Читать книгу Quantum Mechanical Foundations of Molecular Spectroscopy - Max Diem - Страница 27

The principles derived in the previous section can easily by expanded to a two‐dimensional (2D) case. Here, an electron would be confined in a box with dimensions L_x in the x‐direction and L_y in the y‐direction, with zero potential energy inside the box and infinitely high potential energy outside the box:

(2.42)

The Hamiltonian for this system is

(2.43)

and the total wavefunction ψ_{x, y} can be written as

(2.44)

where A as before is an amplitude (normalization) constant. The total energy of the system is

(2.45)

Figure 2.4 Wavefunctions of the two‐dimensional particle in a box for (a) n_x = 1 and n_y = 2 and (b) n_x = 2 and n_y = 1.

For a square box with L_x = L_y = L, the energy expression simplifies to

(2.46)

The wavefunctions can now be represented as shown in Figure 2.4 for the cases n_x = 2 and n_y = 1 and n_x = 1 and n_y = 2. These wavefunctions represent the standing wave on a square drum. Notice that the energy eigenvalues for these two cases are the same:

(2.47)

When two or more energy eigenvalues for different combination of quantum numbers are the same, these energy states are said to be degenerate. Here, for n_x = 2 and n_y = 1 and n_x = 1 and n_y = 2, the same energy eigenvalues are obtained; consequently, E₂₁ and E₁₂ are degenerate. This is a common occurrence in quantum mechanics, as will be seen later in the discussion of the hydrogen atom (Chapter 7), where all the three 2p orbitals, the five 3d orbitals, and the seven 4f orbitals are found to be degenerate.