The Particle in a Three-Dimensional Box Separation of Variables

Let us now consider the three-dimensional analog of the square well of Section 2-1. This would be a three-dimensional box with zero potential inside and infinite potential outside. As before, the particle has no probability for penetrating beyond the box. Therefore, the Schrodinger equation is just 

This can be expanded and then divided through by XYZ to obtain 

since Hx, for example, operates only on functions of x, but not y or z, we can carry out some limited cancellation. Those functions that are not operated on in a numerator can be canceled against the denominator. Those that are operated on caimot be canceled since these are differential operators [e.g., in (1 /x)dx /dx it is not permissible to cancel 1/x against x before differentiating (llx)dx ldx dx/dx]. Such cancellation gives 

suppose the particle is moving in the box parallel to the x axis so that the variables y and z are not changing. Then, of course, the functions Y and Z are also not changing, so HyY/Y and H Z/Z are both constant. Only HxXjX can vary—but does it vary Not according to Eq. (2-56), which reduces under these conditions to 

Therefore, even though the particle is moving in the x direction, HxX/Xmust also be a constant, which we shall call Ex. Similar reasoning leads to analogous constants Ey and E. Furthermore, the behavior of HxX/X must really be independent of whether the particle is moving parallel to the y and z axes. Even if y and z do change, they do not appear in the quantity HxX/X anyway. Thus we may write, without restriction. 

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Section 2-7 The Particle In a Three-Dimensional Box Separation of Variables... 

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