Diffraction and the Uncertainty Principle (II)

In my previous post, I derived and explained the general formula for the pattern generated by a light beam going through a slit or a circular aperture: the diffraction pattern. For light going through an aperture, this generates the so-called Airy pattern. In practice, diffraction causes a blurring of the image, and may make it difficult to distinguish two separate points, as shown below (credit for the image must go to Wikipedia again, I am afraid).

Airy_disk_spacing_near_Rayleigh_criterion

What’s actually going on is that the lens acts as a slit or, if it’s circular (which is usually the case), as an aperture indeed: the wavefront of the transmitted light is taken to be spherical or plane when it exits the lens and interferes with itself, thereby creating the ring-shaped diffraction pattern that we explained in the previous post.

The spatial resolution is also known as the angular resolution, which is quite appropriate, because it refers to an angle indeed: we know the first minimum (i.e. the first black ring) occurs at an angle θ such that sinθ = λ/L, with λ the wavelength of the light and L the lens diameter. It’s good to remind ourselves of the geometry of the situation: below we picture the array of oscillators, and so we know that the first minimum occurs at an angle such that Δ = λ. The second, third, fourth etc minimum occurs at an angle θ such that Δ = 2λ, 3λ, 4λ, etc. However, these secondary minima do not play any role in determining the resolving power of a lens, or a telescope, or an electron microscope, etc, and so you can just forget about them for the time being.

geometry

For small angles (expressed in radians), we can use the so-called small-angle approximation and equate sinθ with θ: the error of this approximation is less than one percent for angles smaller than 0.244 radians (14°), so we have the amazingly simply result that the first minimum occurs at an angle θ such that:

θ = λ/L

Spatial resolution of a microscope: the Rayleigh criterion versus Dawes’ limit 

If we have two point sources right next to each other, they will create two Airy disks, as shown above, which may overlap. That may make it difficult to see them, in a telescope, a microscope, or whatever device. Hence, telescopes, microscopes (using light or electron beams or whatever) have a limited resolving power. How do we measure that?

The so-called Rayleigh criterion regards two point sources as just resolved when the principal diffraction maximum of one image coincides with the first minimum of the other, as shown below. If the distance is greater, the two points are (very) well resolved, and if it is smaller, they are regarded as not resolved. This angle is obviously related to the θ = λ/L angle but it’s not the same: in fact, it’s a slightly wider angle. The analysis involved in calculating the angular resolution in terms of angle, and we use the same symbol θ for it, is quite complicated and so I’ll skip that and just give you the result:

θ = 1.22λ/L

two point sourcesRayleigh criterion

Note that, in this equation, θ stands for the angular resolution, λ for the wavelength of the light being used, and L is the diameter of the (aperture of) the lens. In the first of the three images above, the two points are well separated and, hence, the angle between them is well above the angular resolution. In the second, the angle between just meets the Rayleigh criterion, and in the third the angle between them is smaller than the angular resolution and, hence, the two points are not resolved.

Of course, the Rayleigh criterion is, to some extent, a matter of judgment. In fact, an English 19th century astronomer, named William Rutter Dawes, actually tested human observers on close binary stars of equal brightness, and found they could make out the two stars within an angle that was slightly narrower than the one given by the Rayleigh criterion. Hence, for an optical telescope, you’ll also find the simple θ = λ/L formula, so that’s the formula without the 1.22 factor (of course, λ here is, once again, the wavelength of the observed light or radiation, and L is the diameter of the telescope’s primary lens). This very simple formula allows us, for example, to calculate the diameter of the telescope lens we’d need to build to separate (see) objects in space with a resolution of, for example, 1 arcsec (i.e. 1/3600 of a degree or π/648,000 of a radian). Indeed, if we filter for yellow light only, which has a wavelength of 580 nm, we find L = 580×10−9 m/(π/648,000) = 0.119633×10−6 m ≈ 12 cm. [Just so you know: that’s about the size of the lens aperture of a good telescope (4 or 6 inches) for amateur astronomers–just in case you’d want one. :-)]

This simplified formula is called Dawes’ limit, and you’ll often see it used instead of Rayleigh’s criterion. However, the fact that it’s exactly the same formula as our formula for the first minimum of the Airy pattern should not confuse you: angular resolution is something different.

Now, after this introduction, let me get to the real topic of this post: Heisenberg’s Uncertainty Principle according to Heisenberg.

Heisenberg’s Uncertainty Principle according to Heisenberg

I don’t know about you but, as a kid, I didn’t know much about waves and fields and all that, and so I had difficulty understanding why the resolving power of a microscope or any other magnifying device depended on the frequency or wavelength. I now know my understanding was limited because I thought the concept of the amplitude of an electromagnetic wave had some spatial meaning, like the amplitude of a water or a sound wave. You know what I mean: this false idea that an electromagnetic wave is something that sort of wriggles through space, just like a water or sound wave wriggle through their medium (water and air respectively). Now I know better: the amplitude of an electromagnetic wave measures field strength and there’s no medium (no aether). So it’s not like a wave going around some object, or making some medium oscillate. I am not ashamed to acknowledge my stupidity at the time: I am just happy I finally got it, because it helps to really understand Heisenberg’s own illustration of his Uncertainty Principle, which I’ll present now.

Heisenberg imagined a gamma-ray microscope, as shown below (I copied this from the website of the American Institute for Physics ). Gamma-ray microscopes don’t exist – they’re hard to produce: you need a nuclear reactor or so 🙂 – but, as Heisenberg saw the development of new microscopes using higher and higher energy beams (as opposed to the 1.5-3 eV light in the visible spectrum) so as to increase the angular resolution and, hence, be able to see smaller things, he imagined one could use, perhaps, gamma-rays for imaging. Gamma rays are the hardest radiation, with frequencies of 10 exaherz and more (or >1019 Hz) and, hence, energies above 100 keV (i.e. 100,000 more than photons in the visible light spectrum, and 1000 times more than the electrons used in an average electron microscope). Gamma rays are not the result of some electron jumping from a higher to a lower energy level: they are emitted in decay processes of atomic nuclei (gamma decay). But I am digressing. Back to the main story line. So Heisenberg imagined we could ‘shine’ gamma rays on an electron and that we could then ‘see’ that electron in the microscope because some of the gamma photons would indeed end up in the microscope after their ‘collision’ with the electron, as shown below.

gammaray

The experiment is described in many places elsewhere but I found these accounts often confusing, and so I present my own here. 🙂

What Heisenberg basically meant to show is that this set-up would allow us to gather precise information on the position of the electron–because we would know where it was–but that, as a result, we’d lose information in regard to its momentum. Why? To put it simply: because the electron recoils as a result of the interaction. The point, of course, is to calculate the exact relationship between the two (position and momentum). In other words: what we want to do is to state the Uncertainty Principle quantitatively, not qualitatively.

