In quantum field theory (QFT), particles are indeed described as excitations of quantum fields. However, the concept of a "wave function" in QFT is somewhat different from its counterpart in non-relativistic quantum mechanics, as it incorporates both particle and wave aspects.
In non-relativistic quantum mechanics, the wave function describes the state of a single particle and evolves according to the Schrödinger equation. It gives the probability amplitude of finding the particle in a particular state at a given time.
In QFT, the fundamental entities are quantum fields, which are defined at every point in spacetime. These fields describe the possible excitations or particles that can exist. The field itself is not directly observable, but its excitations correspond to particles that can be observed and detected. The dynamics of the quantum fields are governed by a different equation called the relativistic wave equation, such as the Klein-Gordon equation for scalar fields or the Dirac equation for spinor fields.
In QFT, the particle-wave duality is manifested through the concept of "quantum superposition." Just as in non-relativistic quantum mechanics, particles in QFT can exhibit wave-like properties, such as interference and diffraction. The excitations of the fields can be superposed, leading to wave-like behavior. However, it's important to note that the interpretation of this wave-like behavior is somewhat different in QFT compared to non-relativistic quantum mechanics.
In non-relativistic quantum mechanics, the wave function describes the probability distribution of finding a particle at a particular position. In QFT, on the other hand, the fields are operators that create and annihilate particles, and the wave function is replaced by the quantum field operator itself. The expectation values of these operators provide information about the probability of finding particles in various states.
So, while there are analogies between the wave function in non-relativistic quantum mechanics and the quantum fields in QFT, the mathematical formalism and interpretation differ due to the relativistic nature of QFT and the need to account for the creation and annihilation of particles.