The Particle in an Infinite Box with a Finite Central Barrier

2-4 The Particle in an Infinite Box with a Finite Central Barrier 

Another example of barrier penetration in a stationary state of a system is provided by inserting a barrier of finite height and thickness at the midpoint of the infinite square well of Section 2-1 (see Fig. 2-12). 

The boundary conditions for this problem are easily obtained by obvious extensions of the considerations already discussed. Rather than solve this case directly, we shall make use of our insights from previous systems to deduce the main characteristics of the solutions. Let us begin by considering the case where the barrier is infinitely high. 

Then the problem becomes merely that of two isolated infinite square wells, each well having solutions as described in Sections 2-1 and 2-2. 

There is a simpler way to decide that the symmetric solution has lower energy. As the barrier height becomes lower and lower, the two solutions become more and more separated in energy, but they always remain symmetric or antisymmetric with respect to reflection since the hamiltonian always has reflection symmetry. In the limit when the barrier completely disappears we have a simple square well again (but larger), the lowest solution of which is symmetric. (See Fig. 2-13b.) This lowest symmetric solution must 

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