# FeaturedB "probability of finding the electron" (teaching QM)

1. Aug 6, 2017

### Staff: Mentor

I was trying to help (elsewhere) a student with some QM related problem and I realized something. When discussing QM we underline the fact that electron doesn't behave as a particle, it behaves as a wave. Yet when we explain wave function we say something like "square of the wave function is a density probability of finding the electron" - which seems to suggests to students the electron is a particle that can be found in a given place with a given probability. I feel like it enforces thinking in terms of electron being a pointlike object and as such doesn't help students in understanding the subject.

Won't it be better to explain these things in terms of the charge density? Unless I am missing something very fundamental, charge density and probability density of finding electron are directly proportional to each other, so we are free to choose whether we call the square of the wave function "probability density" or "charge density", aren't we?

Sadly, fact that every QM textbook I know uses the same "probability density" approach probably renders the idea DOA.

(I am actually not sure where is the best place to discuss it, feel free to move the thread).

2. Aug 6, 2017

### A. Neumaier

The problem is more complicated in that for multiple electrons, the charge density is 3-dimensional only while the absolute square of the wave function is 3N-dimensional, where N is the number of electrons.

For details on the problem see several sections in Chapters A6: The structure of physical objects and B2: Photons and Electrons of my theoretical physics FAQ.

3. Aug 6, 2017

### Staff: Mentor

In my experience, that's the point where the explanations first go astray. Saying that it doesn't behave as a particle is fine, but moving from there to "it does behave like a wave" is not logically necessary and brings in a new set of confusions. I would rather say that quantum systems behave according to rules that don't match our classical expectations especially well and then proceed to describing what these rules are.

4. Aug 6, 2017

### Staff: Mentor

Well yes, and textbooks don't do it that way. It also has the advantage of directly confronting what happened when the early pioneers applied the Klein-Gordon equation to problems and why we need QFT - which, of course, they should study at some point - but not at the start.

That said I am a heretic. After many many years of thinking about QM and reading all sorts of textbooks I now believe the following, in order of mathematical sophistication is the best way of teaching its foundations:
http://www.scottaaronson.com/democritus/lec9.html
https://arxiv.org/pdf/quant-ph/0101012.pdf
https://arxiv.org/abs/1402.6562

And don't go into this wave particle duality thing at all - more advanced books like Ballentine don't even mention it - and rightly so. QM is a theory about the statistics of observations. When not observed your guess is as good as mine - it could be a particle guided by a pilot wave, it could be travelling all paths simultaneously - I am sure you know the plethora of conjectures about (called interpretations). It's not required to actually solve problems which is I think a very important point to get across to students. Its not quite shut-up and calculate - its don't worry about it for now and take a course on decoherence, interpretations etc later if it interests you. Just like you teach calculus before analysis and say (to those students on the ball enough to see it) - yes this stuff has issues - but later with analysis is the place to address it.

Thanks
Bill

Last edited: Aug 6, 2017
5. Aug 6, 2017

### mikeyork

Unfortunately that obscures the problem of interference -- why people talked about the wave-like behavior in the first place. We cannot avoid the notion of a "probability amplitude" as the fundamental generator of such a density.

6. Aug 7, 2017

### vanhees71

One must not explain in a wrong way first and then tell the students that it was wrong but put it in the right way from the very beginning. On the quantum level you must give up thinking in terms of classical particles as well as classical fields from the very first moment. To start the QM1 lecture, of course, you have to motivate this, and thus in the first lecture one should give a brief historical overview beginning with Planck's idea about the exchange of energy (and momentum) between the electromagnetic field and charged particles in terms of "quanta" $\hbar \omega$ and $\hbar \vec{k}$ respectively. In fact that's the right picture in some sense when it comes to relativistic QFT, which is the only still valid way to define "photons", but that cannot be the aim of the QM1 lecture.

