Requirements for an Acceptable Wave Function

In solving the particle in a box, we required ij/ to be continuous. We now discuss other requirements the wave function must satisfy. 

Since i/ p dr is a probability, we want to be able to normalize the wave function by choosing a suitable normalization constant N as multiplier of the wave function. If (// is unnormalized and Ntft is normalized, the normalization condition (3.59) gives 

In addition to demanding that tp be continuous, we usually also require that all the partial derivatives dtp/dx, dtp/dy, and so on, be continuous. (See Fig. 3.4.) Referring back to Section 2.2, however, we note that for the particle in a box dtp/dx is discontinuous at the walls of the box tp and dtp/dx are zero everywhere outside the box, but from Eq. (2.23) we see that dtp/dx does not become zero at the walls. The discontinuity in tp  

FIGURE 3.4 Function (a) is continuous and its first derivative is continuous. Function (b) is continuous, but its first derivative has a discontinuity. Function (c) is discontinuous. 

We require that the eigenfunctions of any operator representing a physical quantity meet the above requirements. A function meeting these requirements is said to be well-behaved. 

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