Electron Orbit: Exploring Kenneth W. Ford's Quantum World

[daisey]

I am currently reading Kenneth W. Ford's book entitled "The Quantum World". In chapter 2 there is a section on Length, and in that section he talks about the size of atoms and electrons. He uses an analogy where the nucleus of a Hydrogen atom is the size of a basketball, and the electron orbits the nucleus up to a mile in radius (within that same analogy).

Now, in that analogy Kenneth states the "speck" of the electron completely "fills up" the volume outside the nucleus due to the wave nature of the electron.

My question concerns his use of the term "fills up". I thought that I had learned that electrons orbit the nucleus at specific distances from the nucleus (called "shells" - two electrons max in the first shell, eight in the second, etc). And the electron can jump from one orbit to another if influenced by some outside force (and I believe absorbing or emitting a photon in the process). So, in a Hydrogen atom, since there is only one electron orbiting the nucleus, does that single electron "fill" every "shell" of the atom at once?

In other words, if the distance from an Helium nucleus to the outer portion of the atom is A, there would be many, many electrons filling many discrete orbital "shells" within this distance A. Now if this were a Hydrogen atom, would that single electron fill that same distance "A" (the entire volume) all at once? That would mean the electron is filling all of the orbital "shells" all at once, wouldn't it?

As you can see, I am confused. Maybe I am using incorrect terminology. Thanks in advance for your patience and help.

Daisey

 

[ytuab]

I am currently reading Kenneth W. Ford's book entitled "The Quantum World". In chapter 2 there is a section on Length, and in that section he talks about the size of atoms and electrons. He uses an analogy where the nucleus of a Hydrogen atom is the size of a basketball, and the electron orbits the nucleus up to a mile in radius (within that same analogy).

Now, in that analogy Kenneth states the "speck" of the electron completely "fills up" the volume outside the nucleus due to the wave nature of the electron.

My question concerns his use of the term "fills up". I thought that I had learned that electrons orbit the nucleus at specific distances from the nucleus (called "shells" - two electrons max in the first shell, eight in the second, etc). And the electron can jump from one orbit to another if influenced by some outside force (and I believe absorbing or emitting a photon in the process). So, in a Hydrogen atom, since there is only one electron orbiting the nucleus, does that single electron "fill" every "shell" of the atom at once?

Daisey

I think you have noticed a good point.
One electron can not fill every shell of the atom at once.
The wavefunction does not mean an electron itself. It means only the probability density.
Because the wave function (free field) is spreading in all space as time goes by. (which Shrodinger noticed.)

In QM, it is impossible to image the motion of electron concretely.
Bohr-Sommerfeld model is the last model (which can be imaged).

 

[daisey]

One electron can not fill every shell of the atom at once.
The wavefunction does not mean an electron itself. It means only the probability density.
Because the wave function (free field) is spreading in all space as time goes by. (which Shrodinger noticed.)

OK. Are you saying the wave function can fill every shell of the atom at once?

Assuming an electron of any atom remains in its orbital shell, I can understerstand the wave function of that atom filling the entire orbital shell at once. Be in the example above, I am talking about the entire volume of the atom outside the nucleus

 

[muppet]

The "shell" structure of an atom (as taught in secondary schools) shouldn't be thought of as a picture of an atom; the electron is not really a tiny sphere orbiting the nucleus like a planet in the solar system.
The "shells" in this description describe not the location, but the energy possesed by an electron. (In a hydrogen atom, by the electron.) An electron in a particular shell has a well-defined, precise value of energy, and is said in the language of quantum mechanics to be in a particular "state". (The full technical term is "energy eigenstate", but never mind.) An electron in such a state does not have a well-defined position; rather, it behaves like a sort of smeared-out cloud of charge. This cloud has a mathematical description -the wave function- from which you can extract the probability of finding the particle in some particular place. When you do the sums, it turns out that no matter how far away from the nucleus you get, the chance of finding the electron there is greater than zero. So in a sense, the "cloud" is infinitely big. However, above a certain distance this probability becomes negligibly small, and it becomes possible to talk about the "size" or "shape" of the cloud, by which it is meant "the size or shape of the volume within which there is a 95% chance an electron will be found should you try and measure its definite position". The latter is a bit of a gobful, so people conversant with quantum mechanics tend to talk about it using loose language and mentally translate it into something more precise when they have to.
To complicate matters slightly, you can also identify the radius at which you're most likely to find the electron. For the hydrogen atom in its lowest energy state, this distance is called the Bohr radius. This distance, however, changes depending on what shell you're in, and it is these most probable radii that are plotted in school textbooks as coincentric circles round the nucleus- the value of that diagram is to indicate that
1)The energy levels are discrete
2)You're more likely to to find more energetic electrons further away from the nucleus.
So you shouldn't think of the shells as hard spheres in space, or a wavefunction as something that "fills" them. Rather, the wavefunction is a description of a particular shell, which is best pictured as a fuzzy cloud with a blurry boundary.