Then unfortunately you have to go on with Einstein 1905, who interpreted the em. field in terms of a kind of particles. You can discuss the photoelectric effect and the Compton effect based on this heuristic picture, but you should also tell them that this is not how modern theory describes it. Nevertheless on the heuristic level it was an important step to realize that qualitatively the em. field has both "field (wave) aspects" and "particle aspects", which brought de Broglie to the idea that also for elementary particles (in his days electrons and protons) this might be the correct quantum description. Of course, you should again emphasize right away that this socalled "wave-particle dualism" is overcome with the modern quantum theory, which will be treated in your lecture now.

The next step in this approach is Schrödinger's work on the wave equation, and you should right away start with the non-relativistic treatment of free particles. Indeed, what you suggest was Schrödinger's original idea: You interpret the wave function $\psi(t,\vec{x})$ as a kind of classical field describing the particle, where $|\psi|^2$ is the density of particles. This is of course an analogy from the em. field, where $1/2(\vec{E}^2+\vec{B}^2)$ is the energy density of the field.

However, and this must be very carefully explained, this contradicts observations, because whenever you put a photo plate in the way of the moving electron it will give you a single spot but not a smooth density distribution, i.e., despite the fact that you use a field to describe the electron, again you find a particle aspect of it. The conclusion finally is the up-to-day valid interpretation of the wave function given by Born in a footnote in his famous paper on scattering processes (1926): Appropriately normalized, $|\psi(t,\vec{x})|^2$ is the probability distribution for finding the electron at position $\vec{x}$ at time $t$.

Together with the Schrödinger equation that's the starting point for the entire modern formulation of QT. You'll find the usual operators (first in position representation) with an algebra which looks pretty similar to classical mechanics, the necessity of the wave functions forming a Hilbert space, the unitarity of time evolution (leading to the conservation of total probability, as it should be for a proper stochastic process) and so on. Finally everything ends up with Dirac's representation free formulation, operator algebras (first just the Heisenberg algebra generated from position and momentum, leading to the full non-relativistic description of a spinless particles), the commutator relations (which have their analogon in classical mechanics in the Poisson bracket formulation of Hamilton mechanics), and finally the importance of symmetry principles to get more general descriptions for particles with spin.

Then it's also natural to extend the entire business to many-body problems, including the important argument given in #2: The wave-function of an $N$-body system is a function of time and $3N$-dimensional configuration space and not some single-particle-density/continuum description. This rather follows from coarse graining in the sense of quantum-statistical mechanics, where you can derive several layers of semi-classical descriptions leading from the Kadanoff-Baym equations to transport equations and finally to ideal and viscous hydrodynamics as approximations to describe many-body systems in an efficient way, but the fundamental description is in terms of quantum theory, which is probabilistic in a fundamental way, and so far despite many attempts to overcome it over the last 9 decades, there's no working alternative formulation avoiding probabilities and statistics as fundamental ways to describe how matter behaves in Nature!

7. Aug 7, 2017

8. Aug 8, 2017

### Staff: Mentor

Well the top bit is what they tell you in say up to 10th grade science - but quite a bit isn't true. They start to rectify that in 11 and 12 with some of the other stuff it mentions - but even that is not true.

Just a couple of examples:
1. In some materials electric current is not from moving electrons - it comes from weird so called positively charge quasi particles called holes - in fact that's how transistors work.
2. Spin in general is not from the Dirac equation - its a defining property based on symmetry for fields in general.

Their is a lot of other stuff as well.

To be honest I don't know what to do about teaching QM in the early stages other than that first paper I gave the link to on quantum probability. The more you study, the more you find what was told to you earlier is wrong. Its utterly maddening but unfortunately is the way it is.

Thanks
Bill

9. Aug 8, 2017

### f95toli

It is also worth keeping in mind that there are many systems where electrons DO to a very good approximation behave as a "classical" particle. The "extreme" example would be an electron paul trap (where you can trap a single electron indefinitely) or -in the solid state- single electron transistors/pumps. Especially in the latter case most of the actual physics can be described by thinking of the electron as a particle with a reasonable well defined position and questions like "what is the probability of detecting an electron in the an interval Δx" do actually makes sense.
There are also methods for directly determining the shape/size (in time) of the electron wavepacket (see e.g. Johnson, N., et al. "Ultrafast voltage sampling using single-electron wavepackets." Applied Physics Letters 110.10 (2017): 102105)

10. Aug 8, 2017

bhobba I have just scanned through the first nine pages and i realised there some things that would not go down well with certain members here but i also think there is some pretty good stuff. As far as holes in semiconductors are concerned i think the author did make brief reference to them.

