# A simple explanation of quantum-mechanical operators

I added an Annex to a paper that talks about all of the fancy stuff quantum physicists like to talk about, like scattering matrices and high-energy particle events. The Annex, however, is probably my simplest and shortest summary of the ordinariness of wavefunction math, including a quick overview of what quantum-mechanical operators actually are. It does not make use of state vector algebra or the usual high-brow talk about Gilbert spaces and what have you: you only need to know what a derivative is, and combine it with our realist interpretation of what the wavefunction actually represents.

I think I should do a paper on the language of physics. To show how (i) rotations (i, j, k), (ii) scalars (constants or just numerical values) and (iii) vectors (real vectors (e.g. position vectors) and pseudovectors (e.g. angular frequency or momentum)), and (iv) operators (derivatives of the wavefunction with respect to time and spatial directions) form ‘words’ (e.g. energy and momentum operators), and how these ‘words’ then combine into meaningful statements (e.g. Schroedinger’s equation).

All of physics can then be summed up in a half-page or so. All the rest is thermodynamics š JL

PS: You only get collapsing wavefunctions when adding uncertainty to the models (i.e. our own uncertainty about the energy and momentum). The ‘collapse’ of the wavefunction (let us be precise, the collapse of the (dissipating) wavepacket) thus corresponds to the ‘measurement’ operation. š

PS2: Incidentally, the analysis also gives an even more intuitive explanation of Einstein’s mass-energy equivalence relation, which I summarize in a reply to one of the many ‘numerologist’ physicists on ResearchGate (copied below).

# Should we reinvent wavefunction math?

Preliminary note: This post may cause brain damage. š If you haven’t worked yourself through a good introduction to physics – including the math – you will probably not understand what this is about. So… Well… Sorry. š¦ But if you have… Then this should be very interesting. Let’s go. š

If you know one or two things about quantum math – SchrĆ¶dinger’s equation and all that – then you’ll agree the math is anything but straightforward. Personally, I find the most annoying thing about wavefunction math are those transformation matrices: every time we look at the same thing from a different direction, we need to transform the wavefunction using one or more rotation matrices – and that gets quite complicated !

Now, if you have read any of my posts on this or my other blog, then you know I firmly believe the wavefunction represents somethingĀ realĀ or… Well… Perhaps it’s just the next best thing to reality: we cannot know das Ding an sich, but the wavefunction gives us everything we would want to know about it (linear or angular momentum, energy, and whatever else we have an operator for). So what am I thinking of? Let me first quote Feynman’s summary interpretation ofĀ SchrĆ¶dinger’s equationĀ (Lectures, III-16-1):

āWe can think of SchrĆ¶dingerās equation as describing the diffusion of the probability amplitude from one point to the next. [ā¦] But the imaginary coefficient in front of the derivative makes the behavior completely different from the ordinary diffusion such as you would have for a gas spreading out along a thin tube. Ordinary diffusion gives rise to real exponential solutions, whereas the solutions of SchrĆ¶dingerās equation are complex waves.ā

Feynman further formalizes this in his Lecture on Superconductivity (Feynman, III-21-2), in which he refers to SchrĆ¶dingerās equation as the āequation for continuity of probabilitiesā. His analysis there is centered on the local conservation of energy, which makes me think SchrĆ¶dingerās equation might be an energy diffusion equation. I’ve written about thisĀ ad nauseamĀ in the past, and so I’ll just refer you to one of my papers here for the details, and limit this post to the basics, which are as follows.

The wave equation (so that’s SchrĆ¶dinger’s equation in its non-relativistic form, which is an approximation that is good enough)Ā isĀ written as:The resemblance with the standard diffusion equation (shown below) is, effectively, very obvious:As Feynman notes, it’s just that imaginary coefficient that makes the behavior quite different.Ā HowĀ exactly? Well… You know we get all of those complicated electron orbitals (i.e. the various wave functionsĀ that satisfy the equation) out of SchrĆ¶dinger’s differential equation. We can think of these solutions as (complex)Ā standing waves. They basically represent someĀ equilibriumĀ situation, and the main characteristic of each is theirĀ energy level. I won’t dwell on this because – as mentioned above – I assume you master the math. Now, you know that – if we would want to interpret these wavefunctions as something real (which is surely whatĀ IĀ want to do!) – the real and imaginary component of a wavefunction will be perpendicular to each other. Let me copy the animation for theĀ elementaryĀ wavefunction Ļ(Īø) =Ā aĀ·eāiāĪøĀ =Ā aĀ·eāiā(E/Ä§)Ā·tĀ = aĀ·cos[(E/Ä§)āt]Ā āĀ iĀ·aĀ·sin[(E/Ä§)āt] once more:

So… Well… That 90Ā° angle makes me think of the similarity with the mathematical description of an electromagnetic wave. Let me quickly show you why. For a particle moving in free space ā with no external force fields acting on it ā there is no potential (U = 0) and, therefore, the VĻ term – which is just the equivalent of the theĀ sinkĀ or sourceĀ term S in the diffusion equation – disappears. Therefore, SchrĆ¶dingerās equation reduces to:

āĻ(x, t)/āt =Ā iĀ·(1/2)Ā·(Ä§/meff)Ā·ā2Ļ(x, t)

Now, the key difference with the diffusion equation – let me write it for you once again: āĻ(x, t)/āt = DĀ·ā2Ļ(x, t) – is thatĀ SchrĆ¶dingerās equation gives usĀ twoĀ equations for the price of one. Indeed, because Ļ is a complex-valued function, with aĀ realĀ and anĀ imaginaryĀ part, we get the following equations:

1. Re(āĻ/āt) = ā(1/2)Ā·(Ä§/meff)Ā·Im(ā2Ļ)
2. Im(āĻ/āt) = (1/2)Ā·(Ä§/meff)Ā·Re(ā2Ļ)

Huh?Ā Yes. These equations are easily derived from noting that two complex numbers a +Ā iāb and c +Ā iād are equal if, and only if, their real and imaginary parts are the same. Now, the āĻ/āt =Ā iā(Ä§/meff)āā2Ļ equation amounts to writing something like this: a +Ā iāb =Ā iā(c +Ā iād). Now, remembering thatĀ i2Ā = ā1, you can easily figure out thatĀ iā(c +Ā iād) =Ā iāc +Ā i2ād = ā d +Ā iāc. [Now that we’re getting a bit technical, let me note that theĀ meff is the effective mass of the particle, which depends on the medium. For example, an electron traveling in a solid (a transistor, for example) will have a different effective mass than in an atom. In free space, we can drop the subscript and just write meff = m.] š OK.Ā Onwards !Ā š