Now, the animation above uses the symbol L for the γ-ray wavelength λ, which is confusing because I used L for the diameter of the aperture in my explanation of diffraction above. The animation above also uses a different symbol for the angular resolution: A instead of θ. So let me borrow the diagram used in the Wikipedia article and rephrase the whole situation.

Heisenberg_Microscope

From the diagram above, it’s obvious that, to be scattered into the microscope, the γ-ray photon must be scattered into a cone with angle ε. That angle is obviously related to the angular resolution of the microscope, which is θ = ε/2 = λ/D, with D the diameter of the aperture (i.e. the primary lens). Now, the electron could actually be anywhere, and the scattering angle could be much larger than ε, and, hence, relating D to the uncertainty in position (Δx) is not as obvious as most accounts of this thought experiment make it out to be. The thing is: if the scattering angle is larger than ε, it won’t reach the light detector at the end of the microscope (so that’s the flat top in the diagram above). So that’s why we can equate D with Δx, so we write Δx = ± D/2 = D. To put it differently: the assumption here is basically that this imaginary microscope ‘sees’ an area that is approximately as large as the lens. Using the small-angle approximation (so we write sin(2ε) ≈ 2ε), we can write:

Δx = 2λ/ε

Now, because of the recoil effect, the electron receives some momentum from the γ-ray photon. How much? Well… The situation is somewhat complicated (much more complicated than the Wikipedia article on this very same topic suggests), because the photon keeps some but also gives some of its original momentum. In fact, what’s happening really is Compton scattering: the electron first absorbs the photon, and then emits another with a different energy and, hence, also with different frequency and wavelength. However, what we do now is that the photon’s original momentum was equal to E/c= p = h/λ. That’s just the Planck relation or, if you’d want to look at the photon as a particle, the de Broglie equation.

Now, because we’re doing an analysis in one dimension only (x), we’re only going to look at the momentum in this direction only, i.e. px, and we’ll assume that all of the momentum of the photon before the interaction (or ‘collision’ if you want) was horizontal. Hence, we can write p= h/λ. After the collision, however, this momentum is spread over the electron and the scattered or emitted photon that’s going into the microscope. Let’s now imagine the two extremes:

  1. The scattered photon goes to the left edge of the lens. Hence, its horizontal momentum is negative (because it moves to the left) and the momentum pwill be distributed over the electron and the photon such that p= p’–h(ε/2)/λ’. Why the ε/2 factor? Well… That’s just trigonometry: the horizontal momentum of the scattered photon is obviously only a tiny fraction of its original horizontal momentum, and that fraction is given by the angle ε/2.
  2. The scattered photon goes to the right edge of the lens. In that case, we write p= p”+ h(ε/2)/λ”.

Now, the spread in the momentum of the electron, which we’ll simply write as Δp, is obviously equal to:

Δp = p”– p’= p+ h(ε/2)/λ” – p+ h(ε/2)/λ’ = h(ε/2)/λ” + h(ε/2)/λ’ = h(ε/2)/λ” + h(ε/2)/λ’

That’s a nice formula, but what can we do with it? What we want is a relationship between Δx and Δp, i.e. the position and the momentum of the electron, and of the electron only. That involves another simplification, which is also dealt with very summarily – too summarily in my view – in most accounts of this experiment. So let me spell it out. The angle ε is obviously very small and, hence, we may equate λ’ and λ”. In addition, while these two wavelengths differ from the wavelength of the incoming photon, the scattered photon is, obviously, still a gamma ray and, therefore, we are probably not too far off when substituting both λ’ and λ” for λ, i.e. the frequency of the incoming γ-ray. Now, we can re-write Δx = 2λ/ε as 1/Δx = ε/(2λ). We then get:

Δp = p”– p’= hε/2λ” + hε/2λ’ = 2hε/2λ = 2h/Δx

Now that yields ΔpΔx = 2h, which is an approximate expression of Heisenberg’s Uncertainty Principle indeed (don’t worry about the factor 2, as that’s something that comes with all of the approximations).

A final moot point perhaps: it is obviously a thought experiment. Not only because we don’t have gamma-ray microscopes (that’s not relevant because we can effectively imagine constructing one) but because the experiment involves only one photon. A real microscope would organize a proper beam, but that would obviously complicate the analysis. In fact, it would defeat the purpose, because the whole point is to analyze one single interaction here.

The interpretation

Now how should we interpret all of this? Is this Heisenberg’s ‘proof’ of his own Principle? Yes and no, I’d say. It’s part illustration, and part ‘proof’, I would say. The crucial assumptions here are:

  1. We can analyze γ-ray photons, or any photon for that matter, as particles having some momentum, and when ‘colliding’, or interacting, with an electron, the photon will impart some momentum to that electron.
  2. Momentum is being conserved and, hence, the total (linear) momentum before and after the collision, considering both particles–i.e. (1) the incoming ray and the electron before the interaction and (2) the emitted photon and the electron that’s getting the kick after the interaction–must be the same.
  3. For the γ-ray photon, we can relate (or associate, if you prefer that term) its wavelength λ with its momentum p through the Planck relation or, what amounts to the same for photons (because they have no mass), the de Broglie relation.

Now, these assumptions are then applied to an analysis of what we know to be true from experiment, and that’s the phenomenon of diffraction, part of which is the observation that the resolving power of a microscope is limited, and that its resolution is given by the θ = λ/D equation.

Bringing it all together, then gives us a theory which is consistent with experiment and, hence, we then assume the theory is true. Why? Well… I could start a long discourse here on the philosophy of science but, when everything is said and done, we should admit we don’t any ‘better’ theory.

But, you’ll say: what’s a ‘better’ theory? Well… Again, the answer to that question is the subject-matter of philosophers. As for me, I’d just refer to what’s known as Occam’s razor: among competing hypotheses, we should select the one with the fewest assumptions. Hence, while more complicated solutions may ultimately prove correct, the fewer assumptions that are made, the better. Now, when I was a kid, I thought quantum mechanics was very complicated and, hence, describing it here as a ‘simple’ theory sounds strange. But that’s what it is in the end: there’s no better (read: simpler) way to describe, for example, why electrons interfere with each other, and with themselves, when sending them through one or two slits, and so that’s what all these ‘illustrations’ want to show in the end, even if you think there must be simpler way to describe reality. As said, as a kid, I thought so too. 🙂

Babushka thinking

What is that we are trying to understand? As a kid, when I first heard about atoms consisting of a nucleus with electrons orbiting around it, I had this vision of worlds inside worlds, like a set of babushka dolls, one inside the other. Now I know that this model – which is nothing but the 1911 Rutherford model basically – is plain wrong, even if it continues to be used in the logo of the International Atomic Energy Agency, or the US Atomic Energy Commission. 

