Particle-in-a-Box Exact Solution

The particle-in-a-box problem, which we considered qualitatively in Chapter 5, turns out to be one of the very few cases in which Schrodinger s equation can be exactly solved. For almost all realistic atomic and molecular potentials, chemists and physicists have to rely on approximate solutions of Equation 6.8 generated by complex computer programs. The known exact solutions are extremely valuable because of the insight 

Schrodinger s equation contains the product U(x) p (x). Recall from Chapter 5 that the particle-in-a-box potential is infinite except inside a box which stretches from x = 0 to x = L. So the product U(x)f(x) would be infinite for x 0 or x L unless fix 0) = x/r(x L) = 0. Thus if the energy E is finite, the wavefunction can be nonzero only for 0 x L—in other words, the particle is inside the box. 

Because wavefunctions must be continuous, we also have boundary conditions—the wavefunction must vanish at x = 0 and x = L. 

Equation 6.17 implies either that A = 0, in which case // = 0 everywhere and the wavefunction cannot be normalized (Equation 6.4), or 

The subscript on the wavefunction identifies it as the one with some particular value of n. n = 0 would force the wavefunction to vanish everywhere, so there would be no probability of finding the particle anywhere. Hence we are restricted to n 0. 

---