# of e in atoms

[steevey]

Although I'm pretty sure there's another equation for this which I can't remember, I noticed that in an energy level of say, 1, or the ground state, the wave of a single electron has 1 crest, and one troff, or two vertices total, and the amount of electrons that are allowed in that energy state is 2. Then I go to the second energy state, the wave of a single electron with its maximums and minimums has 4 vertices, and the energy level is 2, and 8 electrons are allowed. At the third energy level, there are 6 maximum and minimum vertices, and the energy is 3, and there are 18 electrons allowed in that state. It would appear as though the amount of electrons allowed in a single state is the number of vertices of a single electron wave in that state multiplied by "n" or the energy level itself. Or simply, "[math]2n^2[/math]"

 

But, I'm not really sure why this occurs, or why it occurs the way it does.

Edited March 3, 2011 by steevey

[mississippichem]

Although I'm pretty sure there's another equation for this which I can't remember, I noticed that in an energy level of say, 1, or the ground state, the wave of a single electron has 1 crest, and one troff, or two vertices total, and the amount of electrons that are allowed in that energy state is 2. Then I go to the second energy state, the wave of a single electron with its maximums and minimums has 4 vertices, and the energy level is 2, and 8 electrons are allowed. At the third energy level, there are 6 maximum and minimum vertices, and the energy is 3, and there are 18 electrons allowed in that state. It would appear as though the amount of electrons allowed in a single state is the number of vertices of a single electron wave in that state multiplied by "n" or the energy level itself. Or simply, "[math]|2n^2|[/math]"

 

But, I'm not really sure why this occurs, or why it occurs the way it does.

 

In short, the Pauli exclusion principle.

 

-for n=1, there is only an s-orbital

-for n=2 there is an s-orbital and a p-orbital

 

The pattern continues:

 

1| s

2| s p

3| s p d

4| s p d f

 

...and so on.

 

Every orbital can hold two electrons. The angular momentum number "[math] \ell [/math]" defines which type of orbital, and for every level n, the allowed [math] \ell [/math] numbers range from 0 to n-1.

 

So for an s-orbital [math] \ell=0 [/math], for a p-orbital [math] \ell=1 [/math] and so on. You can use this to generate that pattern shown above. Then all you have to know is that s-orbitals are non-degenerate and come by themselves, p-orbitals are triply degenerate and come in groups of 3, d-orbitals come in groups of 5, and f-orbitals come in groups of 7. So for example, a set of d-orbitals can hold 10 electrons.

 

The Wikipedia article is a good place to inquire further: Wikipedia: Electron Configuration

[Schrödinger's hat]

I think you're confusing a square well with an atomic 3d well. It's not so much about energy maxima, as the other quantum numbers. The Pauli exclusion principle states that two fermions can't have exactly the same state.

As the energy levels increase, it opens up more variations for angular momentum and magnetic moment numbers. Spin can always be either positive or negative half, so we wind up with:

n is the energy level

angular momentum l can range from 0 up to n-1

magnetic moment m can range from -l to l

spin can always take two values.

So we have for each l value

[math] (2l+1)*2 [/math]

 

or [math]\sum_{l=0}^{n-1}2*(2*l+1)=2*n^2[/math]

 

This is related to what values and wave functions are valid solutions to the Schrödinger equation.

 

Hah, beat me while I was remembering how to sum a simple series.

We can tell who is a chemist and who's a physicist by the time it took me to remember how atoms work.

I guess, in order to avoid confusion I should talk about m a bit.

The s p d f are just the chemists' annoying convention for things that should be numbers *grumbles* (there's actually a good reason involving spectrometers, and a lack of quantum physics at the time for this convention -- but I digress)

Note that mississippichem said that some orbitals are degenerate.

This is true if there is no magnetic field.

The numbers he stated match the 2(l+1) that I was talking about -- s is l=0, so 1 state, p is l=2, 2*2+1=5 and so on-- because they are the number of different values for the magnetic moment.

Understanding exactly what m is is hard and requires staring at a vector diagram for a couple of hours, but you can think of it as the way in which the angular momentum interacts with a magnetic field when it is present.

Even if it there is no magnetic field, they are still different states, so the pauli exclusion principle doesn't exclude them

[steevey]

Ok, I know the pauli exclusion principal doesn't allow matter to have all the same properties in the same system, but I want to know the physical reason why I couldn't have 3 electrons in the first orbital or 300. Would they just repel each other that much? What physical thing is going on to determine the exact number of electrons in a specific orbital?

 

 

If they have greater energies though, shouldn't their positions be more determined too?

Edited March 3, 2011 by steevey

[Schrödinger's hat]

With systems governed by the Schrödinger equation, the Pauli exclusion principle is simply taken as a postulate, with regards to that theory, asking why is akin to asking why F=ma of Newtonian mechanics.

There are reasons once you get into relativistic and non-linear theories, it is related to the principles of symmetry. I do not understand the standard model yet, and explaining this properly is going to be out of my league for at least another year. Even then I think I would struggle to put it in terms you would understand without studying a lot of maths.

