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

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integrate over the spatial variablesX n " i @an(t) @ t Enan(t) \ufffd \ufffd h\u142k(q)j\u142n(q)i X n " i @an(t) @ t Enan(t) \ufffd \ufffd äkn " i @ak(t) @ t Ek ak(t) 0 Replacing the dummy index k by n, we obtain the result an(t) cneÿiEn t=" (3:63) where cn is a constant independent of both q and t. Substitution of equation (3.63) into (3.60) gives equation (3.59), showing that equation (3.59) is indeed the most general form for a solution of the time-dependent Schro¨dinger equation. All solutions may be expressed as the sum over stationary states. 3.8 Parity operator The parity operator \u2014^ is defined by the relation \u2014^\u142(q) \u142(ÿq) (3:64) Thus, the parity operator reverses the sign of each cartesian coordinate. This operator is equivalent to an inversion of the coordinate system through the origin. In one and three dimensions, equation (3.64) takes the form \u2014^\u142(x) \u142(ÿx), \u2014^\u142(r) \u2014^\u142(x, y, z) \u2014^\u142(ÿx, ÿy, ÿz) \u142(ÿr) The operator \u2014^2 is equal to unity since 94 General principles of quantum theory \u2014^2\u142(q) \u2014^(\u2014^\u142(q)) \u2014^\u142(ÿq) \u142(q) Further, we see that \u2014^n\u142(q) \u142(q), n even \u142(ÿq), n odd or \u2014^n 1, n even \u2014^, n odd The operator \u2014^ is linear and hermitian. In the one-dimensional case, the hermiticity of \u2014^ is demonstrated as follows höj\u2014^j\u142i 1 ÿ1 ö\ufffd(x)\u142(ÿx) dx ÿ ÿ1 1 ö\ufffd(ÿx9)\u142(x9) dx9 1 ÿ1 \u142(x9)\u2014^ö\ufffd(x9) dx9 h\u2014^öj\u142i where x in the second integral is replaced by x9 ÿx to obtain the third integral. By applying the same procedure to each coordinate, we can show that \u2014^ is hermitian with respect to multi-dimensional functions. The eigenvalues º of the parity operator \u2014^ are given by \u2014^\u142º(q) º\u142º(q) (3:65) where \u142º(q) are the corresponding eigenfunctions. If we apply \u2014^ to both sides of equation (3.65), we obtain \u2014^2\u142º(q) º\u2014^\u142º(q) º2\u142º(q) Since \u2014^2 1, we see that º2 1 and that the eigenvalues º, which must be real because \u2014^ is hermitian, are equal to either 1 or ÿ1. To find the eigenfunctions \u142º(q), we note that equation (3.65) now becomes \u142º(ÿq) \ufffd\u142º(q) For º 1, the eigenfunctions of \u2014^ are even functions of q, while for º ÿ1, they are odd functions of q. An even function of q is said to be of even parity, while odd parity refers to an odd function of q. Thus, the eigenfunctions of \u2014^ are any well-behaved functions that are either of even or odd parity in their cartesian variables. We show next that the parity operator \u2014^ commutes with the Hamiltonian operator H^ if the potential energy V (q) is an even function of q. The kinetic energy term in the Hamiltonian operator is given by ÿ " 2 2m =2 ÿ " 2 2m @2 @q21 @ 2 @q22 \ufffd \ufffd \ufffd ! 3.8 Parity operator 95 and is an even function of each qk . If the potential energy V (q) is also an even function of each qk , then we have H^(q) H^(ÿq) and [ H^ , \u2014^] f (q) H^(q)\u2014^ f (q)ÿ \u2014^ H^(q) f (q) H^(q) f (ÿq)ÿ H^(ÿq) f (ÿq) 0 Since the function f (q) is arbitrary, the commutator of H^ and \u2014^ vanishes. Thus, these operators have simultaneous eigenfunctions for systems with V (q) V (ÿq). If the potential energy of a system is an even function of the coordinates and if \u142(q) is a solution of the time-independent Schro¨dinger equation, then the function \u142(ÿq) is also a solution. When the eigenvalues of the Hamiltonian operator are non-degenerate, these two solutions are not independent of each other, but are proportional \u142(ÿq) c\u142(q) These eigenfunctions are also eigenfunctions of the parity operator, leading to the conclusion that c \ufffd1. Consequently, some eigenfunctions will be of even parity while all the others will be of odd parity. For a degenerate energy eigenvalue, the several corresponding eigenfunc- tions of H^ may not initially have a definite parity. However, each eigenfunction may be written as the sum of an even part \u142e(q) and an odd part \u142o(q) \u142(q) \u142e(q) \u142o(q) where \u142e(q) 12[\u142(q) \u142(ÿq)] \u142e(ÿq) \u142o(q) 12[\u142(q)ÿ \u142(ÿq)] ÿ\u142o(ÿq) Since any linear combination of \u142(q) and \u142(ÿq) satisfies Schro¨dinger\u2019s equa- tion, the functions \u142e(q) and \u142o(q) are eigenfunctions of H^ . Furthermore, the functions \u142e(q) and \u142o(q) are also eigenfunctions of the parity operator \u2014^, the first with eigenvalue 1 and the second with eigenvalue ÿ1. 