My previous post on the speed of light as an angular velocity was rather cryptic. This post will be a bit more elaborate. Not all that much, however: this stuff is and remains quite dense, unfortunately. 😦 But I’ll do my best to try to explain what I am thinking of. Remember the formula (or *definition*) of the *elementary* wavefunction:

ψ = *a*·*e*^{−i[E·t − p∙x]/ħ} = *a*·cos(**p**∙**x**/ħ − E∙t/ħ) + *i*·*a*·sin(**p**∙**x**/ħ − E∙t/ħ)

How should we interpret this? We know an *actual* particle will be represented by a wave *packet*: a sum of wavefunctions, each with its own amplitude *a*_{k} and its own argument θ_{k} = (E_{k}∙t − **p**_{k}∙**x**)/ħ. But… Well… Let’s see how far we get when analyzing the *elementary *wavefunction itself only.

According to mathematical* *convention, the imaginary unit (*i*) is a 90° angle in the *counter*clockwise direction. However, *Nature* surely cannot be bothered about our convention of measuring phase angles – or *time *itself – clockwise or counterclockwise. Therefore, both right- as well as left-handed polarization may be possible, as illustrated below.

The left-handed elementary wavefunction would be written as:

ψ = *a*·*e*^{i[E·t − p∙x]/ħ} = *a*·cos(**p**∙**x**/ħ − E∙t/ħ) − *i*·*a*·sin(**p**∙**x**/ħ − E∙t/ħ)

In my previous posts, I hypothesized that the two physical possibilities correspond to the angular momentum of our particle – say, an electron – being either positive or negative: *J* = +ħ/2 or, else, *J* = −ħ/2. I will come back to this in a moment. Let us first further examine the functional form of the wavefunction.

We should note that both the *direction *as well as the *magnitude *of the (linear) momentum (**p**) are *relative*: they depend on the orientation and relative velocity of *our* reference frame – which are, in effect, relative to the reference frame of our object. As such, the wavefunction itself is relative: another observer will obtain a different value for both the momentum (p) as well as for the energy (E). Of course, this makes us think of the relativity of the electric and magnetic field vectors (** E** and

**) but… Well… It’s not quite the same because – as I will explain in a moment – the argument of the wavefunction,**

*B**considered as a whole*, is actually invariant under a Lorentz transformation.

Let me elaborate this point. If we consider the reference frame of the particle itself, then the idea of direction and momentum sort of vanishes, as the momentum vector shrinks to the origin itself: **p** = **0**. Let us now look at how the argument of the wavefunction transforms. The E and **p** in the argument of the wavefunction (θ = ω∙t – **k**∙**x** = (E/ħ)∙t – (**p**/ħ)∙**x** = (E∙t – **p**∙**x**)/ħ) are, of course, the energy and momentum as measured in *our *frame of reference. Hence, we will want to write these quantities as E = E* _{v}* and p = p

*= p*

_{v }*∙*

_{v}*v*. If we then use

*natural*time and distance units (hence, the

*numerical*value of

*c*is equal to 1 and, hence, the (relative) velocity is then measured as a fraction of

*c*, with a value between 0 and 1), we can relate the energy and momentum of a moving object to its energy and momentum when at rest using the following relativistic formulas:

E* _{v }*= γ·E

_{0}and p

*= γ·m*

_{v }_{0}∙

*v*= γ·E

_{0}∙

*v*/

*c*

^{2}

The argument of the wavefunction can then be re-written as:

θ = [γ·E_{0}/ħ]∙t – [(γ·E_{0}∙*v*/*c*^{2})/ħ]∙x = (E_{0}/ħ)·(t − *v∙x*/*c*^{2})·γ = (E_{0}/ħ)∙t’

The γ in these formulas is, of course, the Lorentz factor, and t’ is the *proper* time: t’* *= (t − *v∙x*/*c ^{2}*)/√(1−

*v*

^{2}/

*c*

^{2}). Two essential points should be noted here:

**1.** **The argument of the wavefunction is invariant**. There is a primed time (t’) but there is no primed θ (θ’): θ = (E* _{v}*/ħ)·t – (p

*/ħ)·x = (E*

_{v}_{0}/ħ)∙t’.

