Quantum Mechanics of Particle-in-a-Box Models

We are about to show you how to solve the Schrodinger equation for a simple but important model for which we can carry out every step of the complete solution using only simple mathematics. We will convert the equations into graphical form and use the graphs to provide physical interpretations of the solutions. The key point is for you to learn how to achieve a physical understanding from the graphical forms of the solution. Later in this textbook we present the solutions for more complex applications only in graphical form, and you will rely on the skills you develop here to see the physical interpretation for a host of important chemical applications of quantum mechanics. This section is, therefore, one of the most important sections in the entire textbook. 

A quick inspection of the potential energy function tells us the general nature of the solution. Wherever the potential energy V is infinite, the probability of finding the particle must be zero. Hence, p x) and must be zero in these regions  

Inside the box, where V = 0, the Schrodinger equation has the following form  

As shown earlier, the sine and cosine functions are two possible solutions to this equation, because the second derivative of each function is the function itself multiplied by a (negative) constant. 

Now let us apply the conditions defined in Section 4.5 to select the allowed solutions from these possibilities. The boundary conditions require that i/r(x) = 0 at X = 0 and x = L. An acceptable wave function must be continuous at both these points. The cosine function can be eliminated because it cannot satisfy the condition that i/r(x) must be 0 at x = 0. The sine function does, however, satisfy this boundary condition since sin(O) = 0 so 

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