Читать книгу Quantum Mechanical Foundations of Molecular Spectroscopy - Max Diem - Страница 26

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 2.3 and 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_mn in 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 2.2 as 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 2.3b 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.