IAEA logo US_Atomic_Energy_Commission_logo

Electrons are not planet-like things orbiting around some center. If one wants to understand something about the reality of electrons, one needs to familiarize oneself with complex-valued wave functions whose argument represents a weird quantity referred to as a probability amplitude and, contrary to what you may think (unless you read my blog, or if you just happen to know a thing or two about quantum mechanics), the relation between that amplitude and the concept of probability tout court is not very straightforward.

Familiarizing oneself with the math involved in quantum mechanics is not an easy task, as evidenced by all those convoluted posts I’ve been writing. In fact, I’ve been struggling with these things for almost a year now and I’ve started to realize that Roger Penrose’s Road to Reality (or should I say Feynman’s Lectures?) may lead nowhere – in terms of that rather spiritual journey of trying to understand what it’s all about. If anything, they made me realize that the worlds inside worlds are not the same. They are different – very different.

When everything is said and done, I think that’s what’s nagging us as common mortals. What we are all looking for is some kind of ‘Easy Principle’ that explains All and Everything, and we just can’t find it. The point is: scale matters. At the macro-scale, we usually analyze things using some kind of ‘billiard-ball model’. At a smaller scale, let’s say the so-called wave zone, our ‘law’ of radiation holds, and we can analyze things in terms of electromagnetic or gravitational fields. But then, when we further reduce scale, by another order of magnitude really – when trying to get  very close to the source of radiation, or if we try to analyze what is oscillating really – we get in deep trouble: our easy laws do no longer hold, and the equally easy math – easy is relative of course 🙂 – we use to analyze fields or interference phenomena, becomes totally useless.

Religiously inclined people would say that God does not want us to understand all or, taking a somewhat less selfish picture of God, they would say that Reality (with a capital R to underline its transcendental aspects) just can’t be understood. Indeed, it is rather surprising – in my humble view at least – that things do seem to get more difficult as we drill down: in physics, it’s not the bigger things – like understanding thermonuclear fusion in the Sun, for example – but the smallest things which are difficult to understand. Of course, that’s partly because physics leaves some of the bigger things which are actually very difficult to understand – like how a living cell works, for example, or how our eye or our brain works – to other sciences to study (biology and biochemistry for cells, or for vision or brain functionality). In that respect, physics may actually be described as the science of the smallest things. The surprising thing, then, is that the smallest things are not necessarily the simplest things – on the contrary.

Still, that being said, I can’t help feeling some sympathy for the simpler souls who think that, if God exists, he seems to throw up barriers as mankind tries to advance its knowledge. Isn’t it strange, indeed, that the math describing the ‘reality’ of electrons and photons (i.e. quantum mechanics and quantum electrodynamics), as complicated as it is, becomes even more complicated – and, important to note, also much less accurate – when it’s used to try to describe the behavior of  quarks and gluons? Additional ‘variables’ are needed (physicists call these ‘variables’ quantum numbers; however, when everything is said and done, that’s what quantum numbers actually are: variables in a theory), and the agreement between experimental results and predictions in QCD is not as obvious as it is in QED.

Frankly, I don’t know much about quantum chromodynamics – nothing at all to be honest – but when I read statements such as “analytic or perturbative solutions in low-energy QCD are hard or impossible due to the highly nonlinear nature of the strong force” (I just took this one line from the Wikipedia article on QCD), I instinctively feel that QCD is, in fact, a different world as well – and then I mean different from QED, in which analytic or perturbative solutions are the norm. Hence, I already know that, once I’ll have mastered Feynman’s Volume III, it won’t help me all that much to get to the next level of understanding: understanding quantum chromodynamics will be yet another long grind. In short, understanding quantum mechanics is only a first step.

Of course, that should not surprise us, because we’re talking very different order of magnitudes here: femtometers (10–15 m), in the case of electrons, as opposed to attometers (10–18 m) or even zeptometers (10–21 m) when we’re talking quarks. Hence, if past experience (I mean the evolution of scientific thought) is any guidance, we actually should expect an entirely different world. Babushka thinking is not the way forward.

Babushka thinking

What’s babushka thinking? You know what babushkas are, don’t you? These dolls inside dolls. [The term ‘babushka’ is actually Russian for an old woman or grandmother, which is what these dolls usually depict.] Babushka thinking is the fallacy of thinking that worlds inside worlds are the same. It’s what I did as a kid. It’s what many of us still do. It’s thinking that, when everything is said and done, it’s just a matter of not being able to ‘see’ small things and that, if we’d have the appropriate equipment, we actually would find the same doll within the larger doll – the same but smaller – and then again the same doll with that smaller doll. In Asia, they have these funny expression: “Same-same but different.” Well… That’s what babushka thinking all about: thinking that you can apply the same concepts, tools and techniques to what is, in fact, an entirely different ballgame.

First_matryoshka_museum_doll_open

Let me illustrate it. We discussed interference. We could assume that the laws of interference, as described by superimposing various waves, always hold, at every scale, and that it’s just  the crudeness of our detection apparatus that prevents us from seeing what’s going on. Take two light sources, for example, and let’s say they are a billion wavelengths apart – so that’s anything between 400 to 700 meters for visible light (because the wavelength of visible light is 400 to 700 billionths of a meter). So then we won’t see any interference indeed, because we can’t register it. In fact, none of the standard equipment can. The interference term oscillates wildly up and down, from positive to negative and back again, if we move the detector just a tiny bit left or right – not more than the thickness of a hair (i.e. 0.07 mm or so). Hence, the range of angles θ (remember that angle θ was the key variable when calculating solutions for the resultant wave in previous posts) that are being covered by our eye – or by any standard sensor really – is so wide that the positive and negative interference averages out: all that we ‘see’ is the sum of the intensities of the two lights. The terms in the interference term cancel each other out. However, we are still essentially correct assuming there actually is interference: we just cannot see it – but it’s there.

Reinforcing the point, I should also note that, apart from this issue of ‘distance scale’, there is also the scale of time. Our eye has a tenth-of-a-second averaging time. That’s a huge amount of time when talking fundamental physics: remember that an atomic oscillator – despite its incredibly high Q – emits radiation for like 10-8 seconds only, so that’s one-hundred millionths of a second. Then another atom takes over, and another – and so that’s why we get unpolarized light: it’s all the same frequencies (because the electron oscillators radiate at their resonant frequencies), but so there is no fixed phase difference between all of these pulses: the interference between all of these pulses should result in ‘beats’ – as they interfere positively or negatively – but it all cancels out for us, because it’s too fast.

Indeed, while the ‘sensors’ in the retina of the human eye (there are actually four kind of cells there, but so the principal ones are referred to as ‘rod’ and ‘cone’ cells respectively) are, apparently, sensitive enough able to register individual photons, the “tenth-of-a-second averaging” time means that the cells – which are interconnected and ‘pre-process’ light really – will just amalgamate all those individual pulses into one signal of a certain color (frequency) and a certain intensity (energy). As one scientist puts it: “The neural filters only allow a signal to pass to the brain when at least about five to nine photons arrive within less than 100 ms.” Hence, that signal will not keep track of the spacing between those photons.

