Yes, momentum and energy can be described by wave equations in certain physical contexts. In quantum mechanics, particles such as electrons and photons can exhibit wave-particle duality, meaning they can exhibit both particle-like and wave-like properties.
The wave-like nature of particles is described by a mathematical framework called wave mechanics, which is based on the Schrödinger equation. The Schrödinger equation is a wave equation that describes the behavior of quantum particles, including their energy and momentum.
In the context of the Schrödinger equation, the wave function of a particle evolves in time and is characterized by its amplitude and phase. The amplitude of the wave function is related to the probability of finding the particle at a particular location, while the phase of the wave function is related to the particle's momentum.
The energy of a quantum particle is related to the frequency of its wave function, according to the relation E = hf, where E is the energy, h is Planck's constant, and f is the frequency. This is known as the energy-frequency relation.
Similarly, the momentum of a quantum particle is related to the wave number (k) of its wave function, according to the relation p = hk, where p is the momentum and h is Planck's constant. This is known as the momentum-wave number relation.
So, in quantum mechanics, the wave equation (Schrödinger equation) describes the behavior of particles and provides information about their energy and momentum.