**2.** **The E _{0}/ħ coefficient pops up as an angular**

**frequency: E**. We may refer to it as

_{0}/ħ = ω_{0}*the*frequency of the elementary wavefunction.

Now, if you don’t like the concept of *angular* frequency, we can also write: *f*_{0}* *= ω_{0}/2π = (E_{0}/ħ)/2π = E_{0}/h. Alternatively, and perhaps more elucidating, we get the following formula for the *period *of the oscillation:

T_{0}* *= 1/*f*_{0}* *= h/E_{0}

This is interesting, because **we can look at the period as a natural unit of time for our particle**. This period is

*inversely*proportional to the (rest) energy of the particle, and the constant of proportionality is h. Substituting E

_{0 }for m

_{0}·

*c*

^{2}, we may also say it’s

*inversely*proportional to the (rest) mass of the particle, with the constant of proportionality equal to h/

*c*

^{2}. The period of an electron, for example, would be equal to about 8×10

^{−21}s. That’s

*very*small, and it only gets smaller for larger objects ! But what does all of this really

*tell*us? What does it actually

*mean*?

We can look at the sine and cosine components of the wavefunction as an oscillation in *two *dimensions, as illustrated below.

Look at the little green dot going around. Imagine it is some *mass* going around and around. Its circular motion is equivalent to the two-dimensional oscillation. Indeed, instead of saying it moves along a circle, we may also say it moves simultaneously (1) left and right and back again (the cosine) while also moving (2) up and down and back again (the sine).

Now, a mass that rotates about a fixed axis has *angular momentum*, which we can write as the vector cross-product **L** = ** r**×

**p**or, alternatively, as the product of an

*angular*velocity (

**ω**) and rotational inertia (I), aka as the

*moment of inertia*or the

*angular mass*:

**L**= I·

**ω**. [Note we write

**L**and

**ω**in

**boldface**here because they are (axial) vectors. If we consider their magnitudes only, we write L = I·ω (no boldface).]

We can now do some calculations. We already know the angular velocity (ω) is equal to E_{0}/ħ. Now, the magnitude of *r** *in the **L** = * r*×

**p**vector cross-product should equal the

*magnitude*of ψ =

*a·e*

^{−i∙E·t/ħ}, so we write:

*r*=

*a*. What’s next? Well… The momentum (

**p**) is the product of a

*linear*velocity (

*) – in this case, the*

**v***tangential*velocity – and some mass (m):

**p**= m·

*. If we switch to*

**v***scalar*instead of vector quantities, then the (tangential) velocity is given by

*v*=

*r*·ω.

So now we only need to think about what formula we should use for the angular mass. If we’re thinking, as we are doing here, of some *point *mass going around some center, then the formula to use is I = m·*r*^{2}. However, we may also want to think that the two-dimensional oscillation of our point mass actually describes the surface of a *disk*, in which case the formula for I becomes I = m·*r*^{2}/2. Of course, the addition of this 1/2 factor may seem arbitrary but, as you will see, it will give us a more intuitive result. This is what we get:

L = I·ω = (m·*r*^{2}/2)·(E/ħ) = (1/2)·*a*^{2}·(E/*c*^{2})·(E/ħ) = *a*^{2}·E^{2}/(2·ħ·*c*^{2})

Note that our frame of reference is that of the particle itself, so we should actually write ω_{0}, m_{0} and E_{0} instead of ω, m and E. The value of the rest energy of an electron is about 0.510 MeV, or 8.1871×10^{−14} N∙m. Now, this momentum should equal *J* = ±ħ/2. We can, therefore, derive the (Compton scattering) radius of an electron:Substituting the various constants with their numerical values, we find that *a* is equal 3.8616×10^{−13} m, which is the (reduced) Compton scattering radius of an electron. The (tangential) velocity (*v*) can now be calculated as being equal to *v* = *r*·ω = *a*·ω = [ħ·/(m·*c*)]·(E/ħ) = *c*. This is an amazing result. Let us think about it.

