The eigenfunctions for a quantum particle in a box are standing waves because they represent the stationary states of the particle within the box. These eigenfunctions, also known as stationary states or energy eigenstates, are solutions to the time-independent Schrödinger equation for the system.
In the case of a particle in a box, the system is confined to a finite region with boundaries. The eigenfunctions that satisfy the boundary conditions of the box are standing waves, characterized by distinct energy levels. These standing waves represent the allowed energy states of the particle within the box, and they form a complete set of orthogonal functions.
When physicists refer to the particle being in a single eigenstate, they are referring to the idea that the system can be described by a specific eigenfunction with a well-defined energy. This means that the particle's wavefunction is described by a single standing wave, corresponding to a particular energy eigenvalue. The superposition principle allows for the possibility of the wavefunction being a combination of multiple eigenfunctions, but when the system is in a stationary state (eigenstate), it is described by a single eigenfunction.
Wave packets, on the other hand, are used to describe particles with a well-defined momentum and position uncertainty. They represent a superposition of multiple waves with different wavelengths and momenta, which combine to form a localized wave packet that appears to move as a whole. Wave packets are not eigenstates of the energy operator and do not satisfy the boundary conditions of a confined system like a particle in a box.
In summary, the eigenfunctions for a quantum particle in a box are standing waves because they represent the stationary states of the particle within the box, while wave packets are used to describe particles with momentum and position uncertainties.