A matter wave, such as the wave associated with a particle in quantum mechanics, cannot be described as a single wave because it exhibits wave-particle duality. This means that particles, such as electrons or other subatomic particles, can exhibit both wave-like and particle-like properties.
In quantum mechanics, the behavior of particles is described by wavefunctions, which are mathematical functions that represent the probability distribution of finding a particle in a particular state. These wavefunctions are typically represented by complex numbers and are solutions to the Schrödinger equation.
The key point is that the wavefunction of a particle represents the probability amplitude of finding the particle at different positions or having different momenta. It is not a classical wave in the sense of a single oscillating wave that can be visualized like a water wave or a sound wave.
The wave nature of particles is inherently probabilistic. When you measure a particle's position or momentum, you obtain a specific value with a certain probability. The wavefunction describes the probability distribution of these measurement outcomes, but it does not represent a physical wave that occupies a specific region of space.
In experiments, the behavior of particles often displays wave-like interference and diffraction patterns, similar to what you would expect from waves. However, when particles are detected, they are found at specific locations, behaving like localized particles.
So, while matter waves are described by wavefunctions, they do not correspond to single classical waves. Instead, they represent the probabilistic behavior of particles and their wave-like properties.