Location or Momentum

Bruster Rockit: Space Guy!                           by Tim Rickard

A key element of quantum mechanics is Heisenberg’s uncertainty principle, which forbids the simultaneous measurement of the position and momentum of a particle along the same direction, as so aptly illustrated by Tim Rickard above. 

  for a photon, where E is the energy, c is the speed of light and p is the momentum.    So the momentum of a photon is equivalent to the energy of the photon divided by the speed of light or p =  E/c  where E is also related to the frequency of the photon by Planck’s Constant E = hf.   h is Planck’s constant and f is the frequency assigned to the photon.   f is also related to the wavelength of the photon by f = c/λ.  

So E = hc/λ = cp       Therefore    p = h/λ

But we know the values for both h (6.26×10^-34 joules sec.) and for λ if we know the color of the photon.  Usually if we are dealing with coherent light (red laser for example) then we know the wavelength λ very accurately.   Thus we know the momentum very accurately.

There is another factor in this equation – spin angular momentum of the photon which is independent of its frequency.  Spin angular momentum is essentially circular polarization for a photon.  Angular momentum is ±h/2π.   It is the helical momentum of the photon along its flight path.   In order to pin down the momentum we also need to know its angular momentum, but it is a constant that is either spinning one way or the other, no half spins no quarter spins just +h/2π or -h/2π.   

The key for this discussion is that we know the momentum for any photon if we know its wavelength.   p = h/λ and the direction of its spin ±h/2π.   According to Heisenberg’s principle we cannot know the location of the photon if we know its momentum.  Since we do know its momentum we are at a loss to try to pin the location to a particular spot such as through a narrow slot or pinhole.  

Whenever we try to fit a photon through a slot, we are trying to pin down the location as it goes through the slot.  The narrower we make the slot the closer we are trying to pin it down.   Nature resists by causing havoc with our measurements – fuzzy behavior/weird effects.

Pair Production

Pair production is a possible way for nature to slip one by us – putting a photon through both slots simultaneously, thus confounding our measurements completely.   When a photon hits an obstacle such as the thin barrier between the two slots, it melds through the slots around the barrier as in my earlier posts or possibly down-converts to a lower frequency pair of photons (or up-converts to a higher frequency) through pair production (conserving energy by the frequency change).  These pairs recombine on the far side of the barrier through an up (or down) conversion process causing an effective interference due to jiggling in the conversion process. 

Our barrier strip knocks the photon silly, and it responds by splitting up, zipping through the two slits independently, then recombining in a way that looks like interference.

Virtual Photons 

Another type of pair production would be through creation of a virtual photon – a pair with one real and one virtual as also mentioned in an earlier post.   The scenario is the same – barrier knocks photon silly, virtual photon forms, passes through other side, then effectively recombines while interfering with the “real” one.   The original and virtual photons could actually be down converted or up converted photon pairs that recombine by up or down conversion causing interference-like behavior.

In either case, blocking one slit or the other would prevent melding and also prevent pair production as well as the formation of virtual photons.

Pair production through down/up conversion and/or virtual pairs would fit better with particles with mass acting like waves that cause interference when passed through slits.  Even bucky balls and cats could potentially form virtual pairs if moving close to the speed of light.   Well, again, maybe not cats.

 Oldtimer

Thought Experiment – Photons at radio frequencies

I like to do thought experiments.   Many of them lead to dead ends, but I write most of them down anyway because I’ve found that very often I will go down another thought path and end up crossing an earlier one.  Then things get interesting.  The one below includes a thought experiment that dates to Fri, 25 Sep 1998, and I’ve updated it a little to my more recent thoughts.  If you have an idea, keep it around as it may become useful someday.  This one is mostly useful to describe how thought experiments work for me.

Right now I’m still spending some time with the speed of light and with electromagnetic waves, such as from a radio, since both propagate at the speed we call c.   It is easy to visualize a radio wave as a wave because we have always called it that: radio wave.  Duh…, and something radiating in all directions from an antenna is more of a reminder of waves in a pond after we toss a rock in.  But if photons are discrete and quantized (but sometimes seem to act as waves), how do you visualize a radio wave as a quantizable entity? 