The equations above make me think of the equations for an electromagnetic wave in free space (no stationary charges or currents):

1. āB/āt = āāĆE
2. āE/āt =Ā c2āĆB

Now, these equations – and, I must therefore assume, the other equations above as well – effectively describe a propagation mechanism in spacetime, as illustrated below:

You know how it works for the electromagnetic field: it’s the interplay between circulation and flux. Indeed, circulation around some axis of rotation creates a flux in a direction perpendicular to it, and that flux causes this, and then that, and it all goes round and round and round. š Something like that. š I will let you look up how it goes,Ā exactly. The principle is clear enough.Ā Somehow, in this beautiful interplay between linear and circular motion, energy is borrowed from one place and then returns to the other, cycle after cycle.

Now, we know the wavefunction consist of a sine and a cosine: the cosine is the real component, and the sine is the imaginary component. Could they be equally real? Could each represent half of the total energy of our particle? I firmly believe they do. The obvious question then is the following: why wouldn’t we represent them asĀ vectors, just like E and B? I mean… Representing them as vectorsĀ (I meanĀ realĀ vectors here – something with a magnitude and a direction in aĀ realĀ space – as opposed to state vectors from the Hilbert space) wouldĀ showĀ they are real, and there would be no need for cumbersome transformations when going from one representationalĀ baseĀ to another. In fact, that’s why vector notation was invented (sort of): we don’t need to worry about the coordinate frame. It’s much easier to write physical laws in vector notation because… Well… They’re theĀ realĀ thing, aren’t they? š

What about dimensions? Well… I am not sure. However, because we are – arguably – talking about some pointlike charge moving around in those oscillating fields, I would suspect the dimension of the real and imaginary component of the wavefunction will be the same as that of the electric and magnetic field vectors E and B. We may want to recall these:

1. EĀ is measured inĀ newton per coulombĀ (N/C).
2. BĀ is measured in newton per coulomb divided by m/s, so that’s (N/C)/(m/s).

The weird dimension of BĀ is because of the weird force law for the magnetic force. It involves a vector cross product, as shown by Lorentz’ formula:

F = qE + q(vĆB)

Of course, it is onlyĀ oneĀ force (one and the same physical reality), as evidenced by the fact that we can write B as the following vector cross-product: BĀ = (1/c)āexĆE, withĀ exĀ the unit vector pointing in the x-direction (i.e. the direction of propagation of the wave). [Check it, because you may not have seen this expression before. Just take a piece of paper and think about the geometry of the situation.] Hence, we may associate the (1/c)āexĆ operator, which amounts to a rotation by 90 degrees, with the s/m dimension. Now, multiplication by i also amounts to a rotation by 90Ā° degrees. Hence, if we can agree on a suitable convention for the directionĀ of rotation here,Ā we may boldly write:

BĀ = (1/c)āexĆE = (1/c)āiāE

This is, in fact, what triggered my geometric interpretation of SchrĆ¶dingerās equation about a year ago now. I have had little time to work on it, but think I am on the right track. Of course, you should note that, for anĀ electromagnetic wave, the magnitudes of E and B reach their maximum, minimum and zero point simultaneouslyĀ (as shown below). So theirĀ phaseĀ is the same.

In contrast, the phase of the real and imaginary component of the wavefunction is not the same, as shown below.

In fact, because of the Stern-Gerlach experiment, I am actually more thinking of a motion like this:

But that shouldn’t distract you. š The question here is the following: could we possibly think of a new formulation of SchrĆ¶dinger’s equation – usingĀ vectors (again,Ā realĀ vectors – not these weirdĀ state vectors)Ā rather than complex algebra?

I think we can, but then I wonder why theĀ inventorsĀ of the wavefunction – Heisenberg, Born, Dirac, and SchrĆ¶dinger himself, of course – never thought of that. š

Hmm… I need to do some research here. š

Post scriptum: You will, of course, wonder how and why the matter-wave would be different from the electromagnetic wave if my suggestion that the dimension of the wavefunction component is the same is correct. The answer is: the difference lies in the phase difference and then, most probably, the different orientation of the angular momentum. Do we have any other possibilities? š

P.S. 2: I also published this post on my new blog:Ā https://readingeinstein.blog/. However, I thought the followers of this blog should get it first. š

# Re-visiting electron orbitals (III)

Pre-script (dated 26 June 2020): Our ideas have evolved into a full-blown realistic (or classical) interpretation of all things quantum-mechanical. In addition, I note the dark force has amused himself by removing some material. So no use to read this. Read my recent papers instead. š

Original post:

In my previous post, I mentioned that it wasĀ not so obvious (both from a physicalĀ as well as from aĀ mathematicalĀ point of view) to write the wavefunction for electron orbitals – which we denoted as Ļ(x, t), i.e. a function of two variables (or four: one time coordinate and three space coordinates) –Ā as the product of two other functions in one variable only.

[…] OK. The above sentence is difficult to read. Let me write in math. š It isĀ notĀ so obvious to write Ļ(x, t) as:

Ļ(x, t) = eāiĀ·(E/Ä§)Ā·tĀ·Ļ(x)

As I mentioned before, the physicists’ use of the same symbol (Ļ, psi) for both the Ļ(x, t) and Ļ(x) function is quite confusing – because the two functions areĀ veryĀ different:

• Ļ(x, t) is a complex-valued function of twoĀ (real)Ā variables: x and t. OrĀ four, I should say, because xĀ = (x, y, z) – but it’s probably easier to think of x as oneĀ vectorĀ variable – aĀ vector-valued argument, so to speak. And then t is, of course, just aĀ scalarĀ variable. So… Well… A function of twoĀ variables: the position in space (x), and time (t).
• In contrast, Ļ(x) is a real-valuedĀ function ofĀ oneĀ (vector) variable only: x, so that’s the position in space only.

