John Lindsay Orr

Notes on the Schrödinger Equation

The two key ingredients are the Planck equation for the energy of a photon, and the de Brogli9e wavelength of a matter wave.

The Planck equation for the energy, of a photon with frequency is:

The de Broglie wavelength, , of the matter wave asoociated with a particle with momentum is:

It's convenient to recast this in terms of the wave number and the normalized frequency . (Conceptually, represents the number of wave cycles in an interval of length , and represents the numbers of cycles in a time interval of length .) In these terms, the Planck equations and the de Broglie wavelength formulas become respectively where is the adjusted Planck constant, approximately .

The Schrödinger Equation is motivated by looking for a wave-type equation which has as a solution the simplest possible candidate for a matter wave, i.e., a plane wave of the form

Then compute and so

Next we calculate and then, using the classical (non-relativistic) relationship together with the de Broglie formula, we obtain

Putting together the two formulas for we obtain the free Schrödinger Equation:

Finally, since we are interpreting as the energy operator for a free particle, it makes sense that the energy operator for a particle in potential is and so we reassemble the equations for a particle in a potential as