In , is this an eigenfunction of energy (because its a solution to the Schrödinger equation)?
No, the wavepacket is not an eigenfunction of energy: it’s made up of a superposition of waves of lots of different energies, so it doesn’t have a definite energy. Note that when we solve the time-independent Schrödinger equation, we are finding eigenfunction solutions of the Hamiltonian which have definite energies. However you can always write down more solutions by making linear combinations of these eigenfunctions. Eigenfunctions of the Hamiltonian are solutions to the Schrödinger equation, but not all solutions are eigenfunctions.