Now you should cry foul, of course: Ļ(x) is notĀ necessarilyĀ real-valued. It mayĀ be complex-valued. You’re right.Ā You know the formula:Note the derivation of this formula involved a switch from Cartesian to polar coordinates here, so from xĀ = (x, y, z) to rĀ = (r, Īø, Ļ), and that the function is also a function of the twoĀ quantum numbersĀ l and m now, i.e. the orbital angular momentum (l) and its z-component (m) respectively. In my previous post(s), I gave you the formulas for Yl,m(Īø, Ļ) and Fl,m(r) respectively. Fl,m(r) was a real-valued function alright, but the Yl,m(Īø, Ļ) had that eiĀ·mĀ·ĻĀ factor in it. So… Yes. You’re right: the Yl,m(Īø, Ļ) function is real-valued if – and onlyĀ if – m = 0, in which case eiĀ·mĀ·ĻĀ = 1.Ā Let me copy the table from Feynman’s treatment of the topic once again:The Plm(cosĪø) functions are the so-called (associated) Legendre polynomials, and the formula for these functions is rather horrible:Don’t worry about it too much: just note the Plm(cosĪø)Ā is aĀ real-valuedĀ function. The point is the following:theĀ Ļ(x, t) is a complex-valuedĀ function because – andĀ onlyĀ because – we multiply a real-valued envelope function – which depends on positionĀ only – with eāiĀ·(E/Ä§)Ā·tĀ·eiĀ·mĀ·ĻĀ = eāiĀ·[(E/Ä§)Ā·tĀ āĀ mĀ·Ļ].

[…]

Please read the above once again and – more importantly – think about it for a while. š You’ll have to agree with the following:

• As mentioned in my previous post,Ā the eiĀ·mĀ·ĻĀ factor just gives us phase shift: just aĀ re-set of our zero point for measuring time, so to speak, and the whole eāiĀ·[(E/Ä§)Ā·tĀ āĀ mĀ·Ļ]Ā factor just disappears when weāre calculating probabilities.
• The envelope function gives us the basic amplitude – in theĀ classicalĀ sense of the word:Ā the maximum displacement fromĀ theĀ zeroĀ value. And so it’s that eāiĀ·[(E/Ä§)Ā·tĀ āĀ mĀ·Ļ]Ā that ensures the whole expression somehow captures the energyĀ of the oscillation.

Let’s first look at the envelope function again. Let me copy the illustration forĀ n = 5 and lĀ = 2 from aĀ Wikimedia CommonsĀ article.Ā Note the symmetry planes:

• Any plane containing theĀ z-axis is a symmetry plane – like a mirror in which we can reflect one half of theĀ shape to get the other half. [Note that I am talking theĀ shapeĀ only here. Forget about the colors for a while – as these reflect the complex phase of the wavefunction.]
• Likewise, the plane containingĀ bothĀ the x– and the y-axis is a symmetry plane as well.

The first symmetry plane – or symmetryĀ line, really (i.e. theĀ z-axis) – should not surprise us, because the azimuthal angle Ļ is conspicuously absent in the formula for our envelope function if, as we are doing in this article here, we merge theĀ eiĀ·mĀ·ĻĀ factor with the eāiĀ·(E/Ä§)Ā·t, so it’s just part and parcel of what the author of the illustrations above refers to as the ‘complex phase’ of our wavefunction.Ā OK. Clear enough – I hope. š But why is theĀ the xy-plane a symmetry plane too? We need to look at that monstrous formula for the Plm(cosĪø) function here: just note the cosĪø argument in it is being squaredĀ before it’s used in all of the other manipulation. Now, we know that cosĪø = sin(Ļ/2Ā āĀ Īø). So we can define someĀ newĀ angle – let’s just call it Ī± – which is measured in the way we’re used to measuring angle, which is notĀ from the z-axis but from the xy-plane. So we write: cosĪø = sin(Ļ/2Ā āĀ Īø) = sinĪ±. The illustration below may or may not help you to see what we’re doing here.So… To make a long story short, we can substitute the cosĪø argument in the Plm(cosĪø) function for sinĪ± = sin(Ļ/2Ā āĀ Īø). Now, if the xy-plane is a symmetry plane, then we must find the same value for Plm(sinĪ±) and Plm[sin(āĪ±)]. Now, that’s not obvious, because sin(āĪ±) = āsinĪ± ā Ā sinĪ±. However, because the argument in that Plm(x) function is being squared before any other operation (like subtracting 1 and exponentiating the result), it is OK: [āsinĪ±]2Ā = [sinĪ±]2Ā =Ā sin2Ī±. […] OK, I am sure the geeks amongst my readers will be able to explain this more rigorously. In fact, I hope they’ll have a look at it, because there’s also that dl+m/dxl+mĀ operator, and so you should check what happens with the minus sign there. š

[…] Well… By now, you’re probably totally lost, but the fact of the matter is that we’ve got a beautiful result here. Let me highlight the most significant results:

• AĀ definiteĀ energy state of a hydrogen atom (or of an electron orbiting around some nucleus, I should say) appears to us as some beautifully shaped orbital – an envelopeĀ function in three dimensions, really – whichĀ has the z-axis – i.e. the vertical axis – as a symmetry line and the xy-plane as a symmetry plane.
• The eāiĀ·[(E/Ä§)Ā·tĀ āĀ mĀ·Ļ]Ā factor gives us the oscillation within the envelope function. As such, it’s this factor that, somehow,Ā captures the energyĀ of the oscillation.

It’s worth thinking about this. Look at the geometry of the situation again – as depicted below. We’re looking at the situation along the x-axis, in the direction of the origin, which is the nucleus of our atom.

The eiĀ·mĀ·ĻĀ factor just gives us phase shift: just aĀ re-set of our zero point for measuring time, so to speak. Interesting, weird – but probably less relevant than the eāiĀ·[(E/Ä§)Ā·tĀ factor, which gives us the two-dimensional oscillation that captures the energy of the state.

