As we shall see, the solution of the Schrödinger equation for the particle in a box problem reveals some decidedly quantum behavior of the particle that agrees with observation but contrasts sharply with the predictions of classical mechanics. This is a particularily useful illustration because this behaviour is not "forced" on the system, it arises naturally from the initial conditions. It neatly demonstrates that quantum behaviour is a natural outcome of any wave-like system, contrary to the common concept of a "quantum leap" where the behavior is almost magicalal.
The quantum behavior in the box include:
For the 1-dimensional case in the direction, the time-independent Schrödinger equation can be written as:
Now in order find the specific solution for the problem at hand, we must specify the appropriate boundary conditions and find the values for A and B that satisfy those conditions. One usually resorts to one of the following two choices, describing two kind of sytems. The first case, with which we shall pursue our derivation, demands that ψ equal zero at x = 0 and x = L. A handwaving argument to motivate these boundary conditions is that the particle is unlikely to be found at a location with a high potential (the potential repulses the particle), thus the probability of finding the particle, |ψ|^{2}, must be small in these regions and decreases with increasing potential. For the case of an infinite potential, |ψ|^{2} must infinitesimally small or 0, thus ψ must also be zero in this region. In summary,
Substituting the general solution from Equation 3 into Equation 2 and evaluating at x = 0 (ψ = 0), we find that B = 0 (since sin(0) = 0 and cos(0) = 1). It follows that the wavefunction must be of the form:
Finally, substituting the results from Equations 7 and 8 into Equation 3 gives the complete solution for the 1-dimensional particle in a box problem:
Note, that as mentioned previously, only "quantized" energy levels are possible. Also, since n cannot be zero, the lowest energy from Equation 9 is also non-zero. This zero-point energy, as it is called, can be explained in terms of the Uncertainty Principle. Because the particle is constrained within a finite region, the variance in its position is upper-bounded. Thus due to the uncertainty principle the variance in the particle's momentum cannot be zero, so the particle must contain some amount of energy that increases as the length of the box, L, decreases.
Also, since ψ consists of sine waves, for any value of n greater than one, there are regions within the box for which ψ and thus ψ^{2} both equal zero, indicating that for these energy levels, nodes exist in the box where the probability of finding the particle is zero.
For the 2-dimensional case the particle is confined to a rectangular surface of length L_{x} in the x-direction and L_{y} in the y-direction. Again the potential is zero inside the "box" and infinite at the walls. For the region inside the box, where the potential is zero, the two dimensional analogue of Equation 2 applies:
An interesting feature of the above solutions is that when two or more of the lengths are the same (eg. L_{x} = L_{y}), there are multiple wavefunctions corresponding to the same total energy. For example the wavefunction with n_{x} = 2, n_{y} = 1 has the same energy as the wavefunction with n_{x} = 1, n_{y} = 2. This situation is called degeneracy and for the case where exactly two degenerate wavefunctions have the same energy that energy level is said to be doubly degenerate. Degenaracy results from symmetry in the system. For the above case two of the lengths are equal so the system is symmetric with respect to a 90° rotation.
If the potential is zero (or constant) everywhere, one describes a free particle. This leads to some difficulties of normalization of the wavefunction. One way around is to constrain the particle in a finite volume V of arbitrary (large) extension, in which it is free to propagate. It is expected that in the limit of V→ ∞ we recover the free particle while allowing in the intermediate calculations the use of properly normalized states. Also, when describing for instance a particle propagating in a solid, one does not expect spatially localized states but instead completely delocalized states (withing the solid), meaning that the particles propagates inside it (since it can be everywhere with the same probability, conversely to the sine solutions we encountered where the particle has favored locations). This understanding follows from the solutions of the Schrödinger equation for zero potential following from the so-called Von-Karman boundary conditions, i.e., the wavefunction assumes same values on opposite sides of the box but it is not required to be zero here. One can then check that the following solutions obey eq. 1:
The energy remains (cf. eq. 3) but interestingly, now the k are twice as before (cf. eq. 7). This is because in the previous case, n was strictly positive whereas now it can be negative or zero (the ground state). The solutions where the sine does not superpose to itself after a translation of L can not be recovered with exponentials, since in this propagating particle interpretation, the derivative is discontinuous at the border, meaning that the particle acquires infinite velocity here. This shows how the two interpretations bear intrinsically differing behaviours.