I thought that holes were simply locations of electron vacancies in the lattice created by temporary thermal release of electrons from certain sites or by the addition of doping atoms.

Last edited: Aug 8, 2017
11. Aug 8, 2017

Electrons have measurable properties such as mass and that seems to suggest, to me at least, that in terms of those properties electrons do approximate to classical particles.

12. Aug 8, 2017

### Staff: Mentor

Sorry I wasn't 100% clear - it said 'An unfilled spot in this sea acts like a positively charged electron, although it is called a hole rather than a positron'

Its the sort of thing they tell engineers in books on how transistors work.

But a little thought shows its wrong - absences of electrons going in one direction being filled up by electrons as they go is wrong - its the same as electrons travelling in the usual direction. It will not for example explain the Hall effect.

Whats going on is like phonons etc - they are examples of a purely quantum effect that makes them actually act like particles - they are not 'absences of electrons' as many articles will tell you because as I said it does not explain things - its really a nothing explanation. Here is something much better:
https://en.wikipedia.org/wiki/Quasiparticle

But you are correct - it contains some interesting and good stuff.

Thanks
Bill

13. Aug 8, 2017

### vanhees71

Sure, Thomson discovered them in cathode-ray tubes by measuring their charge-mass ratio from the motion in applied electro- and magnetostatic fields. This was, however, not as clear as it might seem, and many physicists before thought about the electrons as some kind of fields, and after the prediction of the wave properties of all kinds of particles within modern quantum theory, indeed Davisson and Germer could demonstrate these wave properties by diffraction experiments

https://en.wikipedia.org/wiki/Davisson–Germer_experiment

This clearly demonstrated the predictons of de Broglie's particle-wave conjecture and is an early demonstration of the correctness of modern quantum theory.

14. Aug 13, 2017

### Isaac0427

Respectfully, as a student, I strongly disagree with this approach. For me, I cannot work with just math and no explanation, it makes me wonder why I am doing what I am doing--I need that explanation, and "you'll learn it in another class later" is not a sufficient explanation for me whatsoever.

Perhaps my two biggest issues with Griffith's book is:
1. He introduces the Schrödinger equation with no explanation at all, or no "derivation" if you will (I know you can't "derive" the Schrödinger equation, but the way I think of it, there are steps to understanding where the equation comes from). I have trouble using an equation if I don't understand it's motivations or just why it exists at all.
2. He does not do much explaining of the wave-particle duality. This makes me ask "why are we doing this with the wave function at all?"

Focusing on number 2, which is the point of this thread (although I say number 1 just as another important note on why explanations are so crutial to me), I don't believe you are justified to use the wave function without explaining why we describe an electron/classical particle as a wave. But, that begs the exact issue the OP is talking about: we still talk about electrons as they can be localized and "found somewhere" like a particle is.

Here's how the wave-particle duality was explained to me, and I believe that it is the best way to explain it:
Waves are non-localized and have interference effects. Particles are localized and do not have interference effects. A "quantum particle" is sometimes not localized (i.e. before observation) but can be localized as well (i.e. there is a probablity of finding the quantum particle somewhere, and you can't "find" non-localized waves in that way. Or at observation, you can know exactly where the particle is, which you also cannot do with a non-localized wave). Additionally, there is an interference effect with quantum particles that does not exist with classical particles. Thus, the quantum particle has some aspects of a classical particle and some aspects of a classical wave.

We then describe the state of the particle as a wave function, but the (most popular/accepted) interpretation of the wavefunction has aspects of the classical particle (as the particle can be found somewhere at observation).