For completeness: The above discussion has been simplified in two respects. Firstly, the idea of a "shell" is slightly more complicated than the 2,8,8... pattern of school chemistry. These shells are divided up into sublevels, some of which have energies very slightly higher than those you'd expect based on that simplistic grouping. The term used to describe these individual sublevels is orbital, and you can see pictures of the different orbitals at http://en.wikipedia.org/wiki/Atomic_orbital.
Secondly, the wavefunction is a more general description of the state than I've made it sound above. As well as the position distribution it describes the energy, momentum, angular momentum, etc. of your system, and it's possible to describe other states that have no definite energy but are "spread across" different energy states.

Apologies if that was a bit more complicated an answer than you'd hoped for...

 

[daisey]

Muppet,

Thanks! I think I understand now.

What your saying is that electrons (or I should say their wave functions) do not stay within a specified distance (or narrow range - i.e their "shell") from the atom when they circle the nucleus. But instead the HIGHEST probability of finding any given electron inside an atom is within a range known as a "shell", which is based in their potential energy. And there is a lower probability that same electron will be found within the atom at all other distances from the nucleus, although at a relatively smaller probability outside their "assigned shell".

In this way every electron's wave function covers the entire space within the atom, outside the nucleus.

Correct?

 

[muppet]

Muppet,

Thanks! I think I understand now.

What your saying is that electrons (or I should say their wave functions) do not stay within a specified distance (or narrow range - i.e their "shell") from the atom when they circle the nucleus. But instead the HIGHEST probability of finding any given electron inside an atom is within a range known as a "shell", which is based in their potential energy. And there is a lower probability that same electron will be found within the atom at all other distances from the nucleus, although at a relatively smaller probability outside their "assigned shell".

In this way every electron's wave function covers the entire space within the atom, outside the nucleus.

Correct?

... Sort of!

Most people would say it's more true to say that the electron doesn't have a definite distance from the nucleus. And it's not just "within the atom", but over all space. But essentially yes To be honest, nobody uses the term "shell" in advanced work- it's generally replaced by "principal quantum number", but obviously that term is completely unhelpful for anyone who doesn't already know what's going on. And anyway, in fact people usually make reference to specific orbitals- a look at those pictures might be enough to convince you that the specific sub-level can make quite a difference ...

In case you hadn't gathered, quantum mechanics is quite unlike any other idea human beings have ever taken seriously

 

[diazona]

Great explanations, muppet ;-)

One thing I would add is that - as far as I know - the radius of a "shell" in the school-textbook sense is actually the average (a.k.a. expected) radius at which we'd expect to find the electron, not necessarily the most likely value - i.e. mean rather than mode. I guess they're about the same radius for most/all orbitals though...

 

[muppet]

Shucks I think it definitely shows in the length of my posts when I've just got back from the pub... :tongue:
That's a valid point about the different definitions of "most probable". If memory serves from a 2nd year QM homework the Bohr radius is the "modally" most likely- i.e. if you find the critical values of the radial density it's minimised at the Bohr radius, and the expectation value (mean) is a little trickier to compute, and I have no idea what it is. But that's from memory, and I'm quite open to being persuaded otherwise...

