Accounting for Wave Character in Mechanical Systems

The de Broglie relationship suggests that in order to obtain a full mechanical description of a free particle (a free particle has no forces acting on it), there must be a wavelength and hence some simple oscillating function associated with the particle s description. This function can be a sine, cosine, or, equivalently, a complex exponential function.  

In the wave equation above, Ao represents the amplitude of the wave and X represents the de Broglie wavelength. Note that when the second derivative 

The complex exponential function e and e (where k = Zit/X in this case) are related to sine and cosine functions as shown in the following mathematical identities (see Equations 1-lOaand 1-lOb)  

Expressing a wavefunction in terms of a Complex exponential can be useful in some cases as will be shown later in the text. 

In such a situation, the function is called an eigenfunction, and the constant is called an eigenvalue. The eigenfunction is a wavefunction and is generally given the symbol, V . 

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