Now, the obvious question is: the oscillation of what, exactly? I am not quite sure but – as I explained in my Deep BlueĀ page – the real and imaginary part of our wavefunction are really like the electric and magnetic field vector of an oscillating electromagnetic field (think of electromagnetic radiation – if that makes it easier). Hence, just like the electric and magnetic field vector represent some rapidly changing forceĀ on a unit charge, the real and imaginary part of our wavefunction must also represent some rapidly changingĀ forceĀ on… Well… I am not quite sure on what though. The unit charge is usually defined as the charge of a proton – rather than an electron – but then forces act on some mass, right? And the massĀ of a proton is hugely different from the mass of an electron. The same electric (or magnetic) force will, therefore, give a hugely different acceleration to both.

So… Well… My guts instinct tells me the real and imaginary part of our wavefunction just represent, somehow, a rapidly changing force on some unit ofĀ mass, but then I am not sure how to define that unit right now (it’s probably notĀ the kilogram!).

Now, there is another thing we should note here: we’re actually sort of de-constructing a rotationĀ (look at the illustration above once again) in two linearly oscillating vectors – one along the z-axis and the other along the y-axis.Ā Hence, in essence, we’re actually talking about something that’s spinning.Ā In other words, we’re actually talking someĀ torqueĀ around the x-axis. In what direction? I think that shouldn’t matter – that we can write E or āE, in other words, but… Well… I need to explore this further – as should you! š

Let me just add one more note on the eiĀ·mĀ·ĻĀ factor. It sort of defines the geometryĀ of the complex phase itself. Look at the illustration below. Click on it to enlarge it if necessary – or, better still, visit the magnificent Wikimedia Commons article from which I get these illustrations. These are the orbitals nĀ = 4 and lĀ = 3. Look at the red hues in particular – or the blue – whatever: focus on one color only, and see how how – for mĀ = Ā±1, we’ve got one appearance of that color only. For mĀ = Ā±1, the same color appears at two ends of the ‘tubes’ – or toriĀ (plural of torus), I should say – just to sound more professional. š For mĀ = Ā±2, the torus consists of three parts – or, in mathematical terms, we’d say the order of its rotational symmetryĀ is equal to 3.Ā Check that Wikimedia Commons article for higher values ofĀ nĀ andĀ l: the shapes become very convoluted, but the observation holds. š

Have fun thinking all of this through for yourself – and please do look at those symmetries in particular. š

Post scriptum: You should do some thinking on whether or not theseĀ mĀ =Ā Ā±1, Ā±2,…, Ā±lĀ orbitals are really different. As I mentioned above, a phase difference is just what it is: a re-set of the t = 0 point. Nothing more, nothing less. So… Well… As far as I am concerned, that’s notĀ aĀ realĀ difference, is it? š As with other stuff, I’ll let you think about this for yourself.

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# Re-visiting uncertainty…

I re-visited the Uncertainty Principle a couple of times already, but here I really want to get at the bottom of the thing? What’s uncertain? The energy? The time? The wavefunction itself? These questions are not easily answered, and I need to warn you: you won’t get too much wiser when you’re finished reading this. I just felt like freewheeling a bit. [NoteĀ that the first part of this post repeats what you’ll find on the Occam page, or my post on Occam’s Razor. But these post doĀ notĀ analyze uncertainty, which is what I will beĀ tryingĀ to do here.]

Let’s first think about the wavefunction itself. Itās tempting to think it actuallyĀ isĀ the particle, somehow. But it isnāt. So what is it then? Wellā¦ Nobody knows. In my previous post, I said I like to think it travelsĀ with the particle, but then doesn’t make much sense either. Itās like a fundamentalĀ propertyĀ of the particle. Like the color of an apple. But where isĀ that color? In the apple, in the light it reflects, in the retina of our eye, or is it in our brain? If you know a thing or two about how perception actually works, you’ll tend to agree the quality ofĀ colorĀ is notĀ in the apple. When everything is said and done, the wavefunction is aĀ mental construct: when learning physics, we start to think of a particle as a wavefunction, but they are two separate things: the particle is reality, the wavefunction is imaginary.

But that’s not what I want to talk about here. It’s about thatĀ uncertainty. Where is the uncertainty? You’ll say: you just said it was in our brain. No. I didn’t say that. It’s not that simple. Let’s look at the basic assumptions of quantum physics:

1. Quantum physics assumes thereās always some randomnessĀ in Nature and, hence, we can measure probabilitiesĀ only. We’ve got randomness in classical mechanics too, but this is different. This is an assumption about how Nature works: we donāt really know whatās happening. We donāt know the internal wheels and gears, so to speak, or the āhidden variablesā, as one interpretation of quantum mechanics would say. In fact, the most commonly accepted interpretation of quantum mechanics says there are no āhidden variablesā.
2. However, as Shakespeare hasĀ one of his charactersĀ say: there is a method in the madness, and the pioneersā I mean Werner Heisenberg, Louis de Broglie, Niels Bohr, Paul Dirac, etcetera ā discovered that method: all probabilities can be found by taking the square of the absolute value of a complex-valued wavefunctionĀ (often denoted by ĪØ), whose argument, or phaseĀ (Īø),Ā is given by the de Broglie relationsĀ Ļ = E/Ä§ and kĀ =Ā p/Ä§. The generic functional form of that wavefunction is:

ĪØ = ĪØ(x, t) =Ā aĀ·eāiĪøĀ = aĀ·eāi(ĻĀ·t ā kĀ āx)Ā = aĀ·eāiĀ·[(E/Ä§)Ā·t ā (p/Ä§)āx]

That should be obvious by now, as Iāve written more than a dozens of posts on this. š I still have trouble interpreting this, howeverāand I am not ashamed, because the Great Ones I just mentioned have trouble with that too. It’s not that complex exponential. ThatĀ eāiĻĀ is a very simple periodic function, consisting of two sine waves rather than just one, as illustrated below. [It’s a sine and a cosine, but they’re the same function: there’s just a phase difference of 90 degrees.]Ā

No. To understand the wavefunction, we need to understand thoseĀ de BroglieĀ relations,Ā Ļ = E/Ä§ and kĀ =Ā p/Ä§, and then, as mentioned, we need to understand the Uncertainty Principle. We need to understand where it comes from. Let’sĀ try to go as far as we can by making a few remarks:

