Particle in a triangle

To conclude this chapter, we present the quantum mechanical results of the particle-in-a-triangle problem. Before going into details, we first need to note that, if the box is a scalene triangle, no analytical solution is known. Indeed, the Schrodinger equation is exactly solvable for only a few triangular systems. In addition, for all these solvable (two-dimensional) cases, the wavefunctions are no longer the simple products of two functions each involving only one variable. 

In the following discussion, some language and notation of group theory are used for convenience. The meaning of this language or notation is made clear in Chapter 6. 

The coordinate system chosen for this problem is shown in Fig. 1.5.5. 

Coordinate system for the problem of a particle in an isosceles right triangle. 

The Schrodinger equation in this case is simply the two-dimensional analog of eqs. (1.5.2) and (1.5.25)  

---