The fundamental equation of quantum mechanics is the schrödinger equation. for many systems, we'll find solutions to this equation involve oscillatory functions, such as sine and cosine. euler's formula provides a convenient way of representing these functions using complex numbers i0 a. using euler's formula, show that: ae ae-ikr bear-be- ) ) a cos (kx) and b sin k = = 2i where a, b, and k are arbitrary constants. -ikx show that a-+beer can be rearranged into the sumacos(k) + βsin(kr) where α and β are arbitrary constants. how do α and β relate to a and b? b. c. show that ae +be can also be written as: csin(kr + φ) where c and φ are constants. derive expressions that relate c and φ to a and b. hint: consider rewriting the sine function above using the result of part (a). d. use euler's formula to find the values of m that satisfy the equation exp(2rim) 1. these values describe the solution to the schrödinger equation for a mass moving in a circle! if you need a refresher on how to do this, take a look at mathchapter a from mcquarrie & simon or section 1-5 of applied mathematics for physical chemistry by james r. barrante.