Visualizing Electrons as Probability Densities

[stevmg]

If electrons were visualized as a clouds and not points, could this be analogous to a continuous probability density function?

In essence, for a given x there is a specific ordinate or y. The probability of achieving that particularx is zero but the probability of achieving anything between x -$$\epsilon$$ and x + $$\epsilon$$ is the area between these two points (x - tex]\epsilon[/tex] and x + $$\epsilon$$ with the total area under the curve as 1.

Am I onto anything here?

 

[Fredrik]

Yes, that's pretty close to what we actually have in QM. The wavefunction is a complex-valued function, and the square of its absolute value is a probability density. To be more precise, if $\psi:\mathbb R\rightarrow\mathbb C$ is the wavefunction, then

$$\int_a^b|\psi(x)|^2 dx$$

is the probability that a position measurement will find the particle in the interval [a,b]. If the measurement does find the particle in that interval, the wavefunction after the measurement is zero outside [a,b], so if we measure the position again, the probability of finding the particle in [a,b] is now 1.

The above is if we pretend that space is one-dimensional. For 3D space, the domain of the wavefunction is $$\mathbb R^3$$, not $$\mathbb R$$, and the integral is a triple integral over some cube or whatever region the measurement is concerned with.

 

[stevmg]

Thank you Fredrik...

Can you recommend a text or source which would get me a little deeper into the subject given that I have a Masters in Math (Statistics.) That's why I thought of the probability density function such as a normal density (but there are other pdf's) where specific abscissas have ordinates but these are not the actual probabilities, only the area in between two of the abscissas represent probability.

And also, is there a source which would explain why complex values are used.

Even in the Relativity Minkowski space there is use of $$\sqrt{-1}$$ so that we have the -c²t² term. I could not find out why this was true (the "negative Pythagorean Theorem") and when I speculate, everybody jumps one me.

Even my original post on this topic (electron cloud) was a chance I took that you smarter folks would not think of me as a dolt.

 

[stevmg]

What I originally meant by the term electron cloud is literally just that - that an electron exists as a cloud in a multitude of points, like an atmospheric cloud, which exist simultaneously.. Thus, if one were to poke at the cloud with a box of a given volume (R³) finding a piece of the cloud in the box did not mean that the cloud did not exist outside the box. The cloud was the multitude of points that existed simultaneously, not in exclusion of each other.

That's what I thought an electron cloud was and that's the only way it makes sense, to me.

Now tell me that I am whacky.

 

[Fredrik]

I can't do that. That description sounds pretty accurate to me. I guess nature is whacky.

I'm not sure what literature I should be recommending. You should probably check out one of the standard introductory textbooks. Griffiths looks good to me, but some people are saying Shankar is better, and Zettili is getting great reviews at Amazon. (I haven't actually studied any of these. My first book was Gasciorowicz, which I'm sure isn't nearly as good as any of these). You should also check out the book by Isham. I really like that one. I usually recommend that people read it on the side, while they're studying one of the standard texts. It's not really suitable as a textbook for an introductory class because it's more about what the theory actually says than about how to calculate stuff. But maybe that's a good thing for you, I don't know.

You should try to find these books at Google Books or Amazon.com, read a few pages in each, and decide for yourself if you like any of them.

I don't have a good answer for the "why complex numbers" question, but I have seen at least three threads about that topic in the past two years. Maybe there's a good answer in one of them.

I wouldn't say that there are complex numbers in SR. Any bilinear function from $$\mathbb R^4\times\mathbb R^4$$ into $$\mathbb R$$ can be expressed as

$$(x,y)\mapsto x^T M y$$

where M is a 4×4 matrix and x and y are 4×1. The Euclidean inner product is the special case defined by M=I (the identity matrix), and the Minkowski form is the special case defined by M=$\eta$, where

$$\eta=\begin{pmatrix}-1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\end{pmatrix}$$

 

[alxm]

You don't have to use complex numbers in QM (or anywhere else in physics, really). But as elsewhere where they're used, it's just the case that the math takes a form where it's convenient to do so. Also, using the existing formalism, if you're interested in the time-independent wavefunction for a system without an external (electrical/magnetic) field, then you have a real wavefunction. E.g. the wavefunction for an atom or molecule in its ground state is real (or at least, to a good approximation).