Photons at Radio Frequencies 

If light and radio are both in the same electromagnetic spectrum, just when do you stop quantizing and start waving?  Stop photoning and start rippling?  Can you just get rid of the waving altogether and talk about photons at any frequency?  The object of this thought experiment is to start with a simple radio wave and see if it can be described as a photon eventually.   In other words, find out if all electromagnetic waves are photons and maybe even decide how big they are.   After all, if they can be shown to be photons always, then the quantum weirdness could explain lots of things, including light diffraction and interference at radio and lower frequencies in a different way than as a wave – particles even.  The object is to take a whack at this duality thing physicists are hung up on.

I am visualizing first a rather coherent radio signal (such as from a radio transmitter generating its carrier frequency) from a typical antenna as it expands in a sphere or bubble front.  I’m thinking of the very first cycle after the carrier is turned on, but it could apply to any peak in the signal as it propagates outward.  The leading edge of the bubble (or any individual peak) as I see it, is an equal-strength signal that covers the surface.    I am visualizing on that bubble (on the surface) countless whorls of small fields rotating in opposite directions and in close proximity to each other.   (I’ve just made them up for thought purposes, hoping that they can become photons later.)

For example, pick one of the circular whorls and it is rotating clockwise and all around it on every side are other whorls/fields rotating counterclockwise, all the same size whatever that is.  Adjacent to any of those you pick are small fields rotating clockwise, the pattern being like a polka-dotted balloon with the black dots rotating one way and the white dots rotating the other.   Between these whorls, the fields are moving in the same direction on all sides.    For example, the one on the left is spinning clockwise and the one next to it on the right is spinning counter clockwise.  In between the whorls, the fields are both moving down – same direction.   The same thing applies for the fields above and below, adjacent fields moving in the same direction.  So far, so good.  These whorls are helping each other out as they move along.

Now, I look at the small rotating field and realize that since the bubble is moving at the speed of light, the rotating field, if it had a crayon, cannot draw a line on the bubble at all, or it would be doing so at faster than the speed of light. Therefore, as each point of the rotating field is drawn on the surface of the bubble, it immediately falls behind the bubble and describes a spiral arc in space that, when looked at in profile, from the top and from the side, could be the sinusoidal magnetic field and its companion electric field that we detect as the field passes us. Any following energy such as for a continuous signal would fall into step with the leading bubble, describing subsequent bubbles behind the first one, but in sync. For now, I am still looking at a single cycle and things are looking better for photons.

Thus, I see countless rotating fields dragging behind the bubble, the bubble that represents the front of the beginning of the radio signal.  I visualize that the size of the rotating fields do not change, but are related to the frequency of the carrier, such that the higher the frequency, the faster they rotate and the smaller they are.   The energy is related to the frequency by Planck’s constant as e = hf.   This means the faster they rotate, the greater the energy.  (Whatever energy these whorls have, it is exceedingly small, but there are lots of them.)  

Now, we need to do a little head scratching.  Can we speculate as to the size of the whorls?  I think we can establish the maximum size of each whorl by assuming that if these are actually photons, then the energy contained in each photon is located in a flattened disk due to relativistic effects as in my drawing in “Speed of Light Regulated“.   If it is rotating around the whorl as in our thought experiment, then no part of the rotating photon can exceed the speed of light.  Therefore, the trip around the circumference of the whorl cannot be faster than the speed of light.

We also have decided to go down a particular path of our thought experiment by assuming that the whorl rotates at the same rate as the frequency of the carrier and so makes a single turn in one wavelength, λ.  We know that  λ=c/f  and also that the circumference = Πd =  λ.   or d = λ/Π.  The diameter of the whorl can’t be more than the wavelength divided by pi.  For a blue photon which has a wavelength of 450nm, the diameter would be d= 143 nm which is quite small, about 1/3 of the wavelength.   For a radio wave of 105 mhz the photon can’t be larger than  0.9 meters, about 1 yard, still about 1/3 of the wavelength, but about 630,000 times larger than for a blue photon.  

There is nothing to say that there can’t be billions upon billions of these photons overlapping each other at every point of the bubble.   In fact, there has to be.   Energy is being poured into the antenna and the output is billions upon billions of photons in ever expanding bubbles.  A photon has energy that we can calculate as e = hf, but h is very small, 6.26×10^-34 joules sec.   For a blue photon this is e = 4.2×10^-14 joules and for a 105mhz photon, e = 6.3 x 10^-28 joules, which is much much smaller.   To put this into perspective it would take 5400 x 10^27 photons (105mh photons) to make one watt-hour of energy.    That’s 5400 billion billion billion photons (roughly) for each watt hour! 

