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

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A^ in equation (3.72) equal first to the position variable x, then the variable y, and finally the variable z, we can show that 98 General principles of quantum theory m dhxi dt hpxi, m dhyi dt hpyi, m dhzi dt hpzi or, in vector notation m dhri dt hpi which is one of the Ehrenfest theorems discussed in Section 2.3. The other Ehrenfest theorem, dhpi dt ÿh=V (r)i may be obtained from equation (3.72) by setting A^ successively equal to p^x, p^y, and p^z. 3.11 Heisenberg uncertainty principle We have shown in Section 3.5 that commuting hermitian operators have simultaneous eigenfunctions and, therefore, that the physical quantities asso- ciated with those operators can be observed simultaneously. On the other hand, if the hermitian operators A^ and B^ do not commute, then the physical observables A and B cannot both be precisely determined at the same time. We begin by demonstrating this conclusion. Suppose that A^ and B^ do not commute. Let Æi and âi be the eigenvalues of A^ and B^, respectively, with corresponding eigenstates jÆii and jâii A^jÆii ÆijÆii (3:73a) B^jâii âijâii (3:73b) Some or all of the eigenvalues may be degenerate, but each eigenfunction has a unique index i. Suppose further that the system is in state jÆ ji, one of the eigenstates of A^. If we measure the physical observable A, we obtain the result Æ j. What happens if we simultaneously measure the physical observable B? To answer this question we need to calculate the expectation value hBi for this system hBi hÆ jjB^jÆ ji (3:74) If we expand the state function jÆ ji in terms of the complete, orthonormal set jâii jÆ ji X i cij âii where ci are the expansion coefficients, and substitute the expansion into equation (3.74), we obtain 3.11 Heisenberg uncertainty principle 99 hBi X i X k c\ufffdk cihâkjB^jâii X i X k c\ufffdk ciâiäki X i jcij2 âi where (3.73b) has been used. Thus, a measurement of B yields one of the many values âi with a probability jcij2. There is no way to predict which of the values âi will be obtained and, therefore, the observables A and B cannot both be determined concurrently. For a system in an arbitrary state Ø, neither of the physical observables A and B can be precisely determined simultaneously if A^ and B^ do not commute. Let \u2dcA and \u2dcB represent the width of the spread of values for A and B, respectively. We define the variance (\u2dcA)2 by the relation (\u2dcA)2 h(A^ÿ hAi)2i (3:75) that is, as the expectation value of the square of the deviation of A from its mean value. The positive square root \u2dcA is the standard deviation and is called the uncertainty in A. Noting that hAi is a real number, we can obtain an alternative expression for (\u2dcA)2 as follows: (\u2dcA)2 h(A^ÿ hAi)2i hA^2 ÿ 2hAiA^ hAi2i hA^2i ÿ 2hAihAi hAi2 hA^2i ÿ hAi2 (3:76) Expressions analogous to equations (3.75) and (3.76) apply for (\u2dcB)2. Since A^ and B^ do not commute, we define the operator C^ by the relation [A^, B^] A^B^ÿ B^A^ iC^ (3:77) The operator C^ is hermitian as discussed in Section 3.3, so that its expectation value hCi is real. The commutator of A^ÿ hAi and B^ÿ hBi may be expanded as follows [A^ÿ hAi, B^ÿ hBi] (A^ÿ hAi)(B^ÿ hBi)ÿ (B^ÿ hBi)(A^ÿ hAi) A^B^ÿ B^A^ iC^ (3:78) where the cross terms cancel since hAi and hBi are numbers and commute with the operators A^ and B^. We use equation (3.78) later in this section. We now introduce the operator A^ÿ hAi iº(B^ÿ hBi) where º is a real constant, and let this operator act on the state function Ø [A^ÿ hAi iº(B^ÿ hBi)]Ø The scalar product of the resulting function with itself is, of course, always positive, so that h[A^ÿ hAi iº(B^ÿ hBi)]Øj[A^ÿ hAi iº(B^ÿ hBi)]Øi > 0 (3:79) Expansion of this expression gives 100 General principles of quantum theory h(A^ÿ hAi)Øj(A^ÿ hAi)Øi º2h(B^ÿ hBi)Øj(B^ÿ hBi)Øi