# Is the constant c (or it's measurement) subject to the uncertainty principle?

1. Jun 17, 2010

### stone1

I appologize if my questions make no sense (very likely as I do not have a physics background).
1. The uncertainty principle states that precision of measurement is identified only down to the Planck's constant. If this is so, how can one measure the speed of light with infinite precision? ANd then how can one claim that such a number (constant) exists?
2. If a beam of light goes through a small hole and then hits a screen, one gets the Airy pattern. Suppose one can measure the time between (a) when a photon is emited and (b) when the photon hits the screen beyond the hole. Whould the time measured depend on where the photon ends up on the screen behind the hole? Constant light speed suggests that it should take longer for photons that hit farther away from the middle of the screen. Is that true or am I missing something?

2. Jun 17, 2010

### mathman

The uncertainty principle refers to the measurement of position and momentum of particles, not the speed of light in a vacuum.

3. Jun 17, 2010

### JesseM

But if a photon traveling through a vacuum is known to have been emitted at a particular position and a particular moment in time (say, because the light source was covered by a shutter that only opened at a single known time), is there any possibility in relativistic quantum physics that a detector could detect it at a distance/time interval such that distance/time is different from c? Or does the uncertainty in momentum reduce solely to an uncertainty in direction (or an uncertainty in the frequency given by f=pc/h) for a photon?

4. Jun 17, 2010

### Staff: Mentor

A photon's momentum is not a function of its speed but its frequency, as you mention.

5. Jun 17, 2010

### stone1

So you are saying that one can measure the speed of a photon with infinate precision, correct? Also, is the exact speed of light (the actual number) derived from theory or is it based on empirical observations?

6. Jun 18, 2010

### f95toli

It was based in empirical observations. But nowadays c has a defined value which is then used to define the meter.

7. Jun 18, 2010

### hpda

There is a way to get a numerical value of c derived from the theory of Electromagnetism. From the Maxwell's Laws, you can get the wave equation for the light, and the parameter $$\frac{1}{\sqrt{\mu_{0}\epsilon_{0}}}$$ is the speed of the light in vacuum, it is to say, the c value. But again, you need to measure both constants, and when doing that, you are subject to the Heissenberg Principle.

The problem of that, I think, is that the Planck Constant is so small that you will get a lot of digits before you reach that limit for the light. So, the actual number you say is very accurate, since we have a lot of numbers in it, but is still subject to the uncertainty principle, and we'll get there some time.

8. Jun 18, 2010

### stone1

Thanks a lot. That helped.

9. Jun 18, 2010

### Staff: Mentor

Hi hpda, welcome to PF.

This is incorrect. If you want to measure the position and momentum of a single particle then you are subject to the uncertainty principle. AFAIK there is no velocity uncertainty principle and in any case it would only apply when trying to simultaneously measure the velocity and its conjugate.

For light the momentum is not a function of velocity so the uncertainty principle is not really relevant for measuring the speed of light regardless of the precision with which the position is known.

Last edited: Jun 18, 2010
10. Jun 20, 2010

### 01030312

as much as i have heard, the velocity operator(or distance/time operator) in dirac's equation, is - (speed of light)*(a dirac matrix). So if I solve the dirac's equation for massless particle, i think I will get a state which is linear combination of dirac matrices, normalised to 1. So if I apply velocity operator for, say x-th component, on it, wont i get, by writing total velocity as vector sum of each component, the magnitude of velocity to be 'c' ? That is, velocity operator , by definition, gives certainity in value of 'c', as the total velocity, quite independent of behaviour of position wave function of photon, like in double slit experiment.

11. Jun 24, 2010

### stone1

This last discussion helps me even more. I guess the motivation of my question is the following: If one were to actually measure the speed of light, one would need a specific initial position in timespace (A) of the photon and a specific end position in timespace (B). Then one needs to calcualte the difference between A and B. Basically my question is "Can one identify both A and B exactly?" Doesn't identify A exactly affect our ability to measure B exactly? Like the airy pattern formed by photon going through a small whole? B is all over the place.

12. Jun 26, 2010

### Staff: Mentor

Let's say that you were trying to accurately measure the position of a single particle. Now, the uncertainty principle says that $$\sigma_x \sigma_p \ge {\hbar}/{2}$$. So, what constraints does the uncertainty principle place on your ability to measure position to arbitrary accuracy? The answer is none whatsoever. The uncertainty principle only prevents you from measuring both position and momentum to arbitrary accuracy. As long as we are not interested in the momentum, we can measure the position to arbitrary accuracy!

In any case, as a practical matter the speed of light is not measured using a single photon. Typically it is measured using interferometry from a coherent source such as a laser. Coherent states don't even have a definite number of photons, but there is an "uncertainty relation" between the number of photons and the phase in a coherent state. Since we are uninterested in the number of photons and only in the phase, again the uncertainty principle does not limit the accuracy.

Last edited: Jun 26, 2010