that quantum statistics start being important when the particle separation is less than the de Broglie wavelength of the particles.

There is no good way to explain QM if you use classical concepts. For instance, your statement "you can't have two electrons in the same place at the same time, but they can have the same energies, either at different times or different places" is not entirely wrong, but not entirely correct either. The only way to understand QM is to just immerse yourself in the formal maths and structure, and then see how classical behaviour emerges from that; rather than trying to shape classical concepts into something that approximates the QM description.

 

[gareth]

Thanks Genneth,

I understand that it is futile to try and understand QM classically, but is it true that the exclusion principle is worked into the maths of QM and not that it is derived theoretically from other features of QM?

In other words, is there anything other than our experimental observations that suggest exclusion is a law of nature?

 

[genneth]

Thanks Genneth,

I understand that it is futile to try and understand QM classically, but is it true that the exclusion principle is worked into the maths of QM and not that it is derived theoretically from other features of QM?

In other words, is there anything other than our experimental observations that suggest exclusion is a law of nature?

There are theoretical arguments from relativistic quantum mechanics -- the spin-statistics theorem alluded to above. It basically states that under some assumptions, particles are either fermions or bosons, and this is linked to their spin. This can be regarded as "deriving" the exclusion principle. Personally (and I do emphasize this), I hold the exclusion principle as an observed phenomenon in higher status and more solid ground than QFT arguments.