In short, information gets lost. But so that, in itself, does not invalidate babushka thinking. Let me visualize it by a non-very-mathematically-rigorous illustration. Suppose that we have some very regular wave train coming in, like the one below: one wave train consisting of three ‘groups’ separated between ‘nodes’.

Graph

All will depend on the period of the wave as compared to that one-tenth-of-a-second averaging time. In fact, we have two ‘periods’: the periodicity of the group – which is related to the concept of group velocity – and, hence, I’ll associate a ‘group wavelength’ and a ‘group period’ with that. [In case you haven’t heard of these terms before, don’t worry: I haven’t either. :-)] Now, if one tenth of a second covers like two or all three of the groups between the nodes (so that means that one tenth of a second is a multiple of the group period Tg), then even the envelope of the wave does not matter much in terms of ‘signal’: our brain will just get one pulse that averages it all out. We will see none of the detail of this wave train. Our eye will just get light in (remember that the intensity of the light is the square of the amplitude, so the negative amplitudes make contributions too) but we cannot distinguish any particular pulse: it’s just one signal. This is the most common situation when we are talking about electromagnetic radiation: many photons arrive but our eye just sends one signal to the brain: “Hey Boss! Light of color X and intensity Y coming from direction Z.”

In fact, it’s quite remarkable that our eye can distinguish colors in light of the fact that the wavelengths of various colors (violet, blue, green, yellow, orange and red) differs 30 to 40 billionths of a meter only! Better still: if the signal lasts long enough, we can distinguish shades whose wavelengths differ by 10 or 15 nm only, so that’s a difference of 1% or 2% only. In case you wonder how it works: Feynman devotes not less than two chapters in his Lectures to the physiology of the eye: not something you’ll find in other physics handbooks! There are apparently three pigments in the cells in our eyes, each sensitive to color in a different way and it is “the spectral absorption in those three pigments that produces the color sense.” So it’s a bit like the RGB system in a television – but then more complicated, of course!

But let’s go back to our wave there and analyze the second possibility. If a tenth of a second covers less than that ‘group wavelength’, then it’s different: we will actually see the individual groups as two or  three separate pulses. Hence, in that case, our eye – or whatever detector (another detector will just have another averaging time – will average over a group, but not over the whole wave train. [Just in case you wonder how we humans compare with our living beings: from what I wrote above, it’s obvious we can see ‘flicker’ only if the oscillation is in the range of 10 or 20 Hz. The eye of a bee is made to see the vibrations of feet and wings of other bees and, hence, its averaging time is much shorter, like a hundredth of a second and, hence, it can see flicker up to 200 oscillations per second! In addition, the eye of a bee is sensitive over a much wider range of ‘color’ – it sees UV light down to a wavelength of 300 nm (where as we don’t see light with a wavelength below 400 nm) – and, to top it all off, it has got a special sensitivity for polarized light, so light that gets reflected or diffracted looks different to the bee.]

Let’s go to the third and final case. If a tenth of a second would cover less than the wavelength of the the so-called carrier wave, i.e. the actual oscillation, then we will be able to distinguish the individual peaks and troughs of the carrier wave!

Of course, this discussion is not limited to our eye as a sensor: any instrument will be able to measure individual phenomena only within a certain range, with an upper and a lower range, i.e. the ‘biggest’ thing it can see, and the ‘smallest’. So that explains the so-called resolution of an optical or an electron microscope: whatever the instrument, it cannot really ‘see’ stuff that’s smaller than the wavelength of the ‘light’ (real light or – in the case of an electron microscope – electron beams) it uses to ‘illuminate’ the object it is looking at. [The actual formula for the resolution of a microscope is obviously a bit more complicated, but this statement does reflect the gist of it.]

However, all that I am writing above, suggests that we can think of what’s going on here as ‘waves within waves’, with the wave between nodes not being any different – in substance that is – as the wave as a whole: we’ve got something that’s oscillating, and within each individual oscillation, we find another oscillation. From a math point of view, babushka thinking is thinking we can analyze the world using Fourier’s machinery to decompose some function (see my posts on Fourier analysis). Indeed, in the example above, we have a modulated carrier wave (it is an example of amplitude modulation – the old-fashioned way of transmitting radio signals), and we see a wave within a wave and, hence, just like the Rutherford model of an atom, you may think there will always be ‘a wave within a wave’.

In this regard, you may think of fractals too: fractals are repeating or self-similar patterns that are always there, at every scale. However, the point to note is that fractals do not represent an accurate picture of how reality is actually structured: worlds within worlds are not the same.

Reality is no onion

Reality is not some kind of onion, from which you peel off a layer and then you find some other layer, similar to the first: “same-same but different”, as they’d say in Asia. The Coast of Britain is, in fact, finite, and the grain of sand you’ll pick up at one of its beaches will not look like the coastline when you put it under a microscope. In case you don’t believe me: I’ve inserted a real-life photo below. The magnification factor is a rather modest 300 times. Isn’t this amazing? [The credit for this nice picture goes to a certain Dr. Gary Greenberg. Please do google his stuff. It’s really nice.]

sand-grains-under-microscope-gary-greenberg-1

In short, fractals are wonderful mathematical structures but – in reality – there are limits to how small things get: we cannot carve a babushka doll out of the cellulose and lignin molecules that make up most of what we call wood. Likewise, the atoms that make up the D-glucose chains in the cellulose will never resemble the D-glucose chains. Hence, the babushka doll, the D-glucose chains that make up wood, and the atoms that make up the molecules within those macro-molecules are three different worlds. They’re not like layers of the same onion. Scale matters. The worlds inside words are different, and fundamentally so: not “same-same but different” but just plain different. Electrons are no longer point-like negative charges when we look at them at close range.

In fact, that’s the whole point: we can’t look at them at close range because we can’t ‘locate’ them. They aren’t particles. They are these strange ‘wavicles’ which we described, physically and mathematically, with a complex wave function relating their position (or their momentum) with some probability amplitude, and we also need to remember these funny rules for adding these amplitudes, depending on whether or not the ‘wavicle’ obeys Fermi or Bose statistics.

Weird, but – come to think of it – not more weird, in terms of mathematical description, than these electromagnetic waves. Indeed, when jotting down all these equations and developing all those mathematical argument, one often tends to forget that we are not talking some physical wave here. The field vector E (or B) is a mathematical construct: it tells us what force a charge will feel when we put it here or there. It’s not like a water or sound wave that makes some medium (water or air) actually move. The field is an influence that travels through empty space. But how can something actually through empty space? When it’s truly empty, you can’t travel through it, can you?

