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

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the expectation value of the momentum can be obtained by an integration in coordinate space. The expectation value of p2 is given by equation (2.16) with f ( p) p2. The expression analogous to (2.17) is hp2i 1 ÿ1 Ø\ufffd(x, t) 1 2ð" p 1 ÿ1 p2 A( p, t) ei( pxÿEt)=" d p \ufffd \ufffd dx From equation (2.7) it can be seen that the quantity in square brackets equals 42 Schro¨dinger wave mechanics " i \ufffd \ufffd2 @2Ø(x, t) @x2 so that hp2i 1 ÿ1 Ø\ufffd(x, t) " i \ufffd \ufffd2 @2 @x2 Ø(x, t) dx (2:19) Similarly, the expectation value of pn is given by hpni 1 ÿ1 Ø\ufffd(x, t) " i @ @x \ufffd \ufffdn Ø(x, t) dx (2:20) Each of the integrands in equations (2.18), (2.19), and (2.20) is the complex conjugate of the wave function multiplied by an operator acting on the wave function. Thus, in the coordinate-space calculation of the expectation value of the momentum p or the nth power of the momentum, we associate with p the operator ("=i)(@=@x). We generalize this association to apply to the expectation value of any function f ( p) of the momentum, so that h f ( p)i 1 ÿ1 Ø\ufffd(x, t) f " i @ @x \ufffd \ufffd Ø(x, t) dx (2:21) Equation (2.21) is equivalent to the momentum-space equation (2.16). We may combine equations (2.14) and (2.21) to find the expectation value of a function f (x, p) of the position and momentum h f (x, p)i 1 ÿ1 Ø\ufffd(x, t) f x, " i @ @x \ufffd \ufffd Ø(x, t) dx (2:22) Ehrenfest\u2019s theorems According to the correspondence principle as stated by N. Bohr (1928), the average behavior of a well-defined wave packet should agree with the classical- mechanical laws of motion for the particle that it represents. Thus, the expectation values of dynamical variables such as position, velocity, momen- tum, kinetic energy, potential energy, and force as calculated in quantum mechanics should obey the same relationships that the dynamical variables obey in classical theory. This feature of wave mechanics is illustrated by the derivation of two relationships known as Ehrenfest\u2019s theorems. The first relationship is obtained by considering the time dependence of the expectation value of the position coordinate x. The time derivative of hxi in equation (2.13) is 2.3 Expectation values of dynamical quantities 43 dhxi dt d dt 1 ÿ1 xjØ(x, t)j2 dx 1 ÿ1 x @ @ t jØ(x, t)j2 dx i" 2m 1 ÿ1 x @ @x Ø\ufffd @Ø @x ÿØ @Ø \ufffd @x \ufffd \ufffd dx where equation (2.12) has been used. Integration by parts of the last integral gives dhxi dt i" 2m x Ø\ufffd @Ø @x ÿØ @Ø \ufffd @x \ufffd \ufffd1 ÿ1 ÿ i" 2m 1 ÿ1 Ø\ufffd @Ø @x ÿØ @Ø \ufffd @x \ufffd \ufffd dx The integrated part vanishes because Ø(x, t) goes to zero as x approaches (\ufffd) infinity. Another integration by parts of the last term on the right-hand side yields dhxi dt 1 m 1 ÿ1 Ø\ufffd " i @ @x \ufffd \ufffd Ø dx According to equation (2.18), the integral on the right-hand side of this equation is the expectation value of the momentum, so that we have hpi m dhxi dt (2:23) Equation (2.23) is the quantum-mechanical analog of the classical definition of momentum, p mv m(dx=dt). This derivation also shows that the associa- tion in quantum mechanics of the operator ("=i)(@=@x) with the momentum is consistent with the correspondence principle. The second relationship is obtained from the time derivative of the expecta- tion value of the momentum hpi in equation (2.18), dhpi dt d dt 1 ÿ1 Ø\ufffd " i @Ø @x dx " i 1 ÿ1 @Ø\ufffd @ t @Ø @x Ø\ufffd @ @x @Ø @ t \ufffd \ufffd dx We next substitute equations (2.11) for the time derivatives of Ø and Ø\ufffd and obtain dhpi dt 1 ÿ1 ÿ"2 2m @2Ø\ufffd @x2 VØ\ufffd \ufffd \ufffd @Ø @x Ø\ufffd @ @x "2 2m @2Ø @x2 ÿ VØ \ufffd \ufffd" # dx ÿ" 2 2m 1 ÿ1 @2Ø\ufffd @x2 @Ø @x dx " 2 2m 1 ÿ1 Ø\ufffd @ 3Ø @x3 dxÿ 1 ÿ1 Ø\ufffdØ dV dx dx (2:24) where