15. Aug 13, 2017

### mikeyork

Most students will have already learned classical mechanics and wave theory -- whether light or sound -- and will have seen demonstrations of interference. So explaining the similarities in quantum theory is the trivial bit. It's the non-classical stuff that must be emphasized as essential to understanding and this means strongly emphasizing that classical analogies break down and lead to false conclusions so that a jump to a new paradigm is essential. "Wave-particle duality" falls into the false conclusion camp and students must be warned against falling into that mind-set. The concept of a "state vector" and an eigenvector instead of a particle and a "probability amplitude" instead of a "wave-function" are the essentials with which learning QM must start.

16. Aug 13, 2017

### Staff: Mentor

Wait until you read Ballintine which you can tackle after Griffiths although I would read Sakuari - Modern QM first.

For example Schrodinger's equation follows from symmetry considerations - nothing to do with waves or anything like that.

The issue in QM is you need to build up to whats actually going on - it cant be given right from the start.

QM is based on two axioms elucidated in Ballintine. In fact there are more but they are so natural is hard to spot. Good exercise once you have studied the book figuring out what they are. I will tel you one - if system 1 has state |a> and 2 |b> then the combined system has state |a>|b>. Its really obvious - but is actually an assumption.

Thanks
Bill

17. Aug 13, 2017

### Staff: Mentor

As I said above it can be derived from symmetry - specifically the very simple assumption that probabilities are frame independent ie they are the same regardless of how fast you are travelling where or when an observation is made. That is so obvious no one would really question it, but strictly speaking you are invoking the principle of relativity.

Thanks
Bill

18. Aug 14, 2017

### Isaac0427

Before my replies to other comments, I realized that I answered the main point in the thread in the middle of a long explanation. So, just to clarify my response:
This is where the wave-particle duality part from post 14 (last 2 paragraphs) comes into play. As it was explained to me, the wave function comes from the wave-like properties of the electron (i.e. interference and being non-localized before observation) and the probability interpretation comes from the particle-like properties of an electron (i.e. being localized at observation and having some momentum and energy if those quantities are observed).

It really bothers me when people go into a thread and jump on other comments or go slightly off topic without first addressing the question at hand, so I figured I'd just clarify my suggestion to the OP.

Let me clarify-- I have no problem with this approach for a non-introductory course in quantum mechanics, which is what Ballintine is for. But, for an introductory course, I feel that more explanation is needed. I do believe the OP is talking about an introductory course.
While technically correct, this means nothing to introductory students. With a beginning student, all the technicalities at the start don't help. First, give a background for what you are doing; talk about Einstein's (mathematical) interpretation of the photoelectric effect and the Davisson-Germer experiment and how it implies some "wave-particle duality," and how DeBroglie's equations help explain this duality by associating wave-like properties and particle-like properties. Then explain that we capture the wave-like nature of a quantum particle by associating its state with a wave function, and that we capture the particle-like properties of a wave function with the physical interpretation of the wavefunction and probabilities and such. Then continue with the math (i.e. eigenvalues, operators, the schrodinger equation, inner products, etc.).

After you've done a bit of that, and the student is familiar with quantum mechanics, you can then go back and explain alternative/technical ways to think about quantum mechanics. However, starting with those technical ways would have confused me, and, in my opinion, would confuse many of other students.

19. Aug 14, 2017

### Staff: Mentor

But That More explanation requires the background you have yet to acquire. There is no getting around it. It must be done in steps and you unlearn and relearn as you go.

I personally believe HS physics should be calculus based on something like Essential University Physics and you study calculus in parallel with it possibly as a combined 2 subject course. The usual non calculus physics taught at HS should stop in grade 10. Then when you reach university you are prepared for the real deal, but until you have that background it simply cant be done.

Isaac, I am sorry that's the way it is - there is no changing it. Professors have arranged the teaching that way for good reason. You then come here and we tell you the non sugar coated truth because you have shown enough interest to be told the truth - but you cant fight against it - its simply the way it is. It must be done in stages - you are in the early Essential University Physics stage if that.