As our bubble expands, the surface “stretches,”  and it is that stretching, as the surface field in dynamically expanding, that causes the field to eventually separate into individual photons as the signal strength falls over huge distances and the wave identity is forever lost – all we have left is photons to try to detect.  The whorls represent in my visualization, the photon/particle aspect of the wave, as the wave is separated into compact quantum induced by the need to tightly spin along the bubble front, each whorl being my visualization of the photon.  

As the field further expands, the various quantum (whorls) begin to separate and the interaction with its neighbors becomes less distinct. Each quantum continues to have the same energy but its neighbors contribute less and less to its effect when exposed to a detector, unless lenses or antennas are used.

If we look at the field as it arrives at a detector (say an antenna), we detect the arrival of the photons as energy buildup on the antenna from one of the peaks involving billions of photons of the carrier followed by a decrease in signal and then a rise to the next peak.  The photon, being on the same order of magnitude as the detecting antenna (by design of the antenna based on electromagnetic theory, not photon theory) is easily captured, but billions upon billions need to arrive in order to make a good signal.   Maybe this dualality of wave / particle can be moved to quantum only – particles.

Enough is enough.  The thought experiment has run its course and it is time to have someone else pick it apart or perhaps add to it.  Well…. after all, it is just a thought experiment, but it’s mine and I’ve now written it down for others to consider or pick at – which should be an easy task.  

Oldtimer

Speed of light regulated

What determines the speed of light? We know that it is a limiting factor for all physical objects. We have heard it time and again – nothing goes faster than c! Nothing.   Can we determine why it is regulated to c?  I think we can.  It is all a matter of relativity.

Suppose we consider the idea that the photon is disk-shaped due to space distortion.  (See figure at left) The photon is traveling at the speed of light and the space distortion equations tell us that, from our perspective, the photon’s dimensions in the direction of travel are greatly shortened, essentially like a very thin pancake set perpendicular to the direction of travel.

We know that the photon is a ball of energy related to its frequency and we know that the frequency determines the color of light that we can actually detect with our eyes. A blue photon has both a higher frequency and energy than a red photon. All the energy is confined to that flat pancake moving along at the speed of light, c.

Now we come to a slight separation from the earlier argument that the clock of the photon is stopped and nothing wiggles in a photon with a stopped clock. That is, in my opinion, true for the photon, but we are talking about the photon here from an observer’s point of view, not the photon’s perspective.  From the observer’s point of view, the photon moves with measurable velocity, measurable frequency, measurable energy, and thus potentially real live vibrational modes as seen by a clever observer. The time experienced by the photon is still zero from start to finish of its journey, but the observer still knows it is moving at a particular pace and also vibrating as it goes.

The photon cannot vibrate in the front to back direction because to do so implies that the vibration mode that goes toward the back lags behind and then it could never catch up without exceeding the speed of light. This implies that the photon vibrates from side to side or possibly either way around the rim of the disk and never front to back (well, maybe a very little, as explained later). The ripples in the disk are shown greatly magnified in the figure of the photon in flight above. Vertically polarized photons vibrate from rim to rim in a vertical fashion, horizontally polarized vibrate side-to-side and circular polarized photons vibrate around the rim, to and fro, and can even be lopsided a little producing an elliptical polarization.  These types of polarization exist in our real world and we can separate photons with various filters. prisims, and crystals.

Now let us suppose that we consider the vibrational modes of the disk in a little more detail. It seems that any vibration would cause at least some ripples along the disk, and that these ripples must involve at least some bunching of energy producing some motion front to back. Suppose these ripples are constrained to some minimum amplitude in order to even exist.  Could it be that these ripples actually limit the speed of the photon to some factor that actually defines c?   They can.

In other words, if the speed of the photon were to try to increase beyond the speed of light, as seen by our (any) frame of reference, the continuing shortening of the disk would reduce the amplitude of the ripples and potentially slow the photon back down to a speed where the ripples can still exist in our frame of reference. This provides a theory of how the speed of light is established and limited to a particular speed, “the speed of light”, for a photon. The speed of light is about 299,792,458 meters per second, usually symbolized by the letter “c”.