• Adding or subtracting two terms in math, (E/Ä§)Ā·t ā (p/Ä§)āx, implies the two terms should have the sameĀ dimension: we can only add apples to apples, and oranges to oranges. We shouldnāt mix them. Now, theĀ (E/Ä§)Ā·t and (p/Ä§)Ā·x termsĀ are actually dimensionless: they are pure numbers. So thatās even better. Just check it: energy is expressed in newtonĀ·meterĀ (energy, or work, is force over distance, remember?) or electronvoltsĀ (1 eVĀ =Ā 1.6Ć10ā19 J = 1.6Ć10ā19 NĀ·m); Planckās constant, as the quantum of action,Ā is expressed in JĀ·s or eVĀ·s; and the unit ofĀ (linear) momentum is 1Ā NĀ·s = 1Ā kgĀ·m/s = 1Ā NĀ·s. E/Ä§ gives a number expressed per second, and p/Ä§ a number expressed per meter. Therefore, multiplying E/Ä§ and p/Ä§ by t and x respectively gives us a dimensionless number indeed.
• Itās also an invariant number, which means weāll always get the same value for it, regardless of our frame of reference. As mentioned above, thatās because theĀ four-vector product pĪ¼xĪ¼Ā =Ā EĀ·t ā pāxĀ is invariant: it doesnāt change when analyzing a phenomenon in oneĀ reference frame (e.g. our inertial reference frame)Ā or another (i.e. in a moving frame).
• Now, Planckās quantum of actionĀ h, or Ä§ āĀ h and Ä§ only differ in their dimension: h is measured in cyclesĀ per second, while Ä§ is measured inĀ radiansĀ per second: both assume we can at least measure oneĀ cycleĀ āĀ is the quantum of energy really. Indeed, if āenergy is the currency of the Universeā, and itās real and/or virtual photons who are exchanging it, then itās good to know the currency unit is h, i.e. the energy thatās associated with one cycleĀ of a photon. [In case you want to see the logic of this, see my post on the physical constantsĀ c, h andĀ Ī±.]
• Itās not only time and space that are related, as evidenced by the fact that t ā x itself is an invariant four-vector, E and p are related too, of course! They are related through the classical velocity of the particle that weāre looking at: E/p = c2/v and, therefore, we can write:Ā EĀ·Ī² = pĀ·c, with Ī² = v/c, i.e. the relativeĀ velocity of our particle, as measured as aĀ ratioĀ of the speed of light.Ā Now, I should add that theĀ t ā xĀ four-vector is invariant only if we measure time and space in equivalent units. Otherwise, we have to write cĀ·t ā x. If we do that, so our unit of distance becomes cĀ meter, rather than one meter, or our unit of time becomes the time that is needed for light to travel one meter, thenĀ cĀ = 1, and the EĀ·Ī² = pĀ·cĀ becomes EĀ·Ī² = p, which we also write as Ī² = p/E: the ratio of the energyĀ and theĀ momentumĀ of our particle is its (relative) velocity.

Combining all of the above, we may want to assume that we are measuring energyĀ andĀ momentum in terms of the Planck constant, i.e. theĀ ānaturalāĀ unit for both. In addition, we may also want to assume that weāre measuring time and distance in equivalent units. Then the equation for the phase of our wavefunctions reduces to:

Īø =Ā (ĻĀ·t ā kĀ āx) = EĀ·tĀ ā pĀ·x

Now,Ā Īø is the argument of a wavefunction, and we can alwaysĀ re-scaleĀ such argument by multiplying or dividing it by someĀ constant. Itās just like writing the argument of a wavefunction asĀ vĀ·tāx or (vĀ·tāx)/vĀ = t āx/vĀ  withĀ vĀ the velocity of the waveform that we happen to be looking at. [In case you have trouble following this argument, please check the post I did for my kids on waves and wavefunctions.] Now, the energy conservation principle tells us the energy of a free particle wonāt change. [Just to remind you, a āfree particleā means it’s in a āfield-freeā space, so our particle is in a region ofĀ uniform potential.] So we can, in this case, treat E as a constant, and divideĀ EĀ·tĀ ā pĀ·x by E, so we get a re-scaled phase for our wavefunction, which Iāll write as:

Ļ =Ā (EĀ·tĀ ā pĀ·x)/E = t ā (p/E)Ā·x = t ā Ī²Ā·x

Alternatively, we could also look at p as some constant, as there is no variation in potential energy that will cause a change in momentum, and the related kinetic energy. Weād then divide by p and weād getĀ (EĀ·tĀ ā pĀ·x)/p = (E/p)Ā·t ā x) = t/Ī² ā x, which amounts to the same, as we can always re-scale by multiplying it with Ī², which would again yield the same t ā Ī²Ā·x argument.

The point is, if we measure energy and momentum in terms of the Planck unit (I mean:Ā in terms of the Planck constant, i.e. theĀ quantum of energy), and if we measure time and distance in ānaturalā units too, i.e. we take the speed of light to be unity, then our Platonic wavefunction becomes as simple as:

Ī¦(Ļ) =Ā aĀ·eāiĻĀ = aĀ·eāi(t ā Ī²Ā·x)

This is a wonderful formula, but let me first answer your most likely question: why would we use a relativeĀ velocity?Wellā¦ Just think of it: when everything is said and done, the whole theory of relativity and, hence, the whole of physics, is based onĀ one fundamental and experimentally verified fact: the speed of light isĀ absolute. In whatever reference frame, we willĀ alwaysĀ measure it asĀ 299,792,458 m/s. Thatās obvious, youāll say, but itās actually the weirdest thing ever if you start thinking about it, and it explains why those Lorentz transformations look so damn complicated. In any case, thisĀ factĀ legitimately establishes cĀ as some kind ofĀ absoluteĀ measure against which all speeds can be measured. Therefore, it is onlyĀ naturalĀ indeed to express a velocity as some number between 0 and 1. Now that amounts to expressing it as theĀ Ī² = v/c ratio.