Oh – you’ll say – but we’ve got these photons, don’t we? Waves are not actually waves: they come in little packets of energy – photons. Yes. You’re right. But, as mentioned above, these photons aren’t little bullets – or particles if you want. They’re as weird as the wave and, in any case, even a billiard ball view of the world is not very satisfying: what happens exactly when two billiard balls collide in a so-called elastic collision? What are the springs on the surface of those balls – in light of the quick reaction, they must resemble more like little explosive charges that detonate on impact, isn’t it? – that make the two balls recoil from each other?

So any mathematical description of reality becomes ‘weird’ when you keep asking questions, like that little child I was – and I still am, in a way, I guess. Otherwise I would not be reading physics at the age of 45, would I? 🙂

Conclusion

Let me wrap up here. All of what I’ve been blogging about over the past few months concerns the classical world of physics. It consists of waves and fields on the one hand, and solid particles on the other – electrons and nucleons. But so we know it’s not like that when we have more sensitive apparatuses, like the apparatus used in that 2012 double-slit electron interference experiment at the University of Nebraska–Lincoln, that I described at length in one of my earlier posts. That apparatus allowed control of two slits – both not more than 62 nanometer wide (so that’s the difference between the wavelength of dark-blue and light-blue light!), and the monitoring of single-electron detection events. Back in 1963, Feynman already knew what this experiment would yield as a result. He was sure about it, even if he thought such instrument could never be built. [To be fully correct, he did have some vague idea about a new science, for which he himself coined the term ‘nanotechnology’, but what we can do today surpasses, most probably, all his expectations at the time. Too bad he died too young to see his dreams come through.]

The point to note is that this apparatus does not show us another layer of the same onion: it shows an entirely different world. While it’s part of reality, it’s not ‘our’ reality, nor is it the ‘reality’ of what’s being described by classical electromagnetic field theory. It’s different – and fundamentally so, as evidenced by those weird mathematical concepts one needs to introduce to sort of start to ‘understand’ it.

So… What do I want to say here? Nothing much. I just had to remind myself where I am right now. I myself often still fall prey to babushka thinking. We shouldn’t. We should wonder about the wood these dolls are made of. In physics, the wood seems to be math. The models I’ve presented in this blog are weird: what are those fields? And just how do they exert a force on some charge? What’s the mechanics behind? To these questions, classical physics does not have an answer really.

But, of course, quantum mechanics does not have a very satisfactory answer either: what does it mean when we say that the wave function collapses? Out of all of the possibilities in that wonderful indeterminate world ‘inside’ the quantum-mechanical universe, one was ‘chosen’ as something that actually happened: a photon imparts momentum to an electron, for example. We can describe it, mathematically, but – somehow – we still don’t really understand what’s going on.

So what’s going on? We open a doll, and we do not find another doll that is smaller but similar. No. What we find is a completely different toy. However – Surprise ! Surprise ! – it’s something that can be ‘opened’ as well, to reveal even weirder stuff, for which we need even weirder ‘tools’ to somehow understand how it works (like lattice QCD, if you’d want an example: just google it if you want to get an inkling of what that’s about). Where is this going to end? Did it end with the ‘discovery’ of the Higgs particle? I don’t think so.

However, with the ‘discovery’ (or, to be generous, let’s call it an experimental confirmation) of the Higgs particle, we may have hit a wall in terms of verifying our theories. At the center of a set of babushka dolls, you’ll usually have a little baby: a solid little thing that is not like the babushkas surrounding it: it’s young, male and solid, as opposed to the babushkas. Well… It seems that, in physics, we’ve got several of these little babies inside: electrons, photons, quarks, gluons, Higgs particles, etcetera. And we don’t know what’s ‘inside’ of them. Just that they’re different. Not “same-same but different”. No. Fundamentally different. So we’ve got a lot of ‘babies’ inside of reality, very different from the ‘layers’ around them, which make up ‘our’ reality. Hence, ‘Reality’ is not a fractal structure. What is it? Well… I’ve started to think we’ll never know. For all of the math and wonderful intellectualism involved, do we really get closer to an ‘understanding’ of what it’s all about?

I am not sure. The more I ‘understand’, the less I ‘know’ it seems. But then that’s probably why many physicists still nurture an acute sense of mystery, and why I am determined to keep reading. 🙂

Post scriptum: On the issue of the ‘mechanistic universe’ and the (related) issue of determinability and indeterminability, that’s not what I wanted to write about above, because I consider that solved. This post is meant to convey some wonder – on the different models of understanding that we need to apply to different scales. It’s got little to do with determinability or not. I think that issue got solved long time ago, and I’ll let Feynman summarize that discussion:

“The indeterminacy of quantum mechanics has given rise to all kinds of nonsense and questions on the meaning of freedom of will, and of the idea that the world is uncertain. […] Classical physics is also indeterminate. It is true, classically, that if we knew the position and the velocity of every particle in the world, or in a box of gas, we could predict exactly what would happen. And therefore the classical world is deterministic. Suppose, however, we have a finite accuracy and do not know exactly where just one atom is, say to one part in a billion. Then as it goes along it hits another atom, and because we did not know the position better than one part in a billion, we find an even larger error in the position after the collision. And that is amplified, of course, in the next collision, so that if we start with only a tiny error it rapidly magnifies to a very great uncertainty. […] Speaking more precisely, given an arbitrary accuracy, no matter how precise, one can find a time long enough that we cannot make predictions valid for that long a time. That length of time is not very large. It is not that the time is millions of years if the accuracy is one part in a billion. The time goes only logarithmically with the error. In only a very, very tiny time – less than the time it took to state the accuracy – we lose all our information. It is therefore not fair to say that from the apparent freedom and indeterminacy of the human mind, we should have realized that classical ‘deterministic’ physics could not ever hope to understand, and to welcome quantum mechanics as a release from a completely ‘mechanistic’ universe. For already in classical mechanics, there was indeterminability from a practical point of view.” (Feynman, Lectures, 1963, p. 38-10)

That really says it all, I think. I’ll just continue to keep my head down – i.e. stay away from philosophy as for now – and try to find a way to open the toy inside the toy. 🙂

Re-visiting the matter wave (II)

My previous post was, once again, littered with formulas – even if I had not intended it to be that way: I want to convey some kind of understanding of what an electron – or any particle at the atomic scale – actually is – with the minimum number of formulas necessary.

We know particles display wave behavior: when an electron beam encounters an obstacle or a slit that is somewhat comparable in size to its wavelength, we’ll observe diffraction, or interference. [I have to insert a quick note on terminology here: the terms diffraction and interference are often used interchangeably, but there is a tendency to use interference when we have more than one wave source and diffraction when there is only one wave source. However, I’ll immediately add that distinction is somewhat artificial. Do we have one or two wave sources in a double-slit experiment? There is one beam but the two slits break it up in two and, hence, we would call it interference. If it’s only one slit, there is also an interference pattern, but the phenomenon will be referred to as diffraction.]