Intuitively, I'd expect them two measures to differ most in discrete systems, where the average value of repeated measurements doesn't correspond to a physically existing state (e.g 3/4 hbar.w would be the expectation value of the energy of a particle in an SHO potential in a superposition of ground and 1st excited states) so my best guess would be that they coincide for an H atom, but that is merely a guess.

 

[daisey]

Wow. I really had it wrong. What your saying, though, actually makes sense.

Your saying there are really no "shells". That term is really only a representation of the electron's potential energy, and every electron's probability wave actually fills the entire atom (outside the nucleus).

Do I have it right now?

 

[RonL]

Wow. I really had it wrong. What your saying, though, actually makes sense.

Your saying there are really no "shells". That term is really only a representation of the electron's potential energy, and every electron's probability wave actually fills the entire atom (outside the nucleus).

Do I have it right now?

Hi daisey,
I'm still trying to make the mental break from the old school illustrations, but my simple logic is making it difficult for me.

Different atoms have a great number of electrons moving around them in some form of motion, I can understand the wave thought, and I like the idea of electron cloud, but in a universe that almost anything that moves is affected by something that causes a reaction, the action will be viewed as an object moving in a curve, or circle, or sphere.

An electron microscope gives images of atoms forming a sheet of gold, or atoms of aluminum on silicone, and other pictures I have seen, all show such precise spheres that I cannot comprehend electrons moving in any manor other than circular orbits.

As I stated I'm at the very beggining of learning, and thanks to wiki, my wrists no longer ache from holding those big and heavy encyclopedias.

Thanks for asking your question.

Ron

 

[daisey]

As I stated I'm at the very beggining of learning, and thanks to wiki, my wrists no longer ache from holding those big and heavy encyclopedias.

I completely agree. And thanks to Muppet for clearing this one up for me. His explanation is not really that hard to understand. What I don't understand why books on the subject retain outdated and (apparently) inaccurate descriptions (of "shells").

 

[muppet]

Wow. I really had it wrong. What your saying, though, actually makes sense.

Your saying there are really no "shells". That term is really only a representation of the electron's potential energy, and every electron's probability wave actually fills the entire atom (outside the nucleus).

Do I have it right now?

Essentially, yes.
As I referred to obliquely above, there is a sense in which "shells" exist, which is the idea of a "principal quantum number". Let me try and explain.
The wavefunction of a particle is the solution of what's called the Schroedinger equation (SE), in the same way that the classical trajectory x(t) of an object is a solution of the equation F=ma.
When you solve the SE for a hydrogen atom, you get a number of solutions which have the same total energy-both potential and kinetic. Have a look again at the table of Atomic orbital images in wikipedia. The number n in the left hand column is the principal quantum no, and the orbitals in those rows all have the same energy as the solutions of the SE.
There is a slight problem, however; one feature that makes the orbitals different from one another is their angular momentum. Classically, a rotating charge gives rise to a magnetic field (think about an electromagnet- electrons move in circular coils, which gives rise to a magnetic field directed along the axis of the coil). A similar effect holds in quantum mechanics, so for non-zero L the electron starts experiencing a magnetic field due to its own motion, which leads to a small correction to its energy which is dependent on L. In more advanced school work (the equivalent of UK A-levels, wherever in the world you happen to live) you probably will cover the existence of these "subshells", and the slight energy differences between them. (Although I very much doubt that your teacher will explain it the way I just tried to )

RonL: The pictures created by electron microscopes are based on the way objects scatter electrons. I won't profess a fantastic understanding of the process, but to the best of my knowledge the prevailing factor is the distance between the atoms, and not the shape of the electron clouds. I think the key physical process at work is diffraction rather than a charge-like interaction. So the shape of the electron clouds is largely irrelevant to the scattering image, and hence isn't reproduced in the computer-generated images. The circular images of the atoms is not a literal "picture" of the individual atoms, but is representative of slight ambiguities in their position; the key value of those images lies in the resolution of detailed structures, not the atoms themselves.

 

[Naty1]

An electron in a particular shell has a well-defined, precise value of energy, and is said in the language of quantum mechanics to be in a particular "state". (The full technical term is "energy eigenstate", but never mind.) An electron in such a state does not have a well-defined position; rather, it behaves like a sort of smeared-out cloud of charge.

