An Analytic Solution The Particle-in-a-Box

Very few systems have analytic solutions (that is, solutions that have a specific mathematical form, either a number or an expression) to the Schrodinger equation. Most of the systems having analytic solutions are defined ideally, much as an ideal gas is defined. This should not be a cause for despair. The few ideal systems whose exact solutions can be determined have applications in the real world, so they are not wasted on ideality Several of these systems were recognized by Schrodinger himself as he developed his equation. 

Unless otherwise noted, all art on this page is Cengage Learning 2014. 

FIGURE 10.5 The particle-in-a-box is the simplest ideal system that is treated by quantum mechanics. It consists of a region between x = 0 and x = a (some length) where the potential energy is zero. Outside of this region (x 0 or x a, the potential energy is o°, so any particle in the box will not be present outside the box. 

The analysis of this system using quantum mechanics is similar to the analysis that we will apply to every system. First, consider the two regions where the potential energy is infinity. According to the Schrodinger equation 

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A I An Analytic Solution The Particle-in-a-Box 305 TABLE 10.1 Operators for various observables and their classical counterparts ... 

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