Letās now go back to that Ī¦(Ļ) =Ā aĀ·eāiĻĀ = aĀ·eāi(t ā Ī²Ā·x)Ā wavefunction. Its temporal frequency Ļ is equal to one, and its spatial frequency k is equal to Ī² = v/c. It couldnāt be simpler but, of course, weāve got this remarkably simple result because we re-scaled the argument of our wavefunction using theĀ energyĀ andĀ momentumĀ itself as the scale factor. So, yes, we can re-write the wavefunction of our particle in a particular elegant and simple form using the only information that we have when looking at quantum-mechanical stuff: energy and momentum, because thatās what everything reduces to at that level.

So… Well… Weāve pretty much explained what quantum physics is all about here.Ā You just need to get used to that complex exponential: eāiĻĀ = cos(āĻ) + iĀ·sin(āĻ) =Ā cos(Ļ) āiĀ·sin(Ļ). It would have been nice if Nature would have given us a simple sine or cosine function. [Remember the sine and cosine function are actually the same, except for a phase difference of 90 degrees: sin(Ļ) = cos(Ļ/2āĻ) = cos(Ļ+Ļ/2). So we can go always from one to the other by shifting the origin of our axis.] Butā¦ Wellā¦ As weāve shown so many times already, a real-valued wavefunction doesnāt explain the interference we observe, be it interference of electrons or whatever other particles or, for that matter, the interference of electromagnetic waves itself, which, as you know, we also need to look at as a stream ofĀ photonsĀ , i.e. light quanta, rather than as some kind of infinitely flexibleĀ aetherĀ thatās undulating, like water or air.

However, the analysis above does notĀ include uncertainty. That’s as fundamental to quantum physics as de Broglie‘s equations, so let’s think about that now.

#### Introducing uncertainty

Our information on the energy and the momentum of our particle will be incomplete: weāll write E = E0Ā Ā± ĻE, and p = p0Ā Ā± Ļp. Huh?Ā No ĪE or ĪE?Ā Well… It’s the same, really, but I am a bit tired of using the Ī symbol, so I am using the Ļ symbol here, which denotes aĀ standard deviationĀ of some density function. It underlines the probabilistic, or statistical, nature of our approach.

The simplest model is that of a two-state system, because it involves two energy levels only: E = E0Ā Ā± A, with A some constant. Large or small, it doesn’t matter. All is relative anyway. šĀ We explained the basics of the two-state system using the example of an ammonia molecule, i.e. an NH3Ā molecule, so it consists on one nitrogen and three hydrogen atoms. We had two baseĀ states in this system: āupā or ādownā, which we denoted asĀ baseĀ stateĀ | 1 āŖ and baseĀ stateĀ | 2 āŖ respectively. This ‘up’ and ‘down’ had nothing to do with the classical or quantum-mechanical notion of spin, which is related to theĀ magneticĀ moment. No. It’s much simpler than that: the nitrogen atom could be either beneath or, else, above the plane of the hydrogens, as shown below, with ‘beneath’ and ‘above’ being defined in regard to the molecule’s direction of rotation around its axis of symmetry.

In any case, for the details, I’ll refer you to the post(s) on it. Here I just want to mention the result. We wroteĀ theĀ amplitudeĀ to find the molecule in either one of these two states as:

• C1Ā =Ā ā© 1 | Ļ āŖ = (1/2)Ā·eā(i/Ä§)Ā·(E0Ā ā A)Ā·tĀ + (1/2)Ā·eā(i/Ä§)Ā·(E0Ā + A)Ā·t
• C2Ā =Ā ā© 2 | Ļ āŖ = (1/2)Ā·eā(i/Ä§)Ā·(E0Ā ā A)Ā·tĀ ā (1/2)Ā·eā(i/Ä§)Ā·(E0Ā + A)Ā·t

That gave us the following probabilities:

If our molecule can be in two states only, and it starts off in one, then the probability that it willĀ remain in that state will gradually decline, while the probability that it flips into the other state will gradually increase.

Now, the point you should note is that we get theseĀ time-dependentĀ probabilitiesĀ onlyĀ because we’re introducingĀ two differentĀ energy levels:Ā E0Ā + A andĀ E0Ā ā A. [Note they separated by an amount equal to 2Ā·A, as I’ll use that information later.] If we’d haveĀ oneĀ energy level onlyĀ ā which amounts to saying that weĀ knowĀ it, and that it’s somethingĀ definiteĀ ā then we’d just have oneĀ wavefunction, which we’d write as:

aĀ·eāiĪøĀ = aĀ·eā(i/Ä§)Ā·(E0Ā·t āĀ pĀ·x)Ā = aĀ·eā(i/Ä§)Ā·(E0Ā·t)Ā·e(i/Ä§)Ā·(pĀ·x)

Note that we can always split our wavefunction in a ātimeā and a āspaceā part, which is quite convenient. In fact, because our ammonia molecule stays where it is, it has no momentum: p = 0. Therefore, its wavefunction reduces to:

aĀ·eāiĪøĀ = aĀ·eā(i/Ä§)Ā·(E0Ā·t)

As simple as it can be. š The point is that a wavefunction like this, i.e. a wavefunction that’s defined by a definiteĀ energy, will alwaysĀ yield a constant and equal probability, both in time as well as in space. That’s just the math of it: |aĀ·eāiĪø|2Ā = a2. Always!Ā If you want to know why, you should think of Euler’s formula and Pythagoras’ Theorem: cos2Īø +sin2Īø = 1. Always!Ā š

That constant probability is annoying, because our nitrogen atom never ‘flips’, and we know it actually does, thereby overcoming a energy barrier: it’s a phenomenon that’s referred to as ‘tunneling’, and it’s real! The probabilities in that graph above are real! Also, if our wavefunction would represent some moving particle, it would imply that the probability to find it somewhereĀ in space is the sameĀ all over space, which implies our particle isĀ everywhereĀ and nowhere at the same time, really.

