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

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a suitably large, but otherwise arbitrary value of º, say ç, and continually apply the lowering operator a^ to the eigenstate jçii, thereby forming a succession of eigenvectors a^jçii, a^2jçii, a^3jçii, . . . with respective eigenvalues çÿ 1, çÿ 2, çÿ 3, . . . We have already shown that if jçii is an eigenfunction of N^ , then a^jçii is also an eigenfunction. By iteration, if a^jçii is an eigenfunction of N^ , then a^2jçii is an eigenfunction, and so forth, so that the members of the sequence are all eigenfunctions. Eventually this procedure gives an eigenfunction a^kjçii with eigenvalue (çÿ k), k being a positive integer, such that 0 < (çÿ k) , 1. The next step in the sequence would yield the eigenfunction a^k1jçii with eigenvalue º (çÿ k ÿ 1) , 0, which is not allowed. Thus, the sequence must terminate by the condition a^k1jçii a^[a^kjçii] 0 112 Harmonic oscillator The only circumstance in which a^ operating on an eigenvector yields the value zero is when the eigenvector corresponds to the eigenvalue º 0, as shown in equation (4.29). Since the eigenvalue of a^kjçii is çÿ k and this eigenvalue equals zero, we have çÿ k 0 and ç must be an integer. The minimum value of º çÿ k is, then, zero. Beginning with º 0, we can apply the operator a^y successively to j0ii to form a series of eigenvectors a^yj0ii, a^y2j0ii, a^y3j0ii, . . . with respective eigenvalues 0, 1, 2, . . . Thus, the eigenvalues of the operator N^ are the set of positive integers, so that º 0, 1, 2, . . . Since the value ç was chosen arbitrarily and was shown to be an integer, this sequence generates all the eigenfunctions of N^ . There are no eigenfunctions corresponding to non- integral values of º. Since º is now known to be an integer n, we replace º by n in the remainder of this discussion of the harmonic oscillator. The energy eigenvalues as related to º in equation (4.25) are now expressed in terms of n by En (n 12)"ø, n 0, 1, 2, . . . (4:30) so that the energy is quantized in units of "ø. The lowest value of the energy or zero-point energy is "ø=2. Classically, the lowest energy for an oscillator is zero. Non-degeneracy of the energy levels To determine the degeneracy of the energy levels or, equivalently, of the eigenvalues of the number operator N^, we must first obtain the eigenvectors j0ii for the ground state. These eigenvectors are determined by equation (4.29). When equation (4.18a) is substituted for a^, equation (4.29) takes the form d dî î \ufffd \ufffd j0ii d dî î \ufffd \ufffd ö0i(î) 0 or dö0i ö0i ÿî dî This differential equation may be integrated to give ö0i(î) ceÿî2=2 eiÆðÿ1=4eÿî2=2 where the constant of integration c is determined by the requirement that the functions öni(î) be normalized and eiÆ is a phase factor. We have used the standard integral (A.5) to evaluate c. We observe that all the solutions for the ground-state eigenfunction are proportional to one another. Thus, there exists 4.2 Quantum treatment 113 only one independent solution and the ground state is non-degenerate. If we arbitrarily set Æ equal to zero so that ö0(î) is real, then the ground-state eigenvector is j0i ðÿ1=4eÿî2=2 (4:31) We next show that if the eigenvalue n of the number operator N^ is non- degenerate, then the eigenvalue n 1 is also non-degenerate. We begin with the assumption that there is only one eigenvector with the property that N^ jni njni and consider the eigenvector j(n 1)ii, which satisfies N^ j(n 1)ii (n 1)j(n 1)ii If we operate on j(n 1)ii with the lowering operator a^, we obtain to within a multiplicative constant c the unique eigenfunction jni, a^j(n 1)ii cjni We next operate on this expression with the adjoint of a^ to give a^y a^j(n 1)ii N^ j(n 1)ii (n 1)j(n 1)ii ca^yjni from which it follows