We assume that the energies and momenta involved are nonrelativistic. The de Broglie wavelength is defined as
where is the kinetic energy of the particle. Initially, , but after the particle enters the region in which it has a potential energy , the kinetic energy becomes . Therefore, the initial de Broglie wavelength must be multiplied by a factor of
in order to obtain the new de Broglie wavelength. Therefore, the correct answer is (E).