# Principles of Quantum Mechanics as Applied to Chemistry and Chemical Physics

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îÿ d dî \ufffd \ufffdn eÿî 2=2 (4:37) This equation may be somewhat simplified if we note that 116 Harmonic oscillator îÿ d dî \ufffd \ufffd eÿî 2=2 îÿ d dî \ufffd \ufffd eî 2=2eÿî 2 îeÿî2=2 ÿ îeî2=2eÿî2 ÿ eî2=2 d dî eÿî 2 ÿeî2=2 d dî eÿî 2 so that îÿ d dî \ufffd \ufffdn eÿî 2=2 (ÿ1)neî2=2 d n dî n eÿî 2 (4:38) Substitution of equation (4.38) into (4.37) gives jni (ÿ1)n(2n n!)ÿ1=2ðÿ1=4eî2=2 d n dî n eÿî 2 (4:39) which may be used to obtain the entire set of eigenfunctions of N^ and H^ . Eigenfunctions in terms of Hermite polynomials It is customary to express the eigenfunctions for the stationary states of the harmonic oscillator in terms of the Hermite polynomials. The infinite set of Hermite polynomials Hn(î) is defined in Appendix D, which also derives many of the properties of those polynomials. In particular, equation (D.3) relates the Hermite polynomial of order n to the nth-order derivative which appears in equation (4.39) H n(î) (ÿ1)neî2 d n dî n eÿî 2 Therefore, we may express the eigenvector jni in terms of the Hermite polynomial Hn(î) by the relation jni ön(î) (2n n!)ÿ1=2ðÿ1=4 Hn(î)eÿî2=2 (4:40) The eigenstates \u142n(x) are related to the functions ön(î) by equation (4.16), so that we have \u142n(x) (2n n!)ÿ1=2 møð" \ufffd \ufffd1=4 Hn(î)e ÿî2=2 î mø " \ufffd \ufffd1=2 x (4:41) For reference, the Hermite polynomials for n 0 to n 10 are listed in Table 4.1. When needed, higher-order Hermite polynomials are most easily obtained from the recurrence relation (D.5). If only a single Hermite polynomial is wanted and the neighboring polynomials are not available, then equation (D.4) may be used. 4.3 Eigenfunctions 117 The functions ön(î) in equation (4.40) are identical to those defined by equation (D.15) and, therefore, form a complete set as shown in equation (D.19). Substituting equation (4.16) into (D.19) and applying the relation (C.5b), we see that the functions \u142n(x) in equation (4.41) form a complete set, so that X1 n0 \u142n(x)\u142n(x9) ä(xÿ x9) (4:42) Physical interpretation The first four eigenfunctions \u142n(x) for n 0, 1, 2, 3 are plotted in Figure 4.2 and the corresponding functions [\u142n(x)]2 in Figure 4.3. These figures also show the outline of the potential energy V (x) from equation (4.5) and the four corresponding energy levels from equation (4.30). The function [\u142n(x)]2 is the probability density as a function of x for the particle in the nth quantum state. The quantity [\u142n(x)]2 dx at any point x gives the probability for finding the particle between x and x dx. We wish to compare the quantum probability distributions with those obtained from the classical treatment of the harmonic oscillator at the same energies. The classical probability density P(y) as a function of the reduced distance y (ÿ1 < y < 1) is given by equation (4.10) and is shown in Figure 4.1. When equations (4.8), (4.14), and (4.30) are combined, we see that the maximum displacement in terms of î for a classical oscillator with energy (n 1 2 )"ø is 2n 1p . For î,ÿ 2n 1p and î. 2n 1p , the classical Table 4.1. Hermite polynomials n Hn(î) 0 1 1 2î 2 4î2 ÿ 2 3 8î3 ÿ 12î 4 16î4 ÿ 48î2 12 5 32î5 ÿ 160î3 120î 6 64î6 ÿ 480î4 720î2 ÿ 120 7 128î7 ÿ 1344î5 3360î3 ÿ 1680î 8 256î8 ÿ 3584î6 13440î4 ÿ 13440î2 1680 9 512î9 ÿ 9216î7 48384î5 ÿ 80640î3 30240î 10 1024î10 ÿ 23040î8 161280î6 ÿ 403200î4 302400î2 ÿ 30240 118 Harmonic oscillator probability for finding the particle is equal to zero. These regions are shaded in Figures 4.2 and 4.3. Each of the quantum probability distributions differs from the corresponding classical distribution in one very