So, in quantum physics, this problem is solved by introducing uncertainty.Ā Introducing some uncertainty about the energy, or about the momentum, is mathematically equivalent to saying that weāre actually looking at a compositeĀ wave, i.e. the sum of a finite or potentially infinite set ofĀ component waves. So we have the sameĀ Ļ = E/Ä§ and kĀ = p/Ä§ relations, but we apply them to nĀ energy levels, or to some continuousĀ rangeĀ of energy levelsĀ ĪE. It amounts to saying that our wave function doesnāt have a specific frequency: it now has n frequencies, or a range of frequenciesĀ ĪĻ =Ā ĪE/Ä§. In our two-state system, n = 2, obviously! So we’veĀ twoĀ energy levels only and so our composite wave consists of two component waves only.

We know what that does: it ensures our wavefunction is being ācontainedā in some āenvelopeā. It becomes a wavetrain, or a kind of beat note, as illustrated below:

[The animation comes from Wikipedia, and shows the difference between theĀ groupĀ andĀ phaseĀ velocity: the green dot shows the group velocity, while the red dot travels at the phase velocity.]

So… OK. That should be clear enough. Let’s now apply these thoughts to our ‘reduced’ wavefunction

Ī¦(Ļ) =Ā aĀ·eāiĻĀ = aĀ·eāi(t ā Ī²Ā·x)

Frankly, I tried to fool you above. If the functional form of the wavefunction isĀ aĀ·eā(i/Ä§)Ā·(EĀ·t āĀ pĀ·x), then we can measure E and p in whatever unit we want, including h or Ä§, but we cannotĀ re-scale the argument of the function, i.e. the phaseĀ Īø, without changingĀ the functional form itself. I explained that in that post for my kids on wavefunctions:, in which I explained we may represent theĀ same electromagneticĀ wave by two different functional forms:

Ā F(ctāx) = G(tāx/c)

So F and G represent the same wave, but they are different wavefunctions. In this regard, you should note that the argument of F is expressed in distance units, as we multiply t with the speed of light (so it’s like our time unit is 299,792,458 m now), while the argument of G is expressed in time units, as we divide x by the distance traveled in one second). But F and G are different functional forms. Just do an example and take a simple sine function: you’ll agree that sin(Īø) ā  sin(Īø/c) for all values of Īø, except 0. Re-scaling changes the frequency, or the wavelength, and it does so quite drastically in this case. š Likewise, you can see thatĀ aĀ·eāi(Ļ/E)Ā = [aĀ·eāiĻ]1/E, so that’s aĀ veryĀ different function. In short, we were a bit too adventurous above. Now, while weĀ canĀ drop the 1/Ä§ in the aĀ·eā(i/Ä§)Ā·(EĀ·t āĀ pĀ·x)Ā function when measuring energy and momentum in units that are numerically equal to Ä§, we’ll just revert to our original wavefunction for the time being, which equals

ĪØ(Īø) =Ā aĀ·eāiĪøĀ = aĀ·eāiĀ·[(E/Ä§)Ā·t āĀ (p/Ä§)Ā·x]

Let’s now introduce uncertainty once again. The simplest situation is that we have two closely spaced energy levels. In theory, the difference between the two can be as small as Ä§, so we’d write: E = E0Ā Ā± Ä§/2. [Remember what I said about the Ā± A: it means the differenceĀ is 2A.] However, we can generalize this and write: E = E0Ā Ā± nĀ·Ä§/2, with n = 1, 2, 3,… This doesĀ notĀ imply any greater uncertaintyĀ ā we still have two states onlyĀ ā but just a largerĀ differenceĀ between the two energy levels.

Let’s also simplify by looking at the ‘time part’ of our equation only, i.e.Ā aĀ·eāiĀ·(E/Ä§)Ā·t. It doesn’t mean we don’t care about the ‘space part’: it just means that we’re onlyĀ looking at how our function variesĀ in timeĀ and so we just ‘fix’ or ‘freeze’ x. Now, the uncertainty is in the energy really but, from a mathematical point of view, we’ve got an uncertainty in the argument of our wavefunction, really.Ā This uncertainty in the argument is, obviously, equal to:

(E/Ä§)Ā·t = [(E0Ā Ā± nĀ·Ä§/2)/Ä§]Ā·t = (E0/Ä§ Ā± n/2)Ā·t =Ā (E0/Ä§)Ā·t Ā± (n/2)Ā·t

So we can write:

aĀ·eāiĀ·(E/Ä§)Ā·tĀ =Ā aĀ·eāiĀ·[(E0/Ä§)Ā·tĀ Ā± (1/2)Ā·t]Ā =Ā aĀ·eāiĀ·[(E0/Ä§)Ā·t]Ā·eiĀ·[Ā±(n/2)Ā·t]

This is valid forĀ anyĀ value of t. What the expression says is that, from a mathematical point of view, introducing uncertainty about the energy is equivalentĀ to introducing uncertainty about the wavefunction itself. It may be equal to aĀ·eāiĀ·[(E0/Ä§)Ā·t]Ā·eiĀ·(n/2)Ā·t, but it may also be equal to aĀ·eāiĀ·[(E0/Ä§)Ā·t]Ā·eāiĀ·(n/2)Ā·t. The phases of the eāiĀ·t/2Ā and eiĀ·t/2Ā factors are separated by a distance equal to t.

So… Well…

[…]

Hmm… I am stuck. How is this going to lead me to theĀ ĪEĀ·Īt =Ā Ä§/2 principle? To anyone out there: can you help? š

[…]

The thing is: you won’t get the Uncertainty Principle by staring at that formula above. It’s a bit more complicated. The idea is that we have some distribution of theĀ observables, like energy and momentum, and that implies some distribution of the associated frequencies, i.e.Ā Ļ for E, and k for p. The Wikipedia article on the Uncertainty Principle gives you a formal derivation of the Uncertainty Principle, using the so-called Kennard formulation of it. You can have a look, but it involves a lot of formalismāwhich is what I wanted to avoid here!

I hope you get the idea though. It’s like statistics. First, we assume weĀ knowĀ the population, and then we describe that population using all kinds of summary statistics. But then we reverse the situation: we don’t know the population but we do haveĀ sampleĀ information, which we also describe using all kinds of summary statistics. Then, based on what we find for the sample, we calculate the estimated statistics for the population itself, like the mean value and the standard deviation, to name the most important ones. So it’s a bit the same here, except that, in quantum mechanics, there may not be anyĀ realĀ value underneath: the mean and the standard deviation represent something fuzzy, rather than something precise.

