All particles can be described as having associated wave-like properties, but not all particles can be said to "vibrate" in a classical sense. The wave-like behavior of particles in quantum mechanics is described by their wavefunctions, which represent the probability amplitudes of finding the particle in different states.
In quantum mechanics, particles are described by wavefunctions that can exhibit wave-like characteristics such as interference and diffraction. These wavefunctions are mathematical representations that describe the probabilities of different outcomes when measuring various properties of the particle, such as position or momentum. The wavefunction is typically expressed as a superposition of different states, each associated with a certain probability amplitude.
When it comes to vibrations, the concept is more relevant in the context of extended objects or systems, such as strings or atoms in a lattice. In these cases, the constituent particles or atoms can be considered to vibrate around their equilibrium positions. These vibrations are described by quantized energy levels and frequencies corresponding to different vibrational modes of the system. The study of these vibrations falls within the field of quantum mechanics, but the terminology and mathematical formalism used to describe these vibrations are different from the wavefunctions associated with individual particles.
While particles themselves do not vibrate in the classical sense, their wave-like behavior and associated wavefunctions do exhibit frequencies and wavelengths. These frequencies and wavelengths are related to the probabilistic nature of finding the particle at different positions or with different momenta. The wave-particle duality in quantum mechanics suggests that particles can exhibit both particle-like and wave-like characteristics simultaneously.
In summary, while particles themselves do not vibrate in the classical sense, they can exhibit wave-like properties described by wavefunctions with associated frequencies and wavelengths. The concept of vibrations is more relevant when discussing extended objects or systems, where the constituent particles or atoms can be considered to vibrate around their equilibrium positions. The wave-like behavior and vibrations are both aspects of the underlying quantum nature of particles and systems, but they are described and understood using different mathematical formalisms.