Quantum Mechanical Foundations of Molecular Spectroscopy. Max DiemЧитать онлайн книгу.

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) upper E left-parenthesis n right-parenthesis equals StartFraction n squared h squared Over 8 m upper L squared EndFraction

Substituting these energy eigenvalues back into Eq. (2.27)

(2.27) normal psi left-parenthesis x right-parenthesis equals upper A sine left-parenthesis StartFraction 2 italic m upper E Over normal h with stroke squared EndFraction right-parenthesis Superscript one half Baseline x

one obtains

(2.34) normal psi Subscript n Baseline left-parenthesis x right-parenthesis equals upper A sine left-parenthesis StartFraction 2 m n squared h squared Over normal h with stroke squared 8 m upper L squared EndFraction right-parenthesis Superscript one half Baseline x equals upper A sine StartFraction n normal pi Over upper L EndFraction x

which are the wave functions for the PiB.

2.3.3 Normalization and Orthogonality of the PiB Wavefunctions

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) integral Subscript 0 Superscript upper L Baseline normal psi Subscript n Superscript 2 Baseline left-parenthesis x right-parenthesis normal d x equals 1 equals upper A squared integral Subscript 0 Superscript upper L Baseline sine squared left-parenthesis StartFraction n normal pi Over upper L EndFraction x right-parenthesis normal d x

Using the integral relationship

(2.36) integral sine squared italic a x normal d x equals StartFraction x Over 2 EndFraction minus StartFraction 1 Over 4 normal a EndFraction sine 2 italic a x

the amplitude A is obtained as follows:

upper A squared integral Subscript 0 Superscript upper L Baseline sine left-parenthesis StartFraction n normal pi x Over upper L EndFraction right-parenthesis squared normal d x equals upper A squared left-bracket StartFraction x Over 2 EndFraction minus StartFraction upper L Over 4 n normal pi EndFraction sine left-parenthesis StartFraction 2 n normal pi x Over upper L EndFraction right-parenthesis right-bracket Subscript x equals 0 Superscript x equals upper L Baseline equals 1

upper A squared left-bracket StartFraction upper L Over 2 EndFraction minus StartFraction upper L Over 4 n normal pi EndFraction sine left-parenthesis StartFraction 2 n normal pi upper L Over upper L EndFraction right-parenthesis minus 0 plus StartFraction upper L Over 4 n normal pi EndFraction sine 0 right-bracket equals upper A squared left-bracket StartFraction upper L Over 2 EndFraction right-bracket equals 1

(2.37) upper A equals StartRoot StartFraction 2 Over upper L EndFraction EndRoot

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

(2.38) normal psi Subscript n Baseline left-parenthesis x right-parenthesis equals StartRoot StartFraction 2 Over upper L EndFraction EndRoot sine left-parenthesis StartFraction n normal pi Over upper L EndFraction right-parenthesis x

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) integral Subscript 0 Superscript upper L Baseline normal psi Subscript n Baseline left-parenthesis x right-parenthesis normal psi Subscript m Baseline left-parenthesis x right-parenthesis normal d x equals normal delta Subscript italic m n Baseline equals StartBinomialOrMatrix 1 if n equals m Choose 0 if n not-equals m EndBinomialOrMatrix

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) StartFraction 2 Over upper L EndFraction integral Subscript 0 Superscript upper L Baseline sine StartFraction n normal pi x Over upper L EndFraction sine StartFraction m normal pi x Over upper L EndFraction normal d x

Schematic illustration of the (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.

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.

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