Quantum Mechanical Foundations of Molecular Spectroscopy. Max Diem

accomplished using the integral relationship

(2.41)

For any two adjacent wavefunction, say, m = 1 and n = 2 or m = 2 and n = 3, the numerator of the first term in Eq. (2.41) contains the sine function of odd multiples of π, whereas the numerator of the second term will contain the sine function of even multiples of π. Since the sine function of odd and even multiples of π is zero, the total integral described by Eq. (2.41) is zero. This argument holds for any case where n ≠ m.

This can also be visualized graphically, as shown in Figure 2.3b for the first two PiB wavefunctions for n = 1 (curve a) and m = 2 (curve b). When multiplied, curve c is obtained. The shaded areas above and below the abscissa of curve c represent the integral in Eq. (2.40) for n = 1 and m = 2 and are equal; therefore, the area under the product curve c is zero.

      Figure 2.3a also shows that the wavefunctions for the states with quantum number larger than 1 have nodal points, or points with no amplitude. This is familiar from classical wave behavior, for example, for a vibrating string. Since the meaning of the squared amplitude of the wavefunction can be visualized for the particle in a box as the probability of finding the electron, these nodal points represent regions in which the electron is not found.

       Example 2.3

      1 What is the probability P of finding a PiB in the center third of the box for n = 1?

      2 What is P for the same range for a classical particle?

      Answer:

      1 The probability P of finding a quantum mechanical particle–wave is given by the square of the amplitude of the wavefunction. Thus,(E2.3.1)The integral over the sin2 function can be evaluated using(E2.3.2)Then the probability is(E.2.3.3)

      2 A classical particle would be found with equal probability anywhere in the box; thus, the probability of finding it in the center third would just 1/3. Note that for higher values of n, the probability of finding it in the center third will decrease.

2.4 The Particle in a Two‐Dimensional Box, the Unbound Particle, and the Particle in a Box with Finite Energy Barriers

      2.4.1 Particle in a 2D Box

      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.

      2.4.2 The Unbound Particle

      Next, the case of a system without the restriction of the boundary conditions (an unbound particle) will be discussed. This discussion starts with the same Hamiltonian used before:

      (2.23)