My thought is that when a photon or other particle is emitted, it probably takes off at the highest possible speed that is limited by the speed at which it can still maintain vibrational modes that can exist within an observer’s frame of reference. This is the speed of light as we know it and the regulator is the relativistic shortening of the disk in the direction of travel as seen by the observer. This shortening reduces the amplitude to a point that is sustainable for the energy it contains. If a photon can vibrate longitudinally, it would still be limited in amplitude to the size constrained by the disk in the same way described above, essentially very little, and regulated by the speed. The photon will always go at the maximum speed it can maintain (and no faster) within a given frame of reference. 

Why photons all travel at the same speed 

So why do all photons travel at the same speed?  Even for two observers traveling in differnt directions both measuring the same speed for a photon crusing by?  First lets consider some facts:  Blue light has a frequency, f, entered on 7.88×10^14 HZ and a corresponding energy e of 5.22 10^ -19 Joules. Red light has a frequency centered on 3.79×10^14 HZ and a corresponding energy e of 2.373 x 10^-19 Joules. Since they have different energies and different frequencies, would they not reach that equilibrium at different speeds?

For the answer, consider this:  The energy and frequency of all photons are related to a simple constant, e= hf.   Where h= Planck’s constant= 6.6262*10 ^-34 J s (Joule second).  So the relationship of the energy of photon to its frequency is a constant.

Or put another way, h = e/f for all photons. The ratio of the energy of a photon to its frequency is a constant for all photons. Thus we can see that the sustainable amplitude is somehow related to h and all photons are regulated to the same speed, which we measure as c in any frame of reference.   For example, if you divide the frequency into the energy for the blue and then the red light photons above, the ratio comes out the same.   The result is h, a constant for all photons.   These relationships are well known in the physics world.

However, the frame of reference is a key element, which means that the regulation to c takes place in any frame of reference because the shortening of the disk is related to the speed within the reference of the observer (any observer and all observers), and thus become regulated to c in all frames of reference. If the frame of reference were within a spaceship traveling at near relativistic speed and attempting to measure the speed of a photon going in its direction, the photon’s speed would still be c in respect to the spaceship. The length contraction relative to the spaceship would just be enough to regulate the speed of light measured by the spaceship to agree with the speed observed on earth.

There is a little of cart before the horse-trading going on here. The equations for space distortion and for time dilation both involve the square root of a term that would be a negative number if the photon exceeded the speed of light. In order for us to consider that the photon might even try to go faster than the speed of light, the equation would need some modification to make things right. It might well be that in order for the photon to reach c it might initially slip into “superluminal” speed, but it would quickly be brought back to within the speed bounds by the disk shortening along the path of flight and the reduction of the amplitude of the energy waves in the disk, the shortening taking place in the frame of reference of the measurer/observer.  Even when there are no observers and no measurement taking place, the photon is not alone.  Other particle exist, even in a vaccum, virtual particles for example.  These make up a frame of reference too, so the photon is always locked in to c. 

All photons strive to go faster than c all the time, but are held back by the relativistic effect of space shortening’s effect on the need to vibrate.

This latter discussion begs a new question. If the vibrational modes could somehow be frozen so that they do not need to vibrate in flight as we observe them, could they then travel at an unregulated speed beyond the speed of light? Consider a particle that starts out at absolute zero. In that case all the parts are locked together and nothing moves and therefore has no vibration to sustain. What is to regulate the speed of that particle? Can we then reach superluminal speeds for such a particle?  I don’t think so because to get it up to speed, energy must be applied.  In the case of a photon, the energy comes from the change in states of an electron around an atom or a collision of some sort that generates a photon.  Once we have energy for a massless particle, it has to cruise along at c.

It may be possible that a photon in flight passing thorough from another dimension/universe might have motion relative to us moving so fast that there is no effective vibration taking place during the time of its passage, effectively frozen during its passage.   Such a particle might zip by at superlumal speed.  Of course we would never know it passed unless it hit something on the way.  Then we would have a mess.  

Physicists call hypothetical particles that travel at superlumal speeds tachyons, (hypothetical so far, that is).

There is one other consideration that acts as a speed regulator.  Something I hinted at above.   c is the speed at which the time and distance experienced by a photon reduces to zero.  I stated that a photon always strives to go faster than c.   Each time it does, it slips into imaginary time and pops back to c, and has to stay there.   Look at it another way.  The photon traveling at c arrives the instant it leaves (from the photon’s perspective).  If it went any faster than c, would it arrive before it left?  I don’t think so and so the photon cannot go any faster.

Hopefully I’ve given you something to think about.  

Oldtimer

Article and drawing, Copyright 2006, 2007,

James A. Tabb