Hmm… I’ll leave you with these thoughts. We’ll develop them further as we will be digging into all much deeper over the coming weeks. š

Post scriptum: I know you expect something more from me, so… Well… Think about the following. If we have some uncertainty about the energy E, we’ll have some uncertainty about the momentum p according to thatĀ Ī² = p/E. [By the way, pleaseĀ thinkĀ about this relationship: it says, all other things being equal (such as the inertia, i.e. theĀ mass, of our particle), that more energy will all go into more momentum. More specifically, note thatĀ āp/āp =Ā Ī² according to this equation. In fact, if we include theĀ massĀ of our particle, i.e. its inertia, as potential energy, then we might say that (1āĪ²)Ā·E isĀ the potential energy of our particle, as opposed to its kinetic energy.] So let’s try to think about that.

Let’sĀ denote the uncertainty about the energy as ĪE. As should be obvious from the discussion above, it can be anything: it can mean twoĀ separate energy levels E = E0Ā Ā± A, or a potentially infiniteĀ setĀ of values. However, even if the set is infinite, we know the various energy levels need to be separated by Ä§, at least. So if the set is infinite, it’s going to be aĀ countableĀ infinite set, like the set of natural numbers, or the set of integers. But let’s stick to our example of two values E = E0Ā Ā± A only, with A = Ä§Ā so E +Ā ĪE =Ā E0Ā Ā± Ä§Ā and, therefore, ĪE = Ā± Ä§. That implies Īp = Ī(Ī²Ā·E) = Ī²Ā·ĪE = Ā± Ī²Ā·Ä§.

Hmm… This is a bit fishy, isn’t it? We said we’d measure the momentum in units of Ä§, but so here we say the uncertainty in the momentum can actually be a fraction of Ä§. […] Well… Yes. Now, the momentum is the product of the mass, as measured by the inertiaĀ of our particle to accelerations or decelerations, and its velocity. If we assume the inertia of our particle, or itsĀ mass, to be constantĀ ā so we say it’s a property of the object that isĀ notĀ subject to uncertainty, which, I admit, is a rather dicey assumption (if all other measurable properties of the particle are subject to uncertainty, then why not its mass?)Ā ā then we can also write: Īp = Ī(mĀ·v) = Ī(mĀ·Ī²) =Ā mĀ·ĪĪ². [Note that we’re not only assuming that the mass is not subject to uncertainty, but also that the velocity is non-relativistic. If not, we couldn’t treat the particle’s mass as a constant.] But let’s be specific here: what we’re saying is that, if ĪE = Ā± Ä§, then Īv = ĪĪ²Ā will be equal toĀ ĪĪ² =Ā Īp/m = Ā± (Ī²/m)Ā·Ä§. The point to note is that we’re no longer sure about the velocityĀ of our particle. Its (relative) velocity is now:

Ī² Ā± ĪĪ²Ā =Ā Ī²Ā Ā± (Ī²/m)Ā·Ä§

But, because velocity is the ratio of distance over time, this introduces an uncertainty about time and distance. Indeed, if its velocity is Ī² Ā± (Ī²/m)Ā·Ä§, then, over some time T, it will travel some distance X = [Ī² Ā± (Ī²/m)Ā·Ä§]Ā·T. Likewise, it we have some distance X, then our particle will need a time equal to TĀ = X/[Ī² Ā± (Ī²/m)Ā·Ä§].

You’ll wonder what I am trying to say because… Well… If we’d just measure X and T precisely, then all the uncertainty is gone and we know if the energy isĀ E0Ā + Ä§ orĀ E0Ā ā Ä§. Well… Yes and no. TheĀ uncertainty is fundamentalĀ ā at least that’s what’s quantum physicists believeĀ ā so our uncertainty about the time and the distance we’re measuring is equally fundamental: we can have eitherĀ of the two values X = [Ī² Ā± (Ī²/m)Ā·Ä§] TĀ = X/[Ī² Ā± (Ī²/m)Ā·Ä§], whenever or wherever we measure. So we have aĀ ĪX andĀ ĪT that are equal to Ā± [(Ī²/m)Ā·Ä§]Ā·T andĀ X/[Ā± (Ī²/m)Ā·Ä§] respectively. We can relate this toĀ ĪE andĀ Īp:

• ĪX = (1/m)Ā·TĀ·Īp
• ĪT = X/[(Ī²/m)Ā·ĪE]

You’ll grumble: this still doesn’t give us the Uncertainty Principle in its canonical form. Not at all, really. I know… I need to do some more thinking here. But I feel I am getting somewhere. š Let me know if you see where, and if you think you can get any further. š

The thing is: you’ll have to read a bit more about Fourier transforms and why and how variables like time and energy, or position and momentum, are so-called conjugate variables. As you can see, energy and time, and position and momentum, are obviously linked through the EĀ·t and pĀ·xĀ products in theĀ E0Ā·t āĀ pĀ·xĀ sum. That says a lot, and it helps us to understand, in a more intuitive way, why the ĪEĀ·Īt and ĪpĀ·ĪxĀ products should obey the relation they are obeying, i.e. the Uncertainty Principle, which we write asĀ ĪEĀ·Īt ā„ Ä§/2 and ĪpĀ·Īx ā„ Ä§/2. But so provingĀ involves more than just staring at that ĪØ(Īø) =Ā aĀ·eāiĪøĀ = aĀ·eāiĀ·[(E/Ä§)Ā·t āĀ (p/Ä§)Ā·x]Ā relation.

Having said, it helps to think about how that EĀ·t ā pĀ·x sum works. For example, think about two particles, a and b, with different velocity and mass, but with the same momentum, so paĀ = pbĀ ā maĀ·vaĀ = maĀ·vaĀ ā ma/vbĀ = mb/va. The spatial frequency of the wavefunction Ā would be the same for both but the temporalĀ frequency would be different, because their energy incorporates the rest mass and, hence, because maĀ ā  mb, we also know that EaĀ ā  Eb.Ā So… It all works out but, yes, I admit it’s all very strange, and it takes a long time and a lot of reflection to advance our understanding.