Now, about those electron clouds. The location-probability density/charge density can be described as a charge cloud, as Fredrik pointed out. This is a consequence of QM. But it shouldn't be confused with the QM description, which is the wavefunction. The probability density is the absolute-square of the wavefunction. You can derive all observable information about a system from the wavefunction, but the charge density does not contain as much information. (It has some information, most notably the energy of the system, which is the foundation of Density Functional Theory. But we don't know exactly how to calculate E from ρ yet.)

The problem with viewing the electrons as a static density-cloud is, while it's perfectly correct for how the probability of their observed locations and charge is distributed, it's very misleading when it comes to thinking about their energy. Although the 'cloud' is static, they still move! (to whatever extent you consider 'motion' to have meaning within QM; it's not classical motion) The electrons have kinetic energy, and they exhibit the dynamical effects of motion. So the fact that they don't have a definite location doesn't allow you to avoid the many-body problem inherent to electron motion.

So electrons 'move' and interact with each other as point particles. But their resulting location forms a static density cloud. It's difficult to wrap your head around it. But it's the really just another version of the infamous "wave-particle duality" confusion that underlies all of QM.

 

[stevmg]

Fredrik - Thank you for the input and book referrals. I will undoubtedly settle on one of the books.

alxm - Thank you for the explanation as to why the electron cloud is there and then it isn't (constant state of "motion.") Thus, my analogy with an atmospheric cloud doesn't entirely hold up.

Could it be analogous to the well known topological (mathematical) entity of a "non-differentiable curve.." i.e. - at every point on the curve (in this case this would mean position in space and time) the approach to such a point has a "jump discontinuity" with the path immediately exiting that point - no matter how infinitesmally close in time we "moved" from point to point such that, in effect, we jump from point to point (no matter how close the points are) with a jump discontinuity and there is no smooth or continuous path followed by the electron in time.

The only way I could conceive of such a phenomenon would be to visualize a "force" field which coalesces into matter points each instant in time but where there is such a coalescence is purely random. So the electron "field" if you will is spread out in space around the nucleus, but where and when each instantaneous point in the field "pops sup" is totally at random (but may follow some probability distribution).

Again, am I making any kind of sense?

 

[SpectraCat]

Fredrik - Thank you for the input and book referrals. I will undoubtedly settle on one of the books.

alxm - Thank you for the explanation as to why the electron cloud is there and then it isn't (constant state of "motion.") Thus, my analogy with an atmospheric cloud doesn't entirely hold up.

Could it be analogous to the well known topological (mathematical) entity of a "non-differentiable curve.." i.e. - at every point on the curve (in this case this would mean position in space and time) the approach to such a point has a "jump discontinuity" with the path immediately exiting that point - no matter how infinitesmally close in time we "moved" from point to point such that, in effect, we jump from point to point (no matter how close the points are) with a jump discontinuity and there is no smooth or continuous path followed by the electron in time.

The only way I could conceive of such a phenomenon would be to visualize a "force" field which coalesces into matter points each instant in time but where there is such a coalescence is purely random. So the electron "field" if you will is spread out in space around the nucleus, but where and when each instantaneous point in the field "pops sup" is totally at random (but may follow some probability distribution).

Again, am I making any kind of sense?

Kind of, but remember in QM, we can't really ask or answer any questions about electrons (or other QM entities) without doing a "measurement". That is, we know how the electron is distributed probabilistically, but until we ask "where is the electron *now*", or "how fast is the electron going *now*", by making measurements, we cannot discuss those discrete realities in a meaningful way.

The exception of course is asking about observables that commute with the Hamiltonian ... we can speak meaningfully about the precise energy or angular momentum of an electron in, say, a 1s orbital of an H-atom, since that orbital is an eigenstate of both the Hamiltonian, and the angular momentum operator.

However, if we ask about observables that don't commute with the Hamiltonian, such as position or momentum, or if we are talking about a superposition state (i.e. not an eigenstate of the Hamiltonian), then we have to talk about average values and probability distributions in the general case. We can also actually carry out measurements to get discrete values for a particular case, and those discrete values will be sampled from the aforementioned probability distributions.