We also know that the wavelength we are talking about it here is not the wavelength of some electromagnetic wave, like light. It’s the wavelength of a de Broglie wave, i.e. a matter wave: such wave is represented by an (oscillating) complex number – so we need to keep track of a real and an imaginary part – representing a so-called probability amplitude Ψ(x, t) whose modulus squared (│Ψ(x, t)│2) is the probability of actually detecting the electron at point x at time t. [The purists will say that complex numbers can’t oscillate – but I am sure you get the idea.]

You should read the phrase above twice: we cannot know where the electron actually is. We can only calculate probabilities (and, of course, compare them with the probabilities we get from experiments). Hence, when the wave function tells us the probability is greatest at point x at time t, then we may be lucky when we actually probe point x at time t and find it there, but it may also not be there. In fact, the probability of finding it exactly at some point x at some definite time t is zero. That’s just a characteristic of such probability density functions: we need to probe some region Δx in some time interval Δt.

If you think that is not very satisfactory, there’s actually a very common-sense explanation that has nothing to do with quantum mechanics: our scientific instruments do not allow us to go beyond a certain scale anyway. Indeed, the resolution of the best electron microscopes, for example, is some 50 picometer (1 pm = 1×10–12 m): that’s small (and resolutions get higher by the year), but so it implies that we are not looking at points – as defined in math that is: so that’s something with zero dimension – but at pixels of size Δx = 50×10–12 m.

The same goes for time. Time is measured by atomic clocks nowadays but even these clocks do ‘tick’, and these ‘ticks’ are discrete. Atomic clocks take advantage of the property of atoms to resonate at extremely consistent frequencies. I’ll say something more about resonance soon – because it’s very relevant for what I am writing about in this post – but, for the moment, just note that, for example, Caesium-133 (which was used to build the first atomic clock) oscillates at 9,192,631,770 cycles per second. In fact, the International Bureau of Standards and Weights re-defined the (time) second in 1967 to correspond to “the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the Caesium-133 atom at rest at a temperature of 0 K.”

Don’t worry about it: the point to note is that when it comes to measuring time, we also have an uncertainty. Now, when using this Caesium-133 atomic clock, this uncertainty would be in the range of ±9.2×10–9 seconds (so that’s nanoseconds: 1 ns = 1×10–9 s), because that’s the rate at which this clock ‘ticks’. However, there are other (much more plausible) ways of measuring time: some of the unstable baryons have lifetimes in the range of a few picoseconds only (1 ps = 1×10–12 s) and the really unstable ones – known as baryon resonances – have lifetimes in the 1×10–22 to 1×10–24 s range. This we can only measure because they leave some trace after these particle collisions in particle accelerators and, because we have some idea about their speed, we can calculate their lifetime from the (limited) distance they travel before disintegrating. The thing to remember is that for time also, we have to make do with time pixels  instead of time points, so there is a Δt as well. [In case you wonder what baryons are: they are particles consisting of three quarks, and the proton and the neutron are the most prominent (and most stable) representatives of this family of particles.]  

So what’s the size of an electron? Well… It depends. We need to distinguish two very different things: (1) the size of the area where we are likely to find the electron, and (2) the size of the electron itself. Let’s start with the latter, because that’s the easiest question to answer: there is a so-called classical electron radius re, which is also known as the Thompson scattering length, which has been calculated as:

r_\mathrm{e} = \frac{1}{4\pi\varepsilon_0}\frac{e^2}{m_{\mathrm{e}} c^2} = 2.817 940 3267(27) \times 10^{-15} \mathrm{m}As for the constants in this formula, you know these by now: the speed of light c, the electron charge e, its mass me, and the permittivity of free space εe. For whatever it’s worth (because you should note that, in quantum mechanics, electrons do not have a size: they are treated as point-like particles, so they have a point charge and zero dimension), that’s small. It’s in the femtometer range (1 fm = 1×10–15 m). You may or may not remember that the size of a proton is in the femtometer range as well – 1.7 fm to be precise – and we had a femtometer size estimate for quarks as well: 0.7 m. So we have the rather remarkable result that the much heavier proton (its rest mass is 938 MeV/csas opposed to only 0.511 MeV MeV/c2, so the proton is 1835 times heavier) is 1.65 times smaller than the electron. That’s something to be explored later: for the moment, we’ll just assume the electron wiggles around a bit more – exactly because it’s lighterHere you just have to note that this ‘classical’ electron radius does measure something: it’s something ‘hard’ and ‘real’ because it scatters, absorbs or deflects photons (and/or other particles). In one of my previous posts, I explained how particle accelerators probe things at the femtometer scale, so I’ll refer you to that post (End of the Road to Reality?) and move on to the next question.

The question concerning the area where we are likely to detect the electron is more interesting in light of the topic of this post (the nature of these matter waves). It is given by that wave function and, from my previous post, you’ll remember that we’re talking the nanometer scale here (1 nm = 1×10–9 m), so that’s a million times larger than the femtometer scale. Indeed, we’ve calculated a de Broglie wavelength of 0.33 nanometer for relatively slow-moving electrons (electrons in orbit), and the slits used in single- or double-slit experiments with electrons are also nanotechnology. In fact, now that we are here, it’s probably good to look at those experiments in detail.

The illustration below relates the actual experimental set-up of a double-slit experiment performed in 2012 to Feynman’s 1965 thought experiment. Indeed, in 1965, the nanotechnology you need for this kind of experiment was not yet available, although the phenomenon of electron diffraction had been confirmed experimentally already in 1925 in the famous Davisson-Germer experiment. [It’s famous not only because electron diffraction was a weird thing to contemplate at the time but also because it confirmed the de Broglie hypothesis only two years after Louis de Broglie had advanced it!]. But so here is the experiment which Feynman thought would never be possible because of technology constraints:

Electron double-slit set-upThe insert in the upper-left corner shows the two slits: they are each 50 nanometer wide (50×10–9 m) and 4 micrometer tall (4×10–6 m). [The thing in the middle of the slits is just a little support. Please do take a few seconds to contemplate the technology behind this feat: 50 nm is 50 millionths of a millimeter. Try to imagine dividing one millimeter in ten, and then one of these tenths in ten again, and again, and once again, again, and again. You just can’t imagine that, because our mind is used to addition/subtraction and – to some extent – with multiplication/division: our mind can’t deal with with exponentiation really – because it’s not a everyday phenomenon.] The second inset (in the upper-right corner) shows the mask that can be moved to close one or both slits partially or completely.

Now, 50 nanometer is 150 times larger than the 0.33 nanometer range we got for ‘our’ electron, but it’s small enough to show diffraction and/or interference. [In fact, in this experiment (done by Bach, Pope, Liou and Batelaan from the University of Nebraska-Lincoln less than two years ago indeed), the beam consisted of electrons with an (average) energy of 600 eV and a de Broglie wavelength of 50 picometer. So that’s like the electrons used in electron microscopes. 50 pm is 6.6 times smaller than the 0.33 nm wavelength we calculated for our low-energy (70 eV) electron – but then the energy and the fact these electrons are guided in electromagnetic fields explain the difference. Let’s go to the results.