Here is another example. Take Newtons Laws. Law 1 follows from law 2 and law 2 is a definition. The physical content of law 3 is momentum conservation which via Noether follows from spatial symmetry ie the laws of physics are the same it doesn't matter where you are. But Newton's laws are only valid in inertial frames that have that property by definition of what an inertial frame is. It would seem totally circular. It isn't really, but requires some advanced material like Landau - Mechanics to understand whats going on. We could tell that to students - but really for them it would confuse more than illuminate - so we do not - we don't even tell it to everyone at university eg we don't tell it to engineers because they don't need to know it to do their job. But really the whole thing is a great big mess. Now for the clanger - virtually nobody picks it up - they don't nut it out that its seemingly rotten at its foundations even though its in plain sight. Science sometime is like that. Is just the way it is.

BTW if you want to discuss the Newtons Laws thing go over to the classical mechanics section even though its full resolution involves QM - but here is not the place to delve into it.

Thanks
Bill

20. Aug 14, 2017

### Staff: Mentor

But its wrong. We tell introductory students something like that but there is no getting around it, its wrong.

You have come here because you presumably are interested in math/physics. We tell you the truth - not some sugar coated version we tell beginning students. Cognate on the Newtons Laws issue I told you about.

Thanks
Bill

21. Aug 14, 2017

### Isaac0427

I do come here for the non-sugar coated version of physics--and that's my point. IMO, when you teach introductory quantum physics, you start with the "sugar coated" version. If they ask questions about it, and want to know more, give them the "real truth." While the "sugar coated" version is wrong, and later in the course the "real truth" should be explained, it is nevertheless useful, and to my knowledge there are plenty of elements of truth in there.

For example, there are properties of electrons that are fairly analogous to a classical particle, and properties that are fairly analogous to a classical wave.

If I am correct (which I may not be), the main issue with the wave-particle duality is that it places classical labels on something fundamentally non-classical, and thus any classical labels will be fundamentally misguided.

Assuming that is correct, I still think classical analogies must be used to lead into quantum physics. Once you do that, and the students are comfortable with their understanding of quantum physics, you can explain why classical analogies don't work. But, many students, including me, have trouble with the math of physics without having some intuition on it. That is why you help them develop a classical intuition, which will help them form a more non-classical mathematical intuition (I don't know if that makes sense--it does in my head).

I don't know if this would have changed had I started learning QM traditionally (i.e. with an instructor) and at an older age. I know learning changes with age, but I can give you and the OP my perspective. Take it or leave it as you may.
Just on a side note, in Michigan most high schools do not actually require that students take physics. Mine thankfully does, but the required mainstream physics is for 11th grade and it's mostly conceptual and project-based with a bit of math here and there. We do have honors and AP physics as well, but those are strictly optional (I "shadowed" honors last year and will be taking AP C this year). And, the ordinary student will take the 11th grade physics and no other physics through all of high school. I know this is not the thread to talk about this in, but I figured I'd mention it.

22. Aug 14, 2017

### Staff: Mentor

I think it's difficult to make this comparison because AFAIK nobody has tried teaching QM any other way than by starting with the "sugar coated" version (at least not on any kind of significant scale). Everybody teaches classical physics first, then introduces QM using classical analogies, and then people have to unlearn the analogies later on. But the fact that everyone has done it that way up to now does not necessarily mean it could not be done some other way. Unfortunately all of us in this discussion have already learned enough QM that we are no longer suitable subjects for any such experiment.

23. Aug 15, 2017

### vanhees71

As soon as I have to introduce it to somebody, I start thinking about the best way to do so myself, and I've not yet found the ideal way. From my own experience learning QT, I only know that one must not start with "old quantum mechanics" without clear disclaimers of its shortcomings. My own history with the greatest discovery in science since Newton is the traditional one (at least traditional in Germany): One starts at High School with the photoelectric effect and the claim it would prove the necessity of photons, i.e., a particle picture of electromagnetic radiation. This is plain wrong, and has to be unlearnt later!

Then one learns about the Bohr-Sommerfeld model of hydrogen, starting with an inconsistent picture from the very beginning: On the one hand one claims that it's just a miniature Kepler problem to be solved, ignoring the fact that accelerated charges radiate and thus loose there mechanical energy very rapidly, so that the hydrogen atom should collapse within fractions of a second, something contradicting everyday experience with the stability of matter. Then you introduce an ad-hoc assumption by Bohr (and in full mathematical glory Sommerfeld) and claim that radiation occurs only when the atom undergoes transitions from one of the Bohr-Sommerfeld states to another, and this is even a discontinuous process. Last but not least the model suggests that a hydrogen atom in its ground state should be like a little disk rather than a little sphere. All this is, of course, utterly wrong and has to be unlearnt later.

