Particle in a Finite Box and the Harmonic Oscillator

When we solved the system in which a particle is confined to an infinite box (that is, an infinite square well), we saw that quantum numbers arose naturally through the enforcement of continuity conditions (that the wavefunction ψ must go to zero at x=0 and x=L). Quantization of energy and position (namely, nodes at which the particle cannot exist) are directly to these quantum numbers, whose values are n=1, 2, ..., ∞, representing an infinite number of energy levels.

A particle in a finite box, however, can tunnel into the walls, in the same fashion that we saw earlier with the two barrier problems. Solving this system is not difficult but, unfortunately, has no analytical solution and must be solved either numerically or, as was done in class, graphically. On the other hand, the wavefunctions are essentially just those from the infinite box but are allowed to bleed into the wall (with the caveat that higher energy states tunnel further than the lower energy states). To summarize the major differences between the particle in a finite box and one in an infinite box:

• only a finite number of energy levels exist [called bound states]
  
• tunneling into the barrier is possible
  
• higher energy states are less tightly bound than lower states
• a particle given enough energy can break free [in other words, unbound]

The next quantum system to investigate is the one-dimensional harmonic oscillator, whose potential [from Hooke's Law] is V=1/2kx². Plugging this into the Schrödinger Equation leads to, after some well-chosen substitutions, a differential equation solved by Hermite in the mid-1800's, and we obtain the wavefunction: ψ(x) = N_vH_ve^-q2/2, where q = αx, v is the quantum number [v=0, 1, ...] and H_v are the Hermite polynomials. Here we see energy quantization as well, giving E = (v + 1/2)hbarω. This quantum system is the only one to exhibit constant spacing but other results mirror those seen in prior examples: tunneling into classically forbidden zones (where x represents displacement from equilibrium rather than position), a nonzero ground state energy as well as the existence of nodes.

One important distinction from the particle in a box result is that the peaks in the wavefunction are not uniform. For example, for v=2 and larger, it is clear that outside peaks (representing larger displacement from x=0) have higher probability than inside peaks. As n gets large, we see another clear example of the correspondence principle.