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

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density for every point in space and at all times, the wave function must be continuous, single-valued, and finite. Since Ø(x, t) satisfies a differential equation that is second-order in x, its first derivative is also continuous. The wave function may be multiplied by a phase factor eiÆ, where Æ is real, without changing its physical significance since [eiÆØ(x, t)]\ufffd[eiÆØ(x, t)] Ø\ufffd(x, t)Ø(x, t) jØ(x, t)j2 Normalization The particle that is represented by the wave function must be found with probability equal to unity somewhere in the range ÿ1 < x <1, so that Ø(x, t) must obey the relation 1 ÿ1 jØ(x, t)j2 dx 1 (2:9) 38 Schro¨dinger wave mechanics A function that obeys this equation is said to be normalized. If a function Ö(x, t) is not normalized, but satisfies the relation 1 ÿ1 Ö\ufffd(x, t)Ö(x, t) dx N then the function Ø(x, t) defined by Ø(x, t) 1 N p Ö(x, t) is normalized. In order for Ø(x, t) to satisfy equation (2.9), the wave function must be square-integrable (also called quadratically integrable). Therefore, Ø(x, t) must go to zero faster than 1= jxjp as x approaches (\ufffd) infinity. Likewise, the derivative @Ø=@x must also go to zero as x approaches (\ufffd) infinity. Once a wave function Ø(x, t) has been normalized, it remains normalized as time progresses. To prove this assertion, we consider the integral N 1 ÿ1 Ø\ufffdØ dx and show that N is independent of time for every function Ø that obeys the Schro¨dinger equation (2.6). The time derivative of N is dN dt 1 ÿ1 @ @ t jØ(x, t)j2 dx (2:10) where the order of differentiation and integration has been interchanged on the right-hand side. The derivative of the probability density may be expanded as follows @ @ t jØ(x, t)j2 @ @ t (Ø\ufffdØ) Ø\ufffd @Ø @ t Ø @Ø \ufffd @ t Equation (2.6) and its complex conjugate may be written in the form @Ø @ t i" 2m @2Ø @x2 ÿ i " VØ @Ø\ufffd @ t ÿ i" 2m @2Ø\ufffd @x2 i " VØ\ufffd (2:11) so that @jØ(x, t)j2=@ t becomes @ @ t jØ(x, t)j2 i" 2m Ø\ufffd @ 2Ø @x2 ÿØ @ 2Ø\ufffd @x2 \ufffd \ufffd where the terms containing V cancel. We next note that @ @x Ø\ufffd @Ø @x ÿØ @Ø \ufffd @x \ufffd \ufffd Ø\ufffd @ 2Ø @x2 ÿØ @ 2Ø\ufffd @x2 2.2 The wave function 39 so that @ @ t jØ(x, t)j2 i" 2m @ @x Ø\ufffd @Ø @x ÿØ @Ø \ufffd @x \ufffd \ufffd (2:12) Substitution of equation (2.12) into (2.10) and evaluation of the integral give dN dt i" 2m 1 ÿ1 @ @x Ø\ufffd @Ø @x ÿØ @Ø \ufffd @x \ufffd \ufffd dx i" 2m Ø\ufffd @Ø @x ÿØ @Ø \ufffd @x \ufffd \ufffd1 ÿ1 Since Ø(x, t) goes to zero as x goes to (\ufffd) infinity, the right-most term vanishes and we have dN dt 0 Thus, the integral N is time-independent and the normalization of Ø(x, t) does not change with time. Not all wave functions can be normalized. In such cases the quantity jØ(x, t)j2 may be regarded as the relative probability density, so that the ratio a2 a1 jØ(x, t)j2 dx b2 b1 jØ(x, t)j2 dx represents the probability that the particle will be found between a1 and a2 relative to the probability that it will be found between b1 and b2. As an example, the plane wave Ø(x, t) ei( pxÿEt)=" does not approach zero as x approaches (\ufffd) infinity and consequently cannot be normalized. The probability density jØ(x, t)j2 is unity everywhere, so that the particle is equally likely to be found in any region of a specified width. Momentum-space wave function The wave function Ø(x, t) may be represented as a Fourier integral, as shown in equation (2.7), with its Fourier transform A( p, t) given by equation (2.8). The transform A( p, t) is uniquely determined by Ø(x, t) and the wave function Ø(x, t) is uniquely determined by A( p, t). Thus, knowledge of one of these functions is equivalent to knowledge of the other. Since