In our previous posts, we introduced the metaphor of two *springs *or oscillators, whose energy was equal to E = m·ω^{2}. Is this compatible with Einstein’s E = m·*c*^{2} mass-energy equivalence relation? It is. The E = m·*c*^{2} implies E/m = *c*^{2}. We, therefore, can write the following:

ω = E/ħ = m·*c*^{2}/ħ = m·(E/m)·/ħ ⇔ ω = E/ħ

Hence, we should actually have titled this and the previous post somewhat differently: the speed of light appears as a *tangential *velocity. Think of the following: the *ratio *of *c *and ω is equal to *c*/ω = *a*·ω/ω = *a*. Hence, the tangential and angular velocity would be the same if we’d measure distance in units of *a*. In other words, the radius of an electron appears as a *natural* distance unit here: if we’d measure ω in *units of* *a *per second, rather than in radians (which are expressed in the SI unit of distance, i.e. the meter) per second, the two concepts would coincide.

More fundamentally, we may want to look at the radius of an electron as a *natural* *unit of* *velocity*. * Huh? *Yes. Just re-write the

*c*/ω =

*a*as ω =

*c*/

*a*. What does it say? Exactly what I said, right? As such, the radius of an electron is not only a

*norm*for measuring distance but also for time. 🙂

If you don’t quite get this, think of the following. For an electron, we get an angular frequency that is equal to ω = E/ħ = (8.19×10^{−14} N·m)/(1.05×10^{−34} N·m·s) ≈ 7.76×10^{20} *radians *per second. That’s an incredible *velocity*, because radians are expressed in distance units—so that’s in *meter*. However, our mass is not moving along the *unit *circle, but along a much tinier orbit. The *ratio *of the radius of the unit circle and *a *is equal to 1/*a ≈* (1 m)/(3.86×10^{−13} m) ≈ 2.59×10^{12}. Now, if we divide the above-mentioned *velocity *of 7.76×10^{20} *radians *per second by this factor, we get… Right ! The speed of light: 2.998×10^{82} m/s. 🙂

**Post scriptum**: I have no clear answer to the question as to why we should use the I = m·*r*^{2}/2 formula, as opposed to the I = m·*r*^{2} formula. It ensures we get the result we want, but this 1/2 factor is actually rather enigmatic. It makes me think of the 1/2 factor in Schrödinger’s equation, which is also quite enigmatic. In my view, the 1/2 factor should not be there in Schrödinger’s equation. Electron orbitals tend to be occupied by *two *electrons with opposite spin. That’s why their energy levels should be *twice* as much. And so I’d get rid of the 1/2 factor, solve for the energy levels, and then divide them by two again. Or something like that. 🙂 But then that’s just my personal opinion or… Well… I’ve always been intrigued by the difference between the original *printed *edition of the Feynman Lectures and the online version, which has been edited on this point. My printed edition is the third printing, which is dated July 1966, and – on this point – it says the following:

“Don’t forget that m_{eff} has nothing to do with the real mass of an electron. It may be quite different—although in commonly used metals and semiconductors it often happens to turn out to be the same general order of magnitude, about 2 to 20 times the free-space mass of the electron.”

** Two** to twenty times. Not 1 or 0.5 to 20 times. No. Two times. As I’ve explained a couple of times, if we’d define a new effective mass which would be twice the old concept – so m

_{eff}

^{NEW}= 2∙m

_{eff}

^{OLD}– then such re-definition would not only solve a number of paradoxes and inconsistencies, but it will also justify my interpretation of energy as a

*two*-dimensional oscillation of mass.

However, the online edition has been edited here to reflect the current knowledge about the behavior of an electron in a medium. Hence, if you click on the link above, you will read that the effective mass can be “about 0.1 to 30 times” the free-space mass of the electron. Well… This is another topic altogether, and so I’ll sign off here and let you think about it all. 🙂