Max Diem - Quantum Mechanical Foundations of Molecular Spectroscopy

As discussed earlier, the total energy is written as the sum of the kinetic and potential energies, T and V , respectively:

(2.17)

As before, the kinetic energy of the particle is given by

(2.3)

where m is the mass of the electron. Substituting the quantum mechanical momentum operator,

(2.4)

into Eq. (2.3), the kinetic energy operator can be written as

(2.5)

Figure 2.2 Panel (a): Wavefunctions for n = 1, 2, 3, 4, and 5 drawn at their appropriate energy levels. Energy given in units of h 2 / 8 mL 2. Panel (b): Plot of the square of the wavefunctions shown in (a).

The potential energy inside the box is zero; thus, the total energy of the particle inside the box is

(2.18)

Since the potential energy outside the box is infinitely high, the electron cannot be there, and the discussion henceforth will deal with the inside of the box. Thus, one may write the total Hamiltonian of the system as

(2.19)

In the notation of linear algebra, this operator/eigenvector/eigenvalue problem is written as

(2.20)

Equation (2.20)instructs to apply the Hamiltonian of Eq. (2.19)to a set of yet unknown eigenfunctions to obtain the desired energy eigenvalues. The eigenfunctions typically form an n‐dimensional vector space in which the eigenvalues appear along the diagonal. Thus, Eq. (2.20)implies

(2.21)

that is, the Hamiltonian operating on a set of eigenfunctions such that

; ; ; and so forth that is, of course, obtained by carrying out the matrix multiplication indicated in Eq. (2.21).

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 b 2would correspond to 2 mE / ħ 2, 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