There is a relationship between photon momentum
and photon energy
that is consistent with the relation given previously for the relativistic total energy of a particle as
. We know
is zero for a photon, but
is not, so that
becomes
or
To check the validity of this relation, note that
for a photon. Substituting this into
yields
as determined experimentally and discussed above. Thus,
is equivalent to Compton’s result
. For a further verification of the relationship between photon energy and momentum, see
[link] .
Photon detectors
Almost all detection systems talked about thus far—eyes, photographic plates, photomultiplier tubes in microscopes, and CCD cameras—rely on particle-like properties of photons interacting with a sensitive area. A change is caused and either the change is cascaded or zillions of points are recorded to form an image we detect. These detectors are used in biomedical imaging systems, and there is ongoing research into improving the efficiency of receiving photons, particularly by cooling detection systems and reducing thermal effects.
Photon energy and momentum
Show that
for the photon considered in the
[link] .
Strategy
We will take the energy
found in
[link] , divide it by the speed of light, and see if the same momentum is obtained as before.
Solution
Given that the energy of the photon is 2.48 eV and converting this to joules, we get
Discussion
This value for momentum is the same as found before (note that unrounded values are used in all calculations to avoid even small rounding errors), an expected verification of the relationship
. This also means the relationship between energy, momentum, and mass given by
applies to both matter and photons. Once again, note that
is not zero, even when
is.
(a) Calculate the momentum of a photon having a wavelength of
. (b) Find the velocity of an electron having the same momentum. (c) What is the kinetic energy of the electron, and how does it compare with that of the photon?
(a) Calculate the wavelength of a photon that has the same momentum as a proton moving at 1.00% of the speed of light. (b) What is the energy of the photon in MeV? (c) What is the kinetic energy of the proton in MeV?
(a) Find the momentum of a 100-keV x-ray photon. (b) Find the equivalent velocity of a neutron with the same momentum. (c) What is the neutron’s kinetic energy in keV?
Consider a space sail such as mentioned in
[link] . Construct a problem in which you calculate the light pressure on the sail in
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backward on the car. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?
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