Detailed Examination of Particle-in-a-Box Solutions

Having solved the Schrddinger equation for the particle in the infinitely deep square-well potential, we now examine the results in more detail. 

We note also that is proportional to L. This means that the more tightly a particle is confined, the greater is the spacing between the allowed energy levels. Alternatively, as the box is made wider, the separation between energies decreases and, in the limit of an infinitely wide box, disappears entirely. Thus, we associate quantized energies with spatial confinement. 

The energy is proportional to 1/m. This means that the separation between allowed energy levels decreases as m increases. Ultimately, for a macroscopic object, m is so large that the levels are too closely spaced to be distinguished from the continuum of levels expected in classical mechanics. This is an example of the correspondence principle, which, in its most general form, states that the predictions of quantum mechanics must pass smoothly into those of classical mechanics whenever we progress in a continuous way from the microscopic to the macroscopic realm. 

Notice that the lowest possible energy for this system occurs for w = 1 and h E = 

It is possible to show that, for L oo, our particle in a box would have to violate the Heisenberg uncertainty principle to achieve an energy of zero. For, suppose the energy is precisely zero. Then the momentum must be precisely zero too. (In this system, all energy of the particle is kinetic since F = 0 in the box.) If the momentum px is precisely zero, however, our uncertainty in the value of the momentum Ispx is also zero. If Ispx is zero, the uncertainty principle [Eq. (2-46)] requires that the uncertainty in position Ax be infinite. But we know that the particle is between x = 0 and x = L. Hence, our uncertainty is on the order of L, not infinity, and the uncertainty principle is not satisfied. However, when L = oo (the particle is unconstrained), it is possible for the uncertainty principle to be satisfied simultaneously with having E = 0, and this is in satisfying accord with the fact that E = goes to zero as L approaches infinity. 

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