Max Diem - Quantum Mechanical Foundations of Molecular Spectroscopy

(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.

In Eq. (2.34), “A” is an amplitude factor still undefined at this point. To determine “A,” one argues as follows: since the square of the wavefunction is defined as the probability of finding the particle, the square of the wavefunction written in Eq. (2.34), integrated over the length of the box, must be unity, since the particle is known to be in the box. This leads to the normalization condition

(2.35)

Using the integral relationship

(2.36)

the amplitude A is obtained as follows:

(2.37)

Thus, the normalized stationary‐state wavefunctions for the particle in a box can be written in a final form as

(2.38)

The stationary‐state (time‐independent) wavefunctions and energies are depicted in Figure 2.2, panel (a). Although one refers to these wavefunctions as time‐independent, they may be considered as standing waves in which the amplitudes oscillate between the extremes as shown in Figure 2.3and resemble the motion of a plugged string. Time independency then refers to the fact that the system will stay in one of these standing wave patterns forever or until perturbed by electromagnetic radiation.

The probability of finding the particle at any given position x is shown in Figure 2.2, panel (b). These traces are the squares of the wavefunctions and depict that for higher levels of n, the probability of finding the particle moves away from the center to the periphery of the box.

The PiB wavefunctions form an orthonormal vector space, which implies that

(2.39)

δ mnin Eq. (2.39)is referred to as the Kronecker symbol that has the value of 1 if n = m and is zero otherwise. The wavefunctions' normality was established above by normalizing them ( Eqs. (2.36)and (2.37)); in order to demonstrate that they are orthogonal, the integral

(2.40)

Figure 2.3 (a) Representation of the particle‐in‐a‐box wavefunctions shown in Figure 2.2as standing waves. (b) Visualization of the orthogonality of the first two PiB wavefunctions. See text for detail.

needs to be evaluated. This can be 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 2.3bfor 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 2.3aalso 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.