On the other hand, we do have evidence that the electrons possesses some localized character within the orbitals, since we must account for electron correlation to properly calculate the energies of atomic and molecular states. Electron correlation can be qualitatively described as the tendency of like-charged electrons to avoid each other. Thus for any instantaneous snapshot of an atom, if we consider a particular electron, the probability of finding other electrons in its immediate vicinity is lower than the probability of finding them far away. Of course this gets very complicated very quickly when you try to consider the mutual correlation of all the electrons in the atom. The fact is that we don't know how to properly account for this behavior mathematically ... otherwise we would be able to calculate exact energies for atomic and molecular quantum states.

One important point about the above paragraph is that we tend to describe the situation "as if" the electrons were point particles, but there is no direct evidence to support that. One could equally-well think about them as probability waves, with fluctuating regions of high and low probability density. The point-particle view is the one espoused by the de Broglie-Bohm (dBB) interpretation, while the fluctuating probability wave picture is supported by the Copenhagen interpretation. So far, both approaches produce answers that are consistent with experimental measurements, so both can be considered "correct", although they do seem mutually exclusive when you get into the details.

 

[jtbell]

Kind of, but remember in QM, we can't really ask or answer any questions about electrons (or other QM entities) without doing a "measurement". That is, we know how the electron is distributed probabilistically, but until we ask "where is the electron *now*", or "how fast is the electron going *now*", by making measurements, we cannot discuss those discrete realities in a meaningful way.

At least not an a way that (most) everybody agrees on.

This gets into the realm of interpretations of QM, the subject (eventually) of probably about 90% of the posts in this forum. The bottom line is that "we" (the physics community in general) don't agree on what the "electron cloud" really "means," in terms of what the an electron (or other QM object) "really does" before we make a measurement. The various valid interpretations of QM all agree, so far, in their predictions for the results of experiments. In fact, they're constructed to lead to the standard mathematical formalism for predicting the results of experiments.

 

[stevmg]

A simplistic book by James Gleick of some 20+ years ago "Chaos" gets into the philosphilcal aspects of these general physics phenomena. His book is a "gee whiz" and "shucks, look at that!" with all kinds of weird pictures and "Holy S**t" for the lay public but there is a central theme that is daunting:
Initially he develops a chaotic picture from a simple iterated mathematical calculation such that if the observer did not know of the the simple iteration the picture would look complex and without explanation (or "chaotic") but, as stated, the basis of it was very simple. Looking at it backwards it would be very hard to pick out the simplicity from the chaotic pattern if one didn't already know it. He then goes on with a little more complex patterns showing the repeated or iterated operations and the complex or chaotic pictures that were produced.
Einstein did say that "God does not play dice with the Universe." and as such he always felt every phenomenon did have a cause and it was not random or probabilistic. Probability meant that we did not know all things about an event so we could not predict it with 100% certainty (p = 1.0). Bohr's response "Don't tell God what to do" was just a cutesy way of saying that we would never know all things about all phenomena so the "dice game" would always be apparent to us, so there would always be a p < 1.0 in prediction.
Maybe, if someone gets lucky and sees the underlying simplicity of the electron orbitals, the electron path will become predictable if one knows the underlying "driver" even though it is not known now.
Somehow, the undifferentiable curve, which would never be predictable, doesn't fit but if Bohr was right, then that is the monkey wrench that God put in the way of Einstein's cause-effect paradigm.
To those more acquainted with this, is my depiction of the everywhere undifferentiable curve as representative of this correct? Is the electron cloud a form of a 3-D or even 4-D everywhere undifferentiable curve (path)?

 

[stevmg]

I wouldn't say that there are complex numbers in SR. Any bilinear function from $$\mathbb R^4\times\mathbb R^4$$ into $$\mathbb R$$ can be expressed as

$$(x,y)\mapsto x^T M y$$

where M is a 4×4 matrix and x and y are 4×1. The Euclidean inner product is the special case defined by M=I (the identity matrix), and the Minkowski form is the special case defined by M=$\eta$, where

$$\eta=\begin{pmatrix}-1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\end{pmatrix}$$

As I look at the "fourth" Lorentz equation which maps time and distance into a cohesive bundle one does get the feel as to why distance subtracts time from travel and why time subtracts distance from travel and the so-called Minkowski 4-space formalizes it. I also get that if the (ct)²-(R³) is positive, you have a time difference, and if negative, it is a distance difference and that the complex numbers are just for ease of calculation but have no physical meaning.

This took a long time to grasp but the "world line" has more meaning now.