The fundamental equation in quantum mechanics is known as a wave equation. We cannot really call it an equation of motion since the concept of ballistic motion is absent. We solve for a wave function, which can be used to calculate the probability that a particle-wave occupies a given region of space and properties such energy, momentum and so on. For the free particle we can obtain a single solution, but unless we have boundary conditions the solution is not unique. For example, either a sine or cosine function could satisfy the wave equation. The solution could be sin(kx) or cos(kx). Which is it? There is no way to know unless we have some information that specifies the phase. The sine and cosine function differ only in phase and it is arbitrary for a free particle. A second issue concerns the physical location of the particle. If we have a single solution, e.g. sin(kx) then we have a single value of the momentum. The DeBroglie relation tells that the momentum is propotional to the k, since p = h_bar k. If we have a unique momentum then the position of the particle is not specified. It could be anywhere. We will consider this point further when we discuss the Uncertainty Principle.

Relationship between e^ikx, e^-ikx, sin(kx) and cos(kx)

The Euler relations provide a relationship between the complex exponential functions and the sine and cosine function. It is important to understand that the complex exponential function is simply a linear combination of in-phase and out-of-phase components.