Light and the Problem of Measurement
20th Century physics changed science by realizing that you can only know
something if you can measure it. In some sense, measurement is reality.
Which of the following statements correctly express two key limitations on
measurement which profoundly altered how we think about “reality.”
a) You can't measure the position and the time of something with equal accuracy
b) You can only get information from a distant place in the time it takes the
information to travel to you
c) Whenever you make a measurement you must change the system you are trying
d) both a and b are true
e) both b and c are true
Correct Answer – E
When we measure something as trivial as the length of an object, we are attempting
to make two simultaneous measurements at different locations, one at each end of
the object. In practice the fastest we can get the information from the far end will be
the time it takes light to bring the information to us (the light travel time of the length
of the object).
Light is important because it is the fastest way we know to send information.
When we measure the position of something which is very small, we essentially
have to bounce something off it. When the bounced object comes back to us, we
can figure out where the collision took place and therefore where the original object
was. Of course it won't be there anymore, because we gave it some momentum
when we bounced it. It would be best to use something with very little momentum to
bounce off it, such as light.
Light is important because it is very “light.” It has very little momentum, and so will
only noticeably disturb what we are trying to see if the object is very small indeed
(like a sub-atomic particle).
When we measure the length of an object which is at rest with respect to us, we
factor in the fact that light travels at a certain speed and work out the “real” length.
We know that Einstein's theory of relativity says that everyone else in the world
should see light as moving with the same speed, but some other people will still
not agree with our measurement of the length of the object. Which people will not
agree with our measurement?
a) No one will agree with our measurement. All measurement is subjective
b) Only people who are not at the same location in space as us.
c) Only people who are not moving with the same velocity as us.
d) Only people who are not moving with the same acceleration as us.
e) Everyone will agree with our measurement. All measurement is objective.
Correct Answer – C
Anyone who is at a different location can make the same adjustment we did in
order to get the same result for the length of the object, because the time
difference between the arrival of light from the two ends of the object is exactly
the same for those people as for us. But someone who is moving will have
changed position between the time of arrival of the light from the two ends of the
object. Since they measure the same speed of light as we do, regardless of their
state of motion, the fact that the light from one end of the object has to travel a bit
extra (or a bit less) means that they have a different adjustment to make to
estimate the length of the object. The result is that they think the object (which is
at rest with respect to us) has a different length. (Someone who is accelerating
relative to us will also measure the length of the object differently, but that is a
more difficult case to calculate).
Which of the following is true regarding measurements of the object made by
Observers moving relative to us
a) They will measure a shorter length of the object then we measure
b) They will measure a longer length of the object then we measure
c) They will measure a shorter or longer length depending on which direction they
are moving in
d) They will measure the same length that we measure
Correct Answer – A
Now suppose the other people want to get to the bottom of the discrepancy in our two
measurements of the same object. They decide to take the object with them as they fly
by us at the same speed as before and they now measure its length. Now which of us
will see the object as longer
a) We will measure the object as longer
b) They will measure the object as longer
c) We will both measure the object to be the same length
d) It is essentially random who will measure the object as longer
Correct Answer – B
Whoever is at rest with respect to the object will see the object at its “rest length.”
Anyone who is moving with respect to the object they are measuring will always
see the object as shorter than its rest length.
THERE IS NO WAY YOU CAN MEASURE THE OBJECT TO BE ANY GREATER
THAN ITS REST LENGTH.
Now let's try to measure the position of a very light object, such as an electron.
What wavelength of light should we use to give us the most accurate information
about where the object is?
In other words, if the position of the object is x, how can we measure x with the least
error or uncertainty in its position, x?
a) By using a short wavelength beam of light
b) By using a long wavelength beam of light
c) By using a medium wavelength beam of light
d) All wavelengths of light are equally accurate for our purposes
Correct Answer – A
A longer wavelength of light will be diffracted more easily around the particle. Since
the particle will cast a fuzzy shadow in this case, it will be hard to pinpoint its position.
In general the error in measuring the position of an object, using a beam of light with
wavelength , will be x ~ .
But now think of the light not as a wave, but as a particle, a photon. When it scatters
off the electron it will give it some or all of its momentum. What is the momentum of a
photon (how much momentum does it have to give)?
a) p = m v = 0 (since a photon has no rest mass)
b) p = infinity (since the photon moves so much faster than any other particle)
c) p = E/c = h f/c = h/ (since the photon behaves like an electromagnetic wave)
d) p = m v = m c (since the speed of a photon is c)
Correct Answer – C
Although the photon is a particle, it is still part of an electromagnetic wave, which
means its momentum is given by Maxwell’s formula p = E/c = h f /c = h/. This
means that short wavelength photons have a lot of momentum and long wavelength
photons have relatively little momentum.
If we want to accurately predict the position of the electron 1 second from now, what
wavelength of light should we choose?
a) Use a short wavelength photon
b) Use a long wavelength photon
c) Use a medium wavelength photon
d) All photons (of whatever wavelength) are equally accurate for our purposes
Correct Answer – C
How can we predict the position of the electron in the future? We would need to know its
current position and its velocity.
To know its current position, as we already discussed, we'd like to hit it with a very short
wavelength photon, .
But we also know that this will give the electron a random amount of momentum
somewhere between zero and p = h where p is the momentum of the photon. (Note
that there is no way we can control how glancingly or otherwise the photon will strike the
electron, since to do that we'd have to illuminate it with a constant stream of high
frequency photons, knocking it all the way to hell and back).
So if we hit it with a short wavelength photon we introduce a large uncertainty into the
momentum of the electron, p = h/. This translates into an uncertainty in the velocity,
which means we have no idea where it will be in 1 second's time. To lower the
momentum uncertainty we need to use a long wavelength photon that won't hit the
electron as hard. We must sacrifice information now for information in the future. Of
course we can't afford to hit the electron too softly or we will have no idea where it is
In fact Heisenberg's Uncertainy Principle tells us that for an measurement of position
and momentum the the product of the uncertainties must be greater than h
x p > h/ = h
Suppose you want to get a tan, which would be a better choice of tanning lamp?
Lamp A: Produces 50W of heat at the wavelength of Ultraviolet light
Lamp B: Produces 50 W of light at the wavelength of Yellow Light
Lamp C: Produces 100W of heat at the wavelength of Infra-Red light
Correct Answer – A
Total quantity of light energy reaching your skin does not matter much for
developing a tan. For that you need chemical reactions in your skin which are
produced by the impact of individual photons.
Which lamp produces the most photons?
Lamp A has the highest frequency, which means that since E = h f
each photon has more energy than for the other lamps. Since Lamp A produces
the same total energy as Lamp B, but each photon it makes contains more
energy it must produce fewer photons than Lamp B. Since Lamp C has the lowest
frequency and therefore the lowest energy in each photon, it must be producing
by far the most photons since it has higher overall energy output than the other
Even if your photons are low-energy (long-wavelength) you can still produce more
light by making a lot of them.
But Infra-red photons are simply not energetic enough to make your skin tan (if
they were then we could get a tan by sitting close to the front of class, since the
teacher emits infra-red light). Even visible light doesn't give us much of a tan. If
we want to actually make chemical reactions occur we are best advised to put the
available energy into the biggest packets possible, so we should choose UV light
frequencies for our photons.