Because we have found no way to formulate quantum mechanics based on a single real wave fu...

Because we have found no way to formulate quantum mechanics based on a single real wave function, we have a choice to make. In Section 4.3, it is said that our choice of using complex numbers is a convenient one. Show that the free-particle Schrodinger equation (4-8) is equivalent to two real equations involving two real functions, as follows:

 and

where ψ (x, t) is by definition ψ1(x, t) + iψ2(x, t). How is the complex approach chosen in Section 4.3 more convenient than the alternative posed here?