the wave function Ø(x, t) completely describes the physical system that it represents, its Fourier transform A( p, t) also possesses that property. Either function may serve as a complete description of the state of the system. As a consequence, we may interpret the quantity jA( p, t)j2 as the probability density for the momentum at 40 Schro¨dinger wave mechanics time t. By Parseval\u2019s theorem (equation (B.28)), if Ø(x, t) is normalized, then its Fourier transform A( p, t) is normalized, 1 ÿ1 jØ(x, t)j2 dx 1 ÿ1 jA( p, t)j2 d p 1 The transform A( p, t) is called the momentum-space wave function, while Ø(x, t) is more accurately known as the coordinate-space wave function. When there is no confusion, however, Ø(x, t) is usually simply referred to as the wave function. 2.3 Expectation values of dynamical quantities Suppose we wish to measure the position of a particle whose wave function is Ø(x, t). The Born interpretation of jØ(x, t)j2 as the probability density for finding the associated particle at position x at time t implies that such a measurement will not yield a unique result. If we have a large number of particles, each of which is in state Ø(x, t) and we measure the position of each of these particles in separate experiments all at some time t, then we will obtain a multitude of different results. We may then calculate the average or mean value hxi of these measurements. In quantum mechanics, average values of dynamical quantities are called expectation values. This name is somewhat misleading, because in an experimental measurement one does not expect to obtain the expectation value. By definition, the average or expectation value of x is just the sum over all possible values of x of the product of x and the probability of obtaining that value. Since x is a continuous variable, we replace the probability by the probability density and the sum by an integral to obtain hxi 1 ÿ1 xjØ(x, t)j2 dx (2:13) More generally, the expectation value h f (x)i of any function f (x) of the variable x is given by h f (x)i 1 ÿ1 f (x)jØ(x, t)j2 dx (2:14) Since Ø(x, t) depends on the time t, the expectation values hxi and h f (x)i in equations (2.13) and (2.14) are functions of t. The expectation value hpi of the momentum p may be obtained using the momentum-space wave function A( p, t) in the same way that hxi was obtained from Ø(x, t). The appropriate expression is 2.3 Expectation values of dynamical quantities 41 hpi 1 ÿ1 pjA( p, t)j2 d p 1 ÿ1 pA\ufffd( p, t)A( p, t) d p (2:15) The expectation value h f ( p)i of any function f ( p) of p is given by an expression analogous to equation (2.14) h f ( p)i 1 ÿ1 f ( p)jA( p, t)j2 d p (2:16) In general, A( p, t) depends on the time, so that the expectation values hpi and h f ( p)i are also functions of time. Both Ø(x, t) and A( p, t) contain the same information about the system, making it possible to find hpi using the coordinate-space wave function Ø(x, t) in place of A( p, t). The result of establishing such a procedure will prove useful when determining expectation values for functions of both position and momentum. We begin by taking the complex conjugate of A( p, t) in equation (2.8) A\ufffd( p, t) 1 2ð" p 1 ÿ1 Ø\ufffd(x, t)ei( pxÿEt)=" dx Substitution of A\ufffd( p, t) into the integral on the right-hand side of equation (2.15) gives hpi 1 2ð" p 1 ÿ1 Ø\ufffd(x, t) pA( p, t)ei( pxÿEt)=" dx d p 1 ÿ1 Ø\ufffd(x, t) 1 2ð" p 1 ÿ1 pA( p, t)ei( pxÿEt)=" d p \ufffd \ufffd dx (2:17) In order to evaluate the integral over p, we observe that the derivative of Ø(x, t) in equation (2.7), with respect to the position variable x, is @Ø(x, t) @x 1 2ð" p 1 ÿ1 i " pA( p, t)ei( pxÿEt)=" d p Substitution of this observation into equation (2.21) gives the final result hpi 1 ÿ1 Ø\ufffd(x, t) " i @ @x \ufffd \ufffd Ø(x, t) dx (2:18) Thus,