3.9 Hellmann\u2013Feynman theorem A useful expression for evaluating expectation values is known as the Hell- mann\u2013Feynman theorem. This theorem is based on the observation that the Hamiltonian operator for a system depends on at least one parameter º, which can be considered for mathematical purposes to be a continuous variable. For example, depending on the particular system, this parameter º may be the mass of an electron or a nucleus, the electronic charge, the nuclear charge parameter Z, a constant in the potential energy, a quantum number, or even Planck\u2019s constant. The eigenfunctions and eigenvalues of H^(º) also depend on this 96 General principles of quantum theory parameter, so that the time-independent Schro¨dinger equation (3.58) may be written as H^(º)\u142n(º) En(º)\u142n(º) (3:66) The expectation value of H^(º) is, then En(º) h\u142n(º)j H^(º)j\u142n(º)i (3:67) where we assume that \u142n(º) is normalized h\u142n(º)j\u142n(º)i 1 (3:68) To obtain the Hellmann\u2013Feynman theorem, we differentiate equation (3.67) with respect to º d dº En(º) \u142n(º) \ufffd\ufffd\ufffd\ufffd ddº H^(º) \ufffd\ufffd\ufffd\ufffd\u142n(º) * + d dº \u142n(º) \ufffd\ufffd\ufffd\ufffd H^(º)\ufffd\ufffd\ufffd\ufffd\u142n(º) * + \u142n(º) \ufffd\ufffd\ufffd\ufffd H^(º)\ufffd\ufffd\ufffd\ufffd ddº\u142n(º) * + (3:69) Applying the hermitian property of H^(º) to the third integral on the right-hand side of equation (3.69) and then applying (3.66) to the second and third terms, we obtain d dº En(º) \u142n(º) \ufffd\ufffd\ufffd\ufffd ddº H^(º) \ufffd\ufffd\ufffd\ufffd\u142n(º) * + En(º) d dº \u142n(º) \ufffd\ufffd\ufffd\ufffd\u142n(º) * + \u142n(º) \ufffd\ufffd\ufffd\ufffd ddº\u142n(º) * +" # (3:70) The derivative of equation (3.68) with respect to º is d dº \u142n(º) \ufffd\ufffd\ufffd\ufffd\u142n(º) * + \u142n(º) \ufffd\ufffd\ufffd\ufffd ddº\u142n(º) * + 0 showing that the last term on the right-hand side of (3.70) vanishes. We thereby obtain the Hellmann\u2013Feynman theorem d dº En(º) \u142n(º) \ufffd\ufffd\ufffd\ufffd ddº H^(º) \ufffd\ufffd\ufffd\ufffd\u142n(º) * + (3:71) 3.10 Time dependence of the expectation value The expectation value hAi of the dynamical quantity or observable A is, in general, a function of the time t. To determine how hAi changes with time, we take the time derivative of equation (3.46) 3.10 Time dependence of the expectation value 97 dhAi dt d dt hØjA^jØi @Ø @ t \ufffd\ufffd\ufffd\ufffd A^ \ufffd\ufffd\ufffd\ufffdØ * + Ø \ufffd\ufffd\ufffd\ufffdA^ \ufffd\ufffd\ufffd\ufffd @Ø@ t * + Ø \ufffd\ufffd\ufffd\ufffd @A^@ t \ufffd\ufffd\ufffd\ufffdØ * + Equation (3.55) may be substituted for the time derivatives of the wave function to give dhAi dt i " h H^ØjA^jØi ÿ i " hØjA^ H^ jØi * Ø \ufffd\ufffd\ufffd\ufffd @A^@ t \ufffd\ufffd\ufffd\ufffdØ + i " hØj H^A^jØi ÿ i " hØjA^ H^ jØi Ø \ufffd\ufffd\ufffd\ufffd @A^@ t \ufffd\ufffd\ufffd\ufffdØ * + i " hØj[ H^ , A^]jØi Ø \ufffd\ufffd\ufffd\ufffd @A^@ t \ufffd\ufffd\ufffd\ufffdØ * + i " h[ H^ , A^]i @A^ @ t \ufffd \ufffd where the hermiticity of H^ and the definition (equation (3.3)) of the commu- tator have been used. If the operator A^ is not an explicit function of time, then the last term on the right-hand side vanishes and we have dhAi dt i " h[ H^ , A^]i (3:72) If we set A^ equal to unity, then the commutator [ H^ , A^] vanishes and equation (3.72) becomes dhAi dt 0 or d dt hØjA^jØi d dt hØjØi 0 We thereby obtain the result in Section 2.2 that if Ø is normalized, it remains normalized as time progresses. If the operator A^ in equation (3.72) is set equal to H^ , then again the commutator vanishes and we have dhAi dt dhHi dt dE dt 0 Thus, the energy E of the system, which is equal to the expectation value of the Hamiltonian, is conserved if the Hamiltonian does not depend explicitly on time. By setting the operator