Particle in one Dimensional Box

The Particle in a One-Dimensional Box:

The particle in a one-dimensional box model is one of the simplest models for electron behavior. In this model, the role of quantization becomes important in determining the energy eigenvalues of the electron. A situation is considered in which an electron is allowed to move along an arbitrary spatial dimension, one can use the x-axis. The x-axis is divided into three regions, I, II, and III where {I | x<0}, {II | 0L}.

In regions I and III, an infinite potential energy barrier is present. The potential energy of any particle within these regions becomes infinite, meaning in order to exist there, the particle's total mechanical energy must be infinite. Since this will never be true, the particle cannot exist within regions I and III, it is restricted to region II. From this, one knows that the wave function of the electron must be zero everywhere except for {0 < x < L}. Also, remembering that all wave functions are continuous, one knows that the wave function at x=0 and at x=L must be zero; otherwise a discontinuity would occur. The goal presently is to determine the set of possible wave functions associated with the electron, and then to use such to determine the energy eigenvalues. To begin, one looks to the time-independent Schrödinger Equation (the time-dependent form is not necessary since the state of the electron and its surroundings are constant).

eq.1

Notice, the potential energy term is neglected. To begin solving for the wave function set, one must rearrange this into the form:

eq.2

To do so, one multiplies eq.1 by - 2m / h-bar²:

eq.3

The general trigonometric solution to this second-order differential equation is:

eq.4

The generalized wave function set for the electron has now been determined. Now to determine the energy eigenvalues associated with these wave functions. To begin, one looks at the right end point of the one-dimensional box, at x=L. Knowing the wave function is zero at this point, one can write:

eq.5

The sine function produces an output of zero when its input values are multiples of pi:

eq.6

where n is member to the natural number set {1,2,3,...}. The number n is called a quantum number, a number which determines the particular state that the electron occupies. One can write an alternate form of the wave function using eq.6:

eq.7

Normalizing the wave function as described in the section concerning operator mechanics is relatively straight-forward:

eq.8

Rewriting eq.7:

eq.9

Graphically demonstrating the wave functions for n=1,2,3,4:

Rearranging eq.6 to isolate energy:

eq.10

The unusual aspect of this model is that the energy of the electron is quantized. Only a discrete set of values is allowed as opposed to a continuous spectrum of values. This quantization phenomenon is the effect of imposing barriers. A minimum energy value is also present which is named the zero-point energy {E₁ or h²/(8mL²)}. This originates from the quantum number n, which is an integer that is greater than or equal to one. If it were to be zero, the wave function would be zero for every value of x, meaning it would have zero mechanical energy, meaning it would have zero momentum, meaning a violation of the Heisenberg uncertainty principle would occur since the uncertainty of momentum would be zero while the uncertainty of position is nonzero.