The illustration below shows the predicted pattern next to the observed pattern for the two scenarios:

  1. We first close slit 2, let a lot of electrons go through it, and so we get a pattern described by the probability density function P1 = │Φ12. Here we see no interference but a typical diffraction pattern: the intensity follows a more or less normal (i.e. Gaussian) distribution. We then close slit 1 (and open slit 2 again), again let a lot of electrons through, and get a pattern described by the probability density function P2 = │Φ22. So that’s how we get P1 and P2.
  2. We then open both slits, let a whole electrons through, and get according to the pattern described by probability density function P12 = │Φ122, which we get not from adding the probabilities P1 and P2 (hence, P12 ≠  P1 + P2) – as one would expect if electrons would behave like particles – but from adding the probability amplitudes. We have interference, rather than diffraction.

Predicted interference effectBut so what exactly is interfering? Well… The electrons. But that can’t be, can it?

The electrons are obviously particles, as evidenced from the impact they make – one by one – as they hit the screen as shown below. [If you want to know what screen, let me quote the researchers: “The resulting patterns were magnified by an electrostatic quadrupole lens and imaged on a two-dimensional microchannel plate and phosphorus screen, then recorded with a charge-coupled device camera. […] To study the build-up of the diffraction pattern, each electron was localized using a “blob” detection scheme: each detection was replaced by a blob, whose size represents the error in the localization of the detection scheme. The blobs were compiled together to form the electron diffraction patterns.” So there you go.]

Electron blobs

Look carefully at how this interference pattern becomes ‘reality’ as the electrons hit the screen one by one. And then say it: WAW ! 

Indeed, as predicted by Feynman (and any other physics professor at the time), even if the electrons go through the slits one by one, they will interfere – with themselves so to speak. [In case you wonder if these electrons really went through one by one, let me quote the researchers once again: “The electron source’s intensity was reduced so that the electron detection rate in the pattern was about 1 Hz. At this rate and kinetic energy, the average distance between consecutive electrons was 2.3 × 106 meters. This ensures that only one electron is present in the 1 meter long system at any one time, thus eliminating electron-electron interactions.” You don’t need to be a scientist or engineer to understand that, isn’t it?]

While this is very spooky, I have not seen any better way to describe the reality of the de Broglie wave: the particle is not some point-like thing but a matter wave, as evidenced from the fact that it does interfere with itself when forced to move through two slits – or through one slit, as evidenced by the diffraction patterns built up in this experiment when closing one of the two slits: the electrons went through one by one as well!

But so how does it relate to the characteristics of that wave packet which I described in my previous post? Let me sum up the salient conclusions from that discussion:

  1. The wavelength λ of a wave packet is calculated directly from the momentum by using de Broglie‘s second relation: λ = h/p. In this case, the wavelength of the electrons averaged 50 picometer. That’s relatively small as compared to the width of the slit (50 nm) – a thousand times smaller actually! – but, as evidenced by the experiment, it’s small enough to show the ‘reality’ of the de Broglie wave.
  2. From a math point (but, of course, Nature does not care about our math), we can decompose the wave packet in a finite or infinite number of component waves. Such decomposition is referred to, in the first case (finite number of composite waves or discrete calculus) as a Fourier analysis, or, in the second case, as a Fourier transform. A Fourier transform maps our (continuous) wave function, Ψ(x), to a (continuous) wave function in the momentum space, which we noted as φ(p). [In fact, we noted it as Φ(p) but I don’t want to create confusion with the Φ symbol used in the experiment, which is actually the wave function in space, so Ψ(x) is Φ(x) in the experiment – if you know what I mean.] The point to note is that uncertainty about momentum is related to uncertainty about position. In this case, we’ll have pretty standard electrons (so not much variation in momentum), and so the location of the wave packet in space should be fairly precise as well.
  3. The group velocity of the wave packet (vg) – i.e. the envelope in which our Ψ wave oscillates – equals the speed of our electron (v), but the phase velocity (i.e. the speed of our Ψ wave itself) is superluminal: we showed it’s equal to (vp) = E/p =   c2/v = c/β, with β = v/c, so that’s the ratio of the speed of our electron and the speed of light. Hence, the phase velocity will always be superluminal but will approach c as the speed of our particle approaches c. For slow-moving particles, we get astonishing values for the phase velocity, like more than a hundred times the speed of light for the electron we looked at in our previous post. That’s weird but it does not contradict relativity: if it helps, one can think of the wave packet as a modulation of an incredibly fast-moving ‘carrier wave’. 

Is any of this relevant? Does it help you to imagine what the electron actually is? Or what that matter wave actually is? Probably not. You will still wonder: How does it look like? What is it in reality?

That’s hard to say. If the experiment above does not convey any ‘reality’ according to you, then perhaps the illustration below will help. It’s one I have used in another post too (An Easy Piece: Introducing Quantum Mechanics and the Wave Function). I took it from Wikipedia, and it represents “the (likely) space in which a single electron on the 5d atomic orbital of an atom would be found.” The solid body shows the places where the electron’s probability density (so that’s the squared modulus of the probability amplitude) is above a certain value – so it’s basically the area where the likelihood of finding the electron is higher than elsewhere. The hue on the colored surface shows the complex phase of the wave function.

Hydrogen_eigenstate_n5_l2_m1

So… Does this help? 

You will wonder why the shape is so complicated (but it’s beautiful, isn’t it?) but that has to do with quantum-mechanical calculations involving quantum-mechanical quantities such as spin and other machinery which I don’t master (yet). I think there’s always a bit of a gap between ‘first principles’ in physics and the ‘model’ of a real-life situation (like a real-life electron in this case), but it’s surely the case in quantum mechanics! That being said, when looking at the illustration above, you should be aware of the fact that you are actually looking at a 3D representation of the wave function of an electron in orbit. 

Indeed, wave functions of electrons in orbit are somewhat less random than – let’s say – the wave function of one of those baryon resonances I mentioned above. As mentioned in my Not So Easy Piece, in which I introduced the Schrödinger equation (i.e. one of my previous posts), they are solutions of a second-order partial differential equation – known as the Schrödinger wave equation indeed – which basically incorporates one key condition: these solutions – which are (atomic or molecular) ‘orbitals’ indeed – have to correspond to so-called stationary states or standing waves. Now what’s the ‘reality’ of that? 

