The momentum of a photon is $\ p=E/c.$
When a photon reflects off a mirror, it is elastic scattering. Elastic scattering should keep the energy of the photon.
But radiation pressure states, that part of the momentum of the photon will be transferred to the mirror, this is how the photon exerts pressure on the mirror.
Now if $\ p=E/c\ $ and the momentum of the photon changes (part of it gets transferred to the mirror), and the momentum of the photon depends on the wavelength $\ p = h/\lambda$.
Elastic scattering is a form of particle scattering in scattering theory, nuclear physics and particle physics. In this process, the kinetic energy of a particle is conserved in the center-of-mass frame, but its direction of propagation is modified (by interaction with other particles and/or potentials). Furthermore, while the particle's kinetic energy in the center-of-mass frame is constant, its energy in the lab frame is not. Generally, elastic scattering describes a process where the total kinetic energy of the system is conserved.
In Rayleigh scattering a photon penetrates into a medium composed of particles whose sizes are much smaller than the wavelength of the incident photon. In this scattering process, the energy (and therefore the wavelength) of the incident photon is conserved and only its direction is changed. In this case, the scattering intensity is proportional to the fourth power of the reciprocal wavelength of the incident photon.
Now this is a contradiction. How can the energy of the photon be kept, and at the same time how can the photon exert pressure on the mirror, thus losing momentum, and change its wavelength?
$p=E/c$, so the momentum and energy of the photon cannot change without the other. If the photon's energy is kept during elastic scattering (mirror reflection), and the photon still exerts radiation pressure on the mirror, then the photon's momentum has to change (part of it needs to get transferred to the mirror), so the energy needs to change too.
- Does the wavelength of the photon change during elastic scattering (mirror reflection)?