that j(n 1)ii c n 1 a^ yjni Thus, all the eigenvectors j(n 1)ii corresponding to the eigenvalue n 1 are proportional to a^yjni and are, therefore, not independent since they are proportional to each other. We conclude then that if the eigenvalue n is non- degenerate, then the eigenvalue n 1 is non-degenerate. Since we have shown that the ground state is non-degenerate, we see that the next higher eigenvalue n 1 is also non-degenerate. But if the eigenvalue n 1 is non-degenerate, then the eigenvalue n 2 is non-degenerate. By iteration, all of the eigenvalues n of N^ are non-degenerate. From equation (4.30) we observe that all the energy levels En of the harmonic oscillator are non-degenerate. 4.3 Eigenfunctions Lowering and raising operations From equations (4.27) and (4.28) and the conclusions that the eigenvalues of N^ are non-degenerate and are positive integers, we see that a^jni and a^yjni are eigenfunctions of N^ with eigenvalues nÿ 1 and n 1, respectively. Accor- dingly, we may write a^jni cnjnÿ 1i (4:32a) 114 Harmonic oscillator and a^yjni c9njn 1i (4:32b) where cn and c9n are proportionality constants, dependent on the value of n, and to be determined by the requirement that jnÿ 1i, jni, and jn 1i are normal- ized. To evaluate the numerical constants cn and c9n, we square both sides of equations (4.32a) and (4.32b) and integrate with respect to î to obtain 1 ÿ1 ja^önj2 dî jcnj2 1 ÿ1 ö\ufffdnÿ1önÿ1 dî (4:33a) and 1 ÿ1 ja^yönj2 dî jc9nj2 1 ÿ1 ö\ufffdn1ön1 dî (4:33b) The integral on the left-hand side of equation (4.33a) may be evaluated as follows 1 ÿ1 ja^önj2 dî 1 ÿ1 (a^ö\ufffdn )(a^ön) dî hnja^y a^jni hnjN^ jni n Similarly, the integral on the left-hand side of equation (4.33b) becomes 1 ÿ1 ja^yönj2 dî 1 ÿ1 (a^yö\ufffdn )(a^yön) dî hnja^a^yjni hnjN^ 1jni n 1 Since the eigenfunctions are normalized, we obtain jcnj2 n, jc9nj2 n 1 Without loss of generality, we may let cn and c9n be real and positive, so that equations (4.32a) and (4.32b) become a^jni np jnÿ 1i (4:34a) a^yjni n 1p jn 1i (4:34b) If the normalized eigenvector jni is known, these relations may be used to obtain the eigenvectors jnÿ 1i and jn 1i, both of which will be normalized. Excited-state eigenfunctions We are now ready to obtain the set of simultaneous eigenfunctions for the commuting operators N^ and H^ . The ground-state eigenfunction j0i has already been determined and is given by equation (4.31). The series of eigenfunctions j1i, j2i, . . . are obtained from equations (4.34b) and (4.18b), which give jn 1i [2(n 1)]ÿ1=2 îÿ d dî \ufffd \ufffd jni (4:35) Thus, the eigenvector j1i is obtained from j0i 4.3 Eigenfunctions 115 j1i 2ÿ1=2 îÿ d dî \ufffd \ufffd (ðÿ1=4eÿî 2=2) 21=2ðÿ1=4îeÿî2=2 the eigenvector j2i from j1i j2i 1 2 îÿ d dî \ufffd \ufffd (21=2ðÿ1=4îeÿî 2=2) 2ÿ1=2ðÿ1=4(2î2 ÿ 1)eÿî2=2 the eigenvector j3i from j2i j3i 6ÿ1=2 îÿ d dî \ufffd \ufffd (2ÿ1=2ðÿ1=4(2î2 ÿ 1)eÿî2=2) 3ÿ1=2ðÿ1=4(2î3 ÿ 3î)eÿî2=2 and so forth, indefinitely. Each of the eigenfunctions obtained by this procedure is normalized. When equation (4.18a) is combined with (4.34a), we have jnÿ 1i (2n)ÿ1=2 î d dî \ufffd \ufffd jni (4:36) Just as equation (4.35) allows one to go \u2018up the ladder\u2019 to obtain jn 1i from jni, equation (4.36) allows one to go \u2018down the ladder\u2019 to obtain jnÿ 1i from jni. This lowering procedure maintains the normalization of each of the eigenvectors. Another, but completely equivalent, way of determining the series of eigen- functions may be obtained by first noting that equation (4.34b) may be written for the series n 0, 1, 2, . . . as follows j1i a^yj0i j2i 2ÿ1=2 a^yj1i 2ÿ1=2(a^y)2j0i j3i 3ÿ1=2 a^yj2i (3!)ÿ1=2(a^y)3j0i .. . Obviously, the expression for jni is jni (n!)ÿ1=2(a^y)nj0i Substitution of equation (4.18b) for a^y and (4.31) for the ground-state eigen- vector j0i gives jni (2n n!)ÿ1=2ðÿ1=4