The illustration below comes from Wikipedia once again (Wikipedia is an incredible resource for autodidacts like me indeed) and so you can check the article (on stationary states) for more details if needed. Let me just summarize the basics:

  1. A stationary state is called stationary because the system remains in the same ‘state’ independent of time. That does not mean the wave function is stationary. On the contrary, the wave function changes as function of both time and space – Ψ = Ψ(x, t) remember? – but it represents a so-called standing wave.
  2. Each of these possible states corresponds to an energy state, which is given through the de Broglie relation: E = hf. So the energy of the state is proportional to the oscillation frequency of the (standing) wave, and Planck’s constant is the factor of proportionality. From a formal point of view, that’s actually the one and only condition we impose on the ‘system’, and so it immediately yields the so-called time-independent Schrödinger equation, which I briefly explained in the above-mentioned Not So Easy Piece (but I will not write it down here because it would only confuse you even more). Just look at these so-called harmonic oscillators below:

QuantumHarmonicOscillatorAnimation

A and B represent a harmonic oscillator in classical mechanics: a ball with some mass m (mass is a measure for inertia, remember?) on a spring oscillating back and forth. In case you’d wonder what the difference is between the two: both the amplitude as well as the frequency of the movement are different. 🙂 A spring and a ball?

It represents a simple system. A harmonic oscillation is basically a resonance phenomenon: springs, electric circuits,… anything that swings, moves or oscillates (including large-scale things such as bridges and what have you – in his 1965 Lectures (Vol. I-23), Feynman even discusses resonance phenomena in the atmosphere in his Lectures) has some natural frequency ω0, also referred to as the resonance frequency, at which it oscillates naturally indeed: that means it requires (relatively) little energy to keep it going. How much energy it takes exactly to keep them going depends on the frictional forces involved: because the springs in A and B keep going, there’s obviously no friction involved at all. [In physics, we say there is no damping.] However, both springs do have a different k (that’s the key characteristic of a spring in Hooke’s Law, which describes how springs work), and the mass m of the ball might be different as well. Now, one can show that the period of this ‘natural’ movement will be equal to t0 = 2π/ω= 2π(m/k)1/2 or that ω= (m/k)–1/2. So we’ve got a A and a B situation which differ in k and m. Let’s go to the so-called quantum oscillator, illustrations C to H.

C to H in the illustration are six possible solutions to the Schrödinger Equation for this situation. The horizontal axis is position (and so time is the variable) – but we could switch the two independent variables easily: as I said a number of times already, time and space are interchangeable in the argument representing the phase (θ) of a wave provided we use the right units (e.g. light-seconds for distance and seconds for time): θ = ωt – kx. Apart from the nice animation, the other great thing about these illustrations – and the main difference with resonance frequencies in the classical world – is that they show both the real part (blue) as well as the imaginary part (red) of the wave function as a function of space (fixed in the x axis) and time (the animation).

Is this ‘real’ enough? If it isn’t, I know of no way to make it any more ‘real’. Indeed, that’s key to understanding the nature of matter waves: we have to come to terms with the idea that these strange fluctuating mathematical quantities actually represent something. What? Well… The spooky thing that leads to the above-mentioned experimental results: electron diffraction and interference. 

Let’s explore this quantum oscillator some more. Another key difference between natural frequencies in atomic physics (so the atomic scale) and resonance phenomena in ‘the big world’ is that there is more than one possibility: each of the six possible states above corresponds to a solution and an energy state indeed, which is given through the de Broglie relation: E = hf. However, in order to be fully complete, I have to mention that, while G and H are also solutions to the wave equation, they are actually not stationary states. The illustration below – which I took from the same Wikipedia article on stationary states – shows why. For stationary states, all observable properties of the state (such as the probability that the particle is at location x) are constant. For non-stationary states, the probabilities themselves fluctuate as a function of time (and space of obviously), so the observable properties of the system are not constant. These solutions are solutions to the time-dependent Schrödinger equation and, hence, they are, obviously, time-dependent solutions.

StationaryStatesAnimationWe can find these time-dependent solutions by superimposing two stationary states, so we have a new wave function ΨN which is the sum of two others:  ΨN = Ψ1  + Ψ2. [If you include the normalization factor (as you should to make sure all probabilities add up to 1), it’s actually ΨN = (2–1/2)(Ψ1  + Ψ2).] So G and H above still represent a state of a quantum harmonic oscillator (with a specific energy level proportional to h), but so they are not standing waves.

Let’s go back to our electron traveling in a more or less straight path. What’s the shape of the solution for that one? It could be anything. Well… Almost anything. As said, the only condition we can impose is that the envelope of the wave packet – its ‘general’ shape so to say – should not change. That because we should not have dispersion – as illustrated below. [Note that this illustration only represent the real or the imaginary part – not both – but you get the idea.]

dispersion

That being said, if we exclude dispersion (because a real-life electron traveling in a straight line doesn’t just disappear – as do dispersive wave packets), then, inside of that envelope, the weirdest things are possible – in theory that is. Indeed, Nature does not care much about our Fourier transforms. So the example below, which shows a theoretical wave packet (again, the real or imaginary part only) based on some theoretical distribution of the wave numbers of the (infinite number) of component waves that make up the wave packet, may or may not represent our real-life electron. However, if our electron has any resemblance to real-life, then I would expect it to not be as well-behaved as the theoretical one that’s shown below.

example of wave packet

The shape above is usually referred to as a Gaussian wave packet, because of the nice normal (Gaussian) probability density functions that are associated with it. But we can also imagine a ‘square’ wave packet: a somewhat weird shape but – in terms of the math involved – as consistent as the smooth Gaussian wave packet, in the sense that we can demonstrate that the wave packet is made up of an infinite number of waves with an angular frequency ω that is linearly related to their wave number k, so the dispersion relation is ω = ak + b. [Remember we need to impose that condition to ensure that our wave packet will not dissipate (or disperse or disappear – whatever term you prefer.] That’s shown below: a Fourier analysis of a square wave.

Square wave packet

While we can construct many theoretical shapes of wave packets that respect the ‘no dispersion!’ condition, we cannot know which one will actually represent that electron we’re trying to visualize. Worse, if push comes to shove, we don’t know if these matter waves (so these wave packets) actually consist of component waves (or time-independent stationary states or whatever).

[…] OK. Let me finally admit it: while I am trying to explain you the ‘reality’ of these matter waves, we actually don’t know how real these matter waves actually are. We cannot ‘see’ or ‘touch’ them indeed. All that we know is that (i) assuming their existence, and (ii) assuming these matter waves are more or less well-behaved (e.g. that actual particles will be represented by a composite wave characterized by a linear dispersion relation between the angular frequencies and the wave numbers of its (theoretical) component waves) allows us to do all that arithmetic with these (complex-valued) probability amplitudes. More importantly, all that arithmetic with these complex numbers actually yields (real-valued) probabilities that are consistent with the probabilities we obtain through repeated experiments. So that’s what’s real and ‘not so real’ I’d say.

Indeed, the bottom-line is that we do not know what goes on inside that envelope. Worse, according to the commonly accepted Copenhagen interpretation of the Uncertainty Principle (and tons of experiments have been done to try to overthrow that interpretation – all to no avail), we never will.