This is one of the simplest quantum mechanical systems. Its Schroedinger equation is easily solved using elementary calculus. This solution is commonly used in elementary courses to illustrate several fundamental concepts of quantum theory.
(Topics 5 through 8 are appropriate to advanced courses.) Besides these important pedagogical uses, the particle in a box is also a model for several important physical appplications.
In addition to the pedagogical uses listed above, the particle in a box problem is also a model for several important applications.
The following sections are devoted to solving the PIB problem, and interpreting the solution. Following this, several related PIB problems are stated, some are solved, and others are exercises for the reader.
The PIB is an idealized physical system consisting of a particle constrained to move in one dimension between infinite potential walls. Thus, the particle is subject to no forces between the walls but infinite force at the walls to prevent its escape from the box. The total energy of the particle is the sum of kinetic and potential terms as represented by the Hamiltonian operator of quantum mechanics: . Here , and h=6.626 10-34 joule sec is Planck's constant, the fundamental constant of quantum mechanics. Two other constants are required to specify the system: m, the particle mass, and L, the length of the box. Then the object is to solve Schroedinger's time-independent equation, Hy = Ey, for the allowed energy levels, E, and states y(x) (energy eigenvalues and eigenfunctions) of the PIB.
The solution is made easier if the coordinate is transformed to x=x/L and at the same time the energy is divided by . With these changes, the Schroedinger equation becomes . Now, wave functions are required to be "well behaved"; usually that means that (a) y is finite, differentiable (at least through first order), and continuous, (b) also must be continuous (except possibly at a finite number of points).
Elementary calculus is enough to find the solutions as shown here. Briefly, the eigenvalues (in dimensionless energy units) are En=n2 for n = 1, 2, ... , and the corresponding wave functions are . Here is a plot to show the energy levels with wave functions superimposed.
The vertical scale is energy (in natural energy units, Enatural) and the energy levels are shown as horizontal lines. The horizontal scale is position, x/L, in the box of length L.Wave functions, y(x), are drawn using the corresponding energy level as the zero of y. Wave functions are normalized, but their vertical scale on the graph is arbitrary. This graph belongs to a system consisting of a mass, m = 4 10-30 kg in a box of length L = 10 Angstrom.
The lowest 4 levels are shown: n = 1, 2, 3, 4. Not counting the nodes at the walls of the box, n is the number of interior nodes in the wave function. The parity of states alternates with n; thus, the lowest state is even, y1(-x) = y1(x) , the first excited state is odd, y2(-x) = -y2(x), and so forth.
Analytical solutions of PIB are not always possible when the potential inside the box is modified. In such cases numerical solutions can be found. For comparison, a numerical solution of the simple PIB problem is given in an rtf file and an interactive mathcad solution file.
The simple PIB system discussed above can be "decorated" in various ways to model important physical systems. In the following problems the potential is V(x) = U(x)(|x|<L) + ¥(|x|>L), that is the potential is infinite outside the range -L<x<L and varies according to the function U(x) inside the range. If U(x) = 0, we have the elementary PIB described above. The solution to each of these problems can be simplified by adopting natural units of length and energy. Complete analytical expressions should be sought but if these are not found, then a numerical solution for the lowest few (say 4 or 5) energy states is acceptable.
Note that the well is from x=-1/3 to x=1/3. The wave function penetrates beyond the well into regions where the potential energy is larger than the total particle energy - an illustration of tunneling.]
Created or up-dated 08/03/99
by R.D. Poshusta