Then, to through out the kid with the bath, you learn to deny the classical-particle picture alltogether and start thinking in terms of fields, using de Broglie's and a quite superficial version of Schrödinger's brillant way to guess the correct equation (within non-relativistic theory) for these "matter waves", claiming wave-particle dualism is the right way to look at the subatomic world. This is, of course, utterly wrong, and has to be unlearnt.

Finally one is taught the modern probabilistic interpretation a la Born, and this is admittedly the most troublesome step, because one has not only to unlearn all our intuition from everyday experience with macroscopic (and thus practically classical) bodies around us but also all this outdated "old quantum mechanics", taught before. At least the latter is unnecessary and should be avoided.

My way out of this trouble so far is to start with a careful qualitative summary of the historical development starting of course from Planck (black-body radiation), Einstein (kinetic derivation of the Planck Black-Body Law with the emphasis on the necessity of spontaneous emission, which is the only hint at a true necessity for electromagnetic-field quantization at this level), de Broglie/Schrödinger (from the very beginning emphasizing the correct probabilistic interpretation a la Born). Then I use the so motivated wave mechanics (it cannot be logically derived from anything more simple since it's one representation of the very fundamental QT) to motivate the representation free formulation a la Dirac. On the way one can treat some applications (finite potential wells and steps; with the infinite potential well as a limiting case, which however is more problematic than one might think, propagation of wave packets in one-dimensional problems, the harmonic oscillator in 1 and more dimensions, the hydrogen-atom energy eigenvalue problem). Finally you give some introduction to many-body physics, introducing Bosons and Fermions and, if time allows, give a glimpse to the equivalent formulation in terms of (non-relativistic) QFT.

In QM 2, one then can treat easily scattering processes (potential scattering, two-body (elastic) scattering, more general processes), and then follows a detailed explanation of symmetry principles, from which all of a sudden QT becomes much less enigmatic compared to the more heuristical arguments used before. Particularly generic non-classical features like the spin (particularly half-integer spin) becomes natural in such a way, and it explains why (non-relativistic) QT looks the way it looks due to the Galilean space-time symmetry. Then you have a great basis to apply the theory to interesting applications, which can be more or less freely chosen to cover the interest of the students. Among them are a full quantum treatment of the Stern-Gerlach experiment, gauge invariance of electrodynamics and the Aharonov-Bohm effect, quantum-interference effects and foundational experimental tests with neutrons, an intro to condensed-matter physics with QFT methods leading to the important notion of quasi-particles, and whatever more you can think of treating in non-relativistic QT.

With this you have a very good foundation to go on and learn relativistic QT (which of course in the 21st century is relativistic QFT from day 1 on, and not old-fashioned cumbersome heuristics in terms of the early attempts to treat it in the "first-quantization formalism").

24. Aug 15, 2017

### Stephen Tashi

The classical idea of a charge density is something that simultaneously exists at various points in space while the idea of a probability density is often connected with repeated measurements of some phenomena taken at different times. The case for the point-like nature of the electron has been presented, but, in general, can we experimentally distinguish between the two types of densities?

What kind of operator does it take to define the measurement of a density that simultaneously exists over a spatial continuum?

Some statistical densities are defined by a finite number of moments. If we have operators O1 and O2 such that O1 measures the mean and O2 measures the variance, then this pair should theoretically be sufficient to measure gaussian distributions. Is that the sort of scheme that should be used?

25. Aug 16, 2017

### haushofer

If I'd teach QM, I'd start with showing them an iceberg in the water, the tip representing our daily experience. Expecting that this experience extrapolates all the way beyond atomic scales is like booking a trip to the brasilian jungle and expect the indian tribes there serving your favorite beer and watching the very same distasteful tv-shows.

Maybe that prepares the right mindset :P