iºh(A^ÿ hAi)Øj(B^ÿ hBi)Øi ÿ iºh(B^ÿ hBi)Øj(A^ÿ hAi)Øi > 0 or, since A^ and B^ are hermitian hØj(A^ÿ hAi)2jØi º2hØj(B^ÿ hBi)2jØi iºhØj[A^ÿ hAi, B^ÿ hBi]jØi > 0 Applying equations (3.75) and (3.78), we have (\u2dcA)2 º2(\u2dcB)2 ÿ ºhCi > 0 If we complete the square of the terms involving º, we obtain (\u2dcA)2 (\u2dcB)2 ºÿ hCi 2(\u2dcB)2 \ufffd \ufffd2 ÿ hCi 2 4(\u2dcB)2 > 0 Since º is arbitrary, we select its value so as to eliminate the second term º hCi 2(\u2dcB)2 (3:80) thereby giving (\u2dcA)2(\u2dcB)2 > 1 4 hCi2 or, upon taking the positive square root, \u2dcA\u2dcB > 1 2 jhCij Substituting equation (3.77) into this result yields \u2dcA\u2dcB > 1 2 jh[A^, B^]ij (3:81) This general expression relates the uncertainties in the simultaneous measure- ments of A and B to the commutator of the corresponding operators A^ and B^ and is a general statement of the Heisenberg uncertainty principle. Position\u2013momentum uncertainty principle We now consider the special case for which A is the variable x (A^ x) and B is the momentum px (B^ ÿi" d=dx). The commutator [A^, B^] may be evaluated by letting it operate on Ø [A^, B^]Ø ÿi" x dØ dx ÿ dxØ dx \ufffd \ufffd i"Ø so that jh[A^, B^]ij " and equation (3.81) gives \u2dcx\u2dcpx > " 2 (3:82) The Heisenberg position\u2013momentum uncertainty principle (3.82) agrees with equation (2.26), which was derived by a different, but mathematically 3.11 Heisenberg uncertainty principle 101 equivalent procedure. The relation (3.82) is consistent with (1.44), which is based on the Fourier transform properties of wave packets. The difference between the right-hand sides of (1.44) and (3.82) is due to the precise definition (3.75) of the uncertainties in equation (3.82). Similar applications of equation (3.81) using the position\u2013momentum pairs y, p^y and z, p^z yield \u2dcy\u2dcpy > " 2 , \u2dcz\u2dcpz > " 2 Since x commutes with the operators p^y and p^z, y commutes with p^x and p^z, and z commutes with p^x and p^y, the relation (3.81) gives \u2dcqi\u2dcpj 0, i 6 j where q1 x, q2 y, q3 z, p1 px, p2 py, p3 pz. Thus, the position coordinate qi and the momentum component pj for i 6 j may be precisely determined simultaneously. Minimum uncertainty wave packet The minimum value of the product \u2dcA\u2dcB occurs for a particular state Ø for which the relation (3.81) becomes an equality, i.e., when \u2dcA\u2dcB 1 2 jh[A^, B^]ij (3:83) According to equation (3.79), this equality applies when [A^ÿ hAi iº(B^ÿ hBi)]Ø 0 (3:84) where º is given by (3.80). For the position\u2013momentum example where A^ x and B^ ÿi" d=dx, equation (3.84) takes the form ÿi" d dx ÿ hpxi \ufffd \ufffd Ø i º (xÿ hxi)Ø for which the solution is Ø ceÿ(xÿhxi)2=2º"eih pxix=" (3:85) where c is a constant of integration and may be used to normalize Ø. The real constant º may be shown from equation (3.80) to be º " 2(\u2dcpx)2 2(\u2dcx) 2 " where the relation \u2dcx\u2dcpx "=2 has been used, and is observed to be positive. Thus, the state function Ø in equation (3.85) for a particle with minimum position\u2013momentum uncertainty is a wave packet in the form of a plane wave exp[ihpxix="] with wave number k0 hpxi=" multiplied by a gaussian modulating function centered at hxi. Wave packets are discussed in Section 102 General principles of quantum theory 1.2. Only the spatial dependence of Ø has been derived in equation (3.85). The state function Ø may also depend on the time through the possible time dependence of the parameters c, º, hxi, and hpxi. Energy\u2013time uncertainty principle We now wish to derive the energy\u2013time uncertainty principle, which is discussed in Section 1.5 and expressed in equation (1.45). We show in Section 1.5 that for a wave packet associated with a free particle moving in the x- direction the product \u2dcE\u2dct is equal to the product \u2dcx\u2dcpx if \u2dcE and \u2dct are defined appropriately. However,