Abstract
This paper investigates the quantum mechanical relationship between time dilation and energy difference of the quantum photon and electron wave functions. The wave functions are studied under two different times and and two energy differences and . The time-dependent Schrödinger equation (TDSE) is used to analyse the wave functions and time dilation is studied in the Galilean frame of reference. A key relation linking Lorentz time dilation to the energy difference between the photon (relativistic kinetic energy) and electron (classical kinetic energy) wave functions is derived analytically. Python simulation is used to compute and visualise the wave function phase, energy ratio, total phase energy, and time dilation as functions of speed v/c.