How does the Heisenberg uncertainty principle apply to photons?

More broadly, Heisenberg posits that we cannot be entirely "sure" of a body's position in space. Obviously, as the body's mass increases, there is a better chance at being "certain".

#Deltax*mDeltanu >= h/(4pi)#

Where #Deltax# is the position's uncertainty, hence #Delta#; likewise with #Deltanu#.

However, a photon is considered to be "massless" with modern theories of physics. I'm not well versed in physics, so I can't elaborate, but if someone can, do so! Let's play with the equation a little:

#Deltax*0*Deltanu >= h/(4pi)#
#Deltax >= h/(4pi*0*c)#
#Deltax >= h/(0)#

As you know, anything divided by #0# is an undefined quantity, hence:

#Deltax >= "undefined"#

There is a different version of the Heisenberg Uncertainty Principle for photons in three dimensions. (In one dimension, they are both #>=# #ℏ//2#.)

As presented in this paper, in their notation, we have from Eq. #5# an uncertainty relation for three dimensions for photons:

#DeltarDeltap >= 4ℏ#

where:

• #r# is the radial position for a given direction in three dimensions (or the "center of energy"). #Deltar# is then the "spread" in the energy of the light from its center.
• #Deltap = h/(Deltalambda) = (hDeltanu)/(c)# is the uncertainty in momentum for photons from the de Broglie relation. That is, #lambda = h/p# and #nu = c/lambda#. After realizing that #Deltalambda = h/(Deltap)#, the rest follows from there.
• #lambda# is the wavelength in #"m"# and #nu# is the frequency in #"s"^(-1)#.
• #c# is of course the speed of light in #"m/s"#.
• #ℏ = h//2pi# is the reduced Planck's constant.

The version for electrons in three dimensions is

#DeltarDeltap >= (3ℏ)/2#,

where here, #Deltap = mDeltav# like usual, since electrons have nonzero rest mass and photons have zero rest mass.

The interesting bit here is that the uncertainty relation for photons only extends "properly" to three dimensions (each dimension multiplied by #ℏ/2#) as the average photon momentum approaches infinity, i.e. as we use very high frequencies of light.

In other words, in the limit as #bb(nu -> oo)#, photons obey the same uncertainty principle as for electrons (except for the definitions of their momenta).