《Lesson_4_SWE》.pdf

Quantum Mechanics for Scientists and Engineers David Miller Schrödinger’s equation From de Broglie to Schrödinger Electrons as waves de Broglie’s hypothesis is that the electron wavelength is given by h  p where p is the electron momentum and h is Planck’s constant h 61034 J s Now we want to use this to help construct a wave equation A Helmholtz wave equation If we are considering only waves of one wavelength for the moment i.e., monochromatic waves we can choose a Helmholtz wave equation 2 d  2 2 k 2   with k dz  which we know works for simple waves with solutions like sin(kz), cos(kz), and exp(ikz) (and sin(–kz), cos(–kz), and exp(–ikz)) A Helmholtz wave equation In three dimensions, we can write this as 2 2 2 2    2  k       2 2 2 x y z which has solutions like sin(k r), cos(k r), and exp(ik r) (and sin(-k r), cos(-k r), and exp(-ik r)) where k and r are vectors From Helmholtz to Schrödinger With de Broglie’s hypothesis  h / p and the definition k 2/  then k 2p / h p /  where we have defined h / 2 so k 2 p2 / 2 Hence we can