Quantum Mechanical Foundations of Molecular Spectroscopy. Max Diem

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target="_blank" rel="nofollow" href="#ulink_3388a178-49af-5b2f-8530-4fd249f53989">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)

      When this differential equation is solved without the previously used boundary conditions

      (2.29)

      the new solutions represent a particle–wave that travels along the positive or negative x‐direction. The most general solution of the differential Eq. (2.23) is

(2.48)

      where b is a constant.

The second derivative of Eq. (2.48) is given by

      (2.49)

      with

(2.50)

      or

(2.51)

Equation (2.51) was obtained by substituting

      (2.52)

into Eq. (2.50). Thus, the unbound particle can be described by a traveling wave (as opposed to a standing wave)

      (2.53)

      carrying a momentum

      (2.54)

      into the positive or negative x‐direction. k is the wave vector defined in Eq. (1.6).

      2.4.3 The Particle in a Box with Finite Energy Barriers

Finally, the particle in a box with a finite energy barrier, V₀, will be discussed qualitatively. This is a situation where the particle is no longer strictly forbidden outside the confinement box and leads to the concept of tunneling, that is, the probability of the electron found outside the box. The shape of the potential function is shown in Figure 2.5b.

      The potential energy for this case is written as

      (2.55)

      and

      (2.56)

      (Notice that the boundaries of the box

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