 

With regards to your second question, you're thinking of the de Broglie wavelength. This only applies to matter waves which are not bound, it also relates to the momentum more than energy (energy is related to momentum, so wavelength is related to kinetic energy in this way). The uncertainty principle does apply in a way, the states are quite well determined energetically, this means that the timing of a photon leaving during an electron transition is well determined, and the photon is consequently close to monochromatic.

[steevey]

With systems governed by the Schrödinger equation, the Pauli exclusion principle is simply taken as a postulate, with regards to that theory, asking why is akin to asking why F=ma of Newtonian mechanics.

There are reasons once you get into relativistic and non-linear theories, it is related to the principles of symmetry. I do not understand the standard model yet, and explaining this properly is going to be out of my league for at least another year. Even then I think I would struggle to put it in terms you would understand without studying a lot of maths.

 

From what it sounds like in this context, your talking about the symmetry of when a wave enters negative values but still appears in reality as though it had positive values. Although the reason this would occur isn't because an electron becomes negatively probable, its just the distance from 0. I could draw a line in the same and distinguish positive and negative sides. If I suddenly step on the negative side, do I suddenly become negative matter? Nope.

 

This is also related to superpositions, and I already know math can describe these systems of exclusion, but I want to know why in reality it appears that way.

Edited March 3, 2011 by steevey

[mooeypoo]

I think you might be misunderstanding what the Pauli Exclusion principle is.

 

http://en.wikipedia....usion_principle

 

You can read a bit about the derivation (a simplistic one, but generally helpful) in the "Connection to quantum state symmetry" section of that article. Symmetry, also, has a few "shapes" to it; try reading a bit about the difference between bosons and fermions, for instance.

 

Boson particles obey Bose-Einstein statistics, while Fermions go by Fermi-Dirac statistics. They differ in behavior and in the type of symmetry that they can have. Fermions are restricted by the Pauli exclusion principle.

 

Electrons are fermions, which are related to an "antisymmetric" wave function: no two fermions can occupy the same quantum state at the same time. Either they have the same orbital but different spins or they have the same spin but different orbitals.

 

Here's another good article to start from: http://en.wikipedia.org/wiki/Fermion

 

If you want to know why this is, you should look at where the Pauli Exclusion principle came from. I can't find an online source for the derivation, but many printed quantum physics books have the explanation of where this came from and why. I have Griffith's "Introduction to Quantum Mechanics" and there's a whole chapter about it with the mathematical principles as well.

 

~moo

 

P.S, this is another good source to read about superposition and spin-states of fermions and bosons: http://en.wikipedia.org/wiki/Spin-statistics_theorem

[steevey]

I think you might be misunderstanding what the Pauli Exclusion principle is.

 

http://en.wikipedia....usion_principle

 

You can read a bit about the derivation (a simplistic one, but generally helpful) in the "Connection to quantum state symmetry" section of that article. Symmetry, also, has a few "shapes" to it; try reading a bit about the difference between bosons and fermions, for instance.

 

Boson particles obey Bose-Einstein statistics, while Fermions go by Fermi-Dirac statistics. They differ in behavior and in the type of symmetry that they can have. Fermions are restricted by the Pauli exclusion principle.

 

Electrons are fermions, which are related to an "antisymmetric" wave function: no two fermions can occupy the same quantum state at the same time. Either they have the same orbital but different spins or they have the same spin but different orbitals.

 

Here's another good article to start from: http://en.wikipedia.org/wiki/Fermion

 

If you want to know why this is, you should look at where the Pauli Exclusion principle came from. I can't find an online source for the derivation, but many printed quantum physics books have the explanation of where this came from and why. I have Griffith's "Introduction to Quantum Mechanics" and there's a whole chapter about it with the mathematical principles as well.

 

~moo

 

P.S, this is another good source to read about superposition and spin-states of fermions and bosons: http://en.wikipedia....tistics_theorem

 

I think there's multiple contexts for symmetry. One can be used to describe a wave function equation, if you add the wave functions of two different particles instead of subtracting them, its symmetrical. The shape of an s orbital is symmetrical. And then there's also super-symmetry which is something about equivalent fermions and bosons which differ in spins or something like that.

[mooeypoo]

I think there's multiple contexts for symmetry. One can be used to describe a wave function equation, if you add the wave functions of two different particles instead of subtracting them, its symmetrical. The shape of an s orbital is symmetrical. And then there's also super-symmetry which is something about equivalent fermions and bosons which differ in spins or something like that.

 

There are multiple contexts because it's a word, and you can use it differently. This, however, is a physical context - a very specific one. If you're mixing terms from different contexts, and instead of helping you understand the principle of your question, you just get more confused; Supersymmetry has nothing to do with this.

 

For that matter, "power" has many contexts too. That doesn't mean that I should start treating the horse-POWER in my car as a super-POWER to calculate how fast my car can go up a hill, or that if I want to understand what electric power is, I should treat it as the power of a point.

Be careful of mixing terminology from different contexts. It will just confuse you.

 

Try not to mix linguistics and physics.

 

 

 

If you add the wave functions instead of subtracting (which is a bit of simplification since you also switch terms, but okay) then in the case of electrons, you will need to have different spins to "break" the symmetry. Because that's how Fermions work. Take a look at the spin statistics theorem and at what Fermions are again.

http://en.wikipedia.org/wiki/Spin-statistics_theorem

 

~moo