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Q & A: uncertainty principle for light

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Most recent answer: 03/06/2013
Q:
An electromagnetic wave is characterized by the fact that it travels at the speed of light. So, since it's velocity is always known, wouldn't the uncertainty principle be inapplicable since it's velocity is a constant? Or would it simply result in the fact that it's position is always unknown? I'm really new to concept of quantum mechanics and it's always fascinated me. This was one question that perplexed me.
- Gurniwaz (age Bal)
Edmonton, Alberta, Canada
A:
Remember that the standard uncertainty relation is for the position and momentum vectors. It applies quite nicely to light, although the speed (not vector velocity) is fixed.

Let's think about a beam going in the z direction on average. It's got some finite area in the xy plane, so the uncertainty relation says it has to has some spread in momenta in the xy plane. That directly means some spread in velocities in the xy plane. The narrower the beam the bigger that spread has to be. That's why you can't keep even a laser beam confined to a narrow area forever.

This limitation is a very familiar aspect of classical optics. For example, in order to get a reasonably narrow spot on the moon (used for some measurements) from a laser on earth, you have to first spread out the laser beam and then collimate it with a big lens. If you start with a narrow beam, the angular spread is so big that you get a very large fuzzy beam hitting the moon.

Now the spread along the z direction is a little less intuitive, since there's virtually no spread in the component of the velocity in the z direction. However, what we're interested in is momentum, p, not velocity. If the beam in the z direction has some finite length, a more-or-less known z-position at some time, then there has to be a range of slightly different wavelengths (λ's) in it. Otherwise the component sine waves couldn't cancel outside that position range. The range of λ's  gives a range of z-momenta for the photons of the beam, the light quanta, even though they're all traveling nearly in the z direction, since p=h/λ. ("h" is Planck's constant.) In terms of quantities that you can (in principle) notice without doing single-photon detection, that range of wavelengths corresponds to a small range of colors.

Mike W.

(published on 03/06/2013)

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