"How can photons have momentum?" (if they do not have mass)
How is this momentum defined (equations)?"
Wikipedia on momentum says:
"In Newtonian mechanics, momentum (...) is the
product of the mass and velocity
of an object."
Interestingly, Wikipedia does not mention,in that context, the De-Broglie wavelength equation which defines momentum alternatively as:
Both equations put in one give mv = h/λ, which shows that p (momentum) is not derived from mass alone. One and the same momentum may come from masses of different velocities or, different from that, from different waves of different lengths. Momentum (and energy) unifies both causes in one and the same effect.
It's an empirical finding that photons have momentum which is being made use of by the solar sail, for instance, and it is a definition of terms to say that photons have "mass", if this is some other way to express the equivalency of both formulas above. To be precise, photons are not said to have mass, they are said to have "rest mass" zero and they are said to have "relativistic mass".
Both terms are intricate, and the following remarks might not explain:
Rest mass as a term literally refers to weights put on both sides of a scale: masses are weighted "at rest" to each other. However, the unit of mass is "kg", and mass is measured by acceleration, by gravitational acceleration (of the earth). Thus, photons' rest mass is considered zero as particles that live at speed of light cannot be accelerated: there do not own "mass" because they cannot be gauged one against another. A particle of wavelength x cannot be put on scales against particle of wavelength y. The term "rest" in fact, in my opinin, may be considered misleading, as it is earth's acceleration, thus "speed not rest" that is being measured. If this is understood it becomes clear that photons which have speed of light cannot be accelerated. The speed of light separates the realm of mass (in strict sens) from the realm of waves. Waves differ in frequency, not velocity, Masses differ in velocity, not frequency. As photons cannot be gauged by different velocities they gain by gravitational (and inertial) acceleartion, they are said to have "rest" mass zero.
On the other hand, there is the term "relativistic" mass. This term is opposed to "rest mass" which is no trivial knowledge, it is just the opposite.
Relativistic mass is a term to acknowledge that restmassless photons do have mass in the form of "relativistic mass" as they own momentum (solar sail impulse) just like Newtonian masses have. It is a mode of putting it.
How can photons have momentum if they do not have mass?
The de Broglie formula shows that mass is not a prerequisite for momentum.
This formula is based on empirical findings; there is no "reason why".
To reconcile these findings with Newtonian formula photons are said to have "rest mass" zero (no accelaration to variation in speed) on one hand, and on the other hand, they are said to have "relativistic" mass (in that sense they do have mass(!)) as they exert mechanical momentum (do not only transfer energy).