the terms in V cancel. The first integral on the right-hand side of equation (2.24) may be integrated by parts twice to give 44 Schro¨dinger wave mechanics 1 ÿ1 @2Ø\ufffd @x2 @Ø @x dx @Ø \ufffd @x @Ø @x \ufffd \ufffd1 ÿ1 ÿ 1 ÿ1 @Ø\ufffd @x @2Ø @x2 dx @Ø \ufffd @x @Ø @x ÿØ\ufffd @ 2Ø @x2 \ufffd \ufffd1 ÿ1 1 ÿ1 Ø\ufffd @ 3Ø @x3 dx The integrated part vanishes because Ø and @Ø=@x vanish at (\ufffd) infinity. The remaining integral cancels the second integral on the right-hand side of equation (2.24), leaving the final result dhpi dt ÿ \ufffd dV dx \ufffd hFi (2:25) where equation (2.1) has been used. Equation (2.25) is the quantum analog of Newton\u2019s second law of motion, F ma, and is in agreement with the correspondence principle. Heisenberg uncertainty principle Using expectation values, we can derive the Heisenberg uncertainty principle introduced in Section 1.5. If we define the uncertainties \u2dcx and \u2dcp as the standard deviations of x and p, as used in statistics, then we have \u2dcx h(xÿ hxi)2i1=2 \u2dcp h( pÿ hpi)2i1=2 The expectation values of x and of p at a time t are given by equations (2.13) and (2.18), respectively. For the sake of simplicity in this derivation, we select the origins of the position and momentum coordinates at time t to be the centers of the wave packet and its Fourier transform, so that hxi 0 and hpi 0. The squares of the uncertainties \u2dcx and \u2dcp are then given by (\u2dcx)2 1 ÿ1 x2Ø\ufffdØ dx (\u2dcp)2 " i \ufffd \ufffd2 1 ÿ1 Ø\ufffd @ 2Ø @x2 dx " i \ufffd \ufffd2 Ø\ufffd @Ø @x " #1 ÿ1 ÿ " i \ufffd \ufffd2 1 ÿ1 @Ø\ufffd @x @Ø @x dx 1 ÿ1 ÿ" i @Ø\ufffd @x \ufffd \ufffd " i @Ø @x \ufffd \ufffd dx where the integrated term for (\u2dcp)2 vanishes because Ø goes to zero as x approaches (\ufffd) infinity. The product (\u2dcx\u2dcp)2 is 2.3 Expectation values of dynamical quantities 45 (\u2dcx\u2dcp)2 1 ÿ1 (xØ\ufffd)(xØ) dx 1 ÿ1 ÿ" i @Ø\ufffd @x \ufffd \ufffd " i @Ø @x \ufffd \ufffd dx Applying Schwarz\u2019s inequality (A.56), we obtain (\u2dcx\u2dcp)2 > 1 4 \ufffd\ufffd\ufffd\ufffd "i 1 ÿ1 xØ\ufffd @Ø @x xØ @Ø \ufffd @x \ufffd \ufffd dx \ufffd\ufffd\ufffd\ufffd2 "24 \ufffd\ufffd\ufffd\ufffd 1ÿ1x @@x (Ø\ufffdØ) dx \ufffd\ufffd\ufffd\ufffd2 " 2 4 \ufffd\ufffd\ufffd\ufffd\ufffdxØ\ufffdØ\ufffd1 ÿ1 ÿ 1 ÿ1 Ø\ufffdØ dx \ufffd\ufffd\ufffd\ufffd2 The integrated part vanishes because Ø goes to zero faster than 1= jxjp , as x approaches (\ufffd) infinity and the remaining integral is unity by equation (2.9). Taking the square root, we obtain an explicit form of the Heisenberg uncer- tainty principle \u2dcx\u2dcp > " 2 (2:26) This expression is consistent with the earlier form, equation (1.44), but relation (2.26) is based on a precise definition of the uncertainties, whereas relation (1.44) is not. 2.4 Time-independent Schro¨dinger equation The first step in the solution of the partial differential equation (2.6) is to express the wave function Ø(x, t) as the product of two functions Ø(x, t) \u142(x)÷(t) (2:27) where \u142(x) is a function of only the distance x and ÷(t) is a function of only the time t. Substitution of equation (2.27) into (2.6) and division by the product \u142(x)÷(t) give i" 1 ÷(t) d÷(t) dt ÿ " 2 2m 1 \u142(x) d2\u142(x) dx2 V (x) (2:28) The left-hand side of equation (2.28) is a function only of t, while the right- hand side is a function only of x. Since x and t are independent variables, each side of equation (2.28) must equal a constant. If this were not true, then the left-hand side could be changed by varying t while the right-hand side remained fixed and so the equality would no longer apply. For reasons that will soon be apparent, we designate this separation constant by E and assume that it is a real number. Equation (2.28) is now separable into two independent differential equations, one for each of the two independent variables x and t. The time-dependent