significant respect. In the quantum solution there is a non-vanishing probability of finding the particle outside the classi- cally allowed region, i.e., in a region where the total energy is less than the potential energy. Since the Hermite polynomial Hn(î) is of degree n, the wave function \u142n(x) has n nodes, a node being a point where a function touches or crosses the x-axis. The quantum probability density [\u142n(x)]2 is zero at a node. Within the classically allowed region, the wave function and the probability density oscillate with n nodes; outside that region the wave function and probability density rapidly approach zero with no nodes. E n e rg y 7 2 hø 5 2 hø 3 2 hø 1 2 hø n 5 3 n 5 2 n 5 1 n 5 0 x 0 \u1420 \u1421 \u1422 \u1423 V Figure 4.2 Wave functions and energy levels for a particle in a harmonic potential well. The outline of the potential energy is indicated by shading. 4.3 Eigenfunctions 119 While the classical particle is most likely to be found near its maximum displacement, the probability density for the quantum particle in the ground state is largest at the origin. However, as the value of n increases, the quantum probability distribution begins to look more and more like the classical probability distribution. In Figure 4.4 the function [\u14230(x)]2 is plotted along with the classical result for an energy 30:5 "ø. The average behavior of the rapidly oscillating quantum curve agrees well with the classical curve. This observation is an example of the Bohr correspondence principle, mentioned in Section 2.3. According to the correspondence principle, classical mechanics and quantum theory must give the same results in the limit of large quantum numbers. E n e rg y 7 2 hø 5 2 hø 3 2 hø 1 2 hø n 5 3 n 5 2 n 5 1 n 5 0 x 0 |\u1420|2 |\u1421|2 |\u1422|2 |\u1423|2 Figure 4.3 Probability densities and energy levels for a particle in a harmonic potential well. The outline of the potential energy is indicated by shading. 120 Harmonic oscillator 4.4 Matrix elements In the application to an oscillator of some quantum-mechanical procedures, the matrix elements of xn and p^n for a harmonic oscillator are needed. In this section we derive the matrix elements hn9jxjni, hn9jx2jni, hn9j p^jni, and hn9j p^2jni, and show how other matrix elements may be determined. The ladder operators a^ and a^y defined in equation (4.18) may be solved for x and for p^ to give x " 2mø \ufffd \ufffd1=2 (a^y a^) (4:43a) p^ i m"ø 2 \ufffd \ufffd1=2 (a^y ÿ a^) (4:43b) From equations (4.34) and the orthonormality of the harmonic oscillator eigenfunctions jni, we find that the matrix elements of a^ and a^y are hn9ja^jni np hn9jnÿ 1i np än9,nÿ1 (4:44a) hn9ja^yjni n 1p hn9jn 1i n 1p än9,n1 (4:44b) The set of equations (4.43) and (4.44) may be used to evaluate the matrix elements of any integral power of x and p^. To find the matrix element hn9jxjni, we apply equations (4.43a) and (4.44) to obtain x 0 |\u14230(x)|2 Figure 4.4 The probability density j\u14230(x)j2 for an oscillating particle in state n 30. The dotted curve is the classical probability density for a particle with the same energy. 4.4 Matrix elements 121 hn9jxjni " 2mø \ufffd \ufffd1=2 (hn9ja^yjni hn9ja^jni) " 2mø \ufffd \ufffd1=2 ( n 1p än9,n1 n p än9,nÿ1) so that hn 1jxjni (n 1)" 2mø r (4:45a) hnÿ 1jxjni n" 2mø r (4:45b) hn9jxjni 0 for n9 6 n 1, nÿ 1 (4:45c) If we replace n by nÿ 1 in equation (4.45a), we obtain hnjxjnÿ 1i n" 2mø r From equation (4.45b) we see that hnÿ 1jxjni hnjxjnÿ 1i Likewise, we can show that hn 1jxjni hnjxjn 1i In general, then, we have hn9jxjni hnjxjn9i To find the matrix element hn9j p^jni, we use equations (4.43b) and (4.44) to give hn9j p^jni i m"ø 2 \ufffd \ufffd1=2 hn9ja^y ÿ a^jni i m"ø 2 \ufffd \ufffd1=2 (