Читать книгу Quantum Mechanical Foundations of Molecular Spectroscopy - Max Diem - Страница 25

Rearranging Eqs. (2.19) and (2.20) yields

(2.22)

which is a simple differential equation that can be used to obtain the eigenfunctions ψ(x):

(2.23)

Any functions fulfilling Eq. (2.23) must be of the form that their second derivative equals to the original function multiplied by a constant. For example, the function

(2.24)

could be solution of the differential Eq. (2.23),

since

(2.25)

Here, the term b² would correspond to 2mE / ħ², and A is a yet undefined amplitude factor. Similarly,

(2.26)

or the sum of Eqs. (2.24) and (2.26) could be acceptable solutions. For the time being, and for reasons that will become obvious shortly, Eq. (2.26) will be used as a trial function to fulfill Eq. (2.23):

(2.27)

and

(2.28)

At this point, it should be pointed out that the solutions of any differential equation depend to a large extent on the boundary conditions: the general solution of the differential equation may or may not describe the physical reality of the system, and it is the boundary conditions that force the solutions to be physically meaningful. In the case of the PiB, the boundary conditions are determined by one of the postulates of quantum mechanics that requires that wavefunctions are continuous. Thus, if the wavefunction outside the box is zero (since the potential energy outside to box is infinitely high and, therefore, the probability of finding the particle outside the box is zero), the wavefunction inside the box also must be zero at the boundaries of the box. Thus, one may write the boundary conditions for the PiB differential equation as

(2.29)

Because of these conditions, the cosine function proposed as possible solutions (Eq. [2.24]) of Eq. (2.23) was rejected, since the cosine function is nonzero at x = 0. Because of the required continuity at x = L, the value of the function

must be zero at x = L as well. This can happen in two ways: The first possibility occurs if the amplitude A is zero. This case is of no further interest, since a zero amplitude of the wavefunction would imply that the particle is not inside the box. The second possibility for the wavefunction to be zero at x = L occurs if

(2.30)

Since the sine function is zero at multiples of π radians, it follows that

(2.31)

Solving Eq. (2.31) for E yields the energy eigenvalues

(2.32)

Equation (2.32) reveals that the energy levels of the particle in a box are quantized, that is, the energy can no longer assume any arbitrary values, but only values of and so on. This is the first appearance of the concept of quantized energy levels in a model system and represents a step of enormous importance for the understanding of quantum mechanics and spectroscopy: by substituting the classical momentum with the momentum operator, quantized energy levels (or stationary states) were obtained. This quantization is a direct consequence of the boundary conditions, which required wavefunctions to be zero at the edge of the box. Since the energy depends on this quantum number n, one usually writes Eq. (2.32) as

(2.33)

Substituting these energy eigenvalues back into Eq. (2.27)

(2.27)

one obtains

(2.34)

which are the wave functions for the PiB.