This gives me the impression that energy has a more precise value than does position..Is that the intent of this comment?

How does the energy have a precise value while the position does not? Aren't such values limited by the Heisenberg uncertainty principle...

 

[daisey]

Although I very much doubt that your teacher will explain it the way I just tried to )

Muppet,

Don't have a teacher. Just books, and the smart folks here, like you. Thanks a bunch!

Daisey

 

[muppet]

This gives me the impression that energy has a more precise value than does position..Is that the intent of this comment?

How does the energy have a precise value while the position does not? Aren't such values limited by the Heisenberg uncertainty principle...

To a first approximation, the energy is usually precise.
Solving the SE for a time-independent potential yields, as it's most general solution, a superposition of energy eigenstates. The effect of the vacuum fluctuations, however, is to repeatedly "measure" the energy, in which process excited states will tend to decay. So in general, the odds are good that the atom will be in the ground state.

Now, energy eigenstates are defined by the condition that they have a definite energy. The reason you can describe the state by a wavefunction at all is that this state- thought of as a vector in hilbert space- can be expressed in a basis of position "eigenstates" (delta functions) with corresponding amplitudes $$\psi(x)$$- as x is uncountably infinite and continuous, the usual sum over states is replaced by an integral. So in the basic treatment, the energy is precise and the position isn't.

Things are slightly more complicated when your atom isn't sat in the perfect isolation of a textbook page, however. As already mentioned, vacuum fluctuations have the effect of a time-dependent perturbation to your potential, as do external sources of radiation (heat, light, etc) so that eigenstates technically no longer exist. However, if you actually carry out the perturbative treatment, the most natural way to characterise the resulting state is as a sinusoidally time-dependent superposition of eigenstates- in other words, a radiating atom oscillates smoothly between energy eigenstates as long as it's being irradiated- hence, very roughly, why we can see things in light, and not in darkness!

The interpretation of the energy-time uncertainty relation is different from that of the position -momentum uncertainty relation, as time is not a dynamical variable in a non-relativistic setting. Instead, it's related to the period for which a particular state "persists" in time. An energy eigenstate has "zero energy uncertainty" and "infinite time uncertainty", which is just a back-of-the-envelope way of reaching the conclusion that energy eigenstates do not change in time, which can be more rigourously shown using the time-evolution operator (when you see that the only change in time is a physically meaningless global phase factor).

 

[daisey]

I have seen pictures of atoms made with a Scanning Tunneling Microscope. There is actually a picture in the book that I mention in my original post in this thread. The picture makes it appear the atom has a defined exterior - almost a covering or shell. Is it safe to say this appearance of a shell is caused by the extent of the magnetic field exerted by the electron(s) orbiting the nucleus of the atom (or by the wave function of the electron)?

 

[RonL]

I have seen pictures of atoms made with a Scanning Tunneling Microscope. There is actually a picture in the book that I mention in my original post in this thread. The picture makes it appear the atom has a defined exterior - almost a covering or shell. Is it safe to say this appearance of a shell is caused by the extent of the magnetic field exerted by the electron(s) orbiting the nucleus of the atom (or by the wave function of the electron)?

Hi daisey,
It looks like we might be on the same page, to me the idea of an electron moving, it's speed of movement producing the magnitude of the field, is like a boat moving across a lake producing a wake, if it changes course and moves in the opposite direction it will be affected by it's own wake.
The electron has to move through it's own field, or the field of any other electrons in particular shells, all in a three dimensional existence.

As I have studied the last few days, I have learned a lot, but things I have learned only reveal the depth of what is still unknown by me.

One other thought in my mind is how these fields interact, the idea of transformer mechanics comes to play where there is no physical contact, but voltages and currents are modified.

I know my thoughts are simple in comparison to some of the well learned people here on PF, but I can't give up the mechanics of our macro world, even on the quantum level. Even one electron at time, distance, and area seems to be impossible to not be affected by it's own movements.

Back to wiki,

RonL