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

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of solutions \u142n(x) given by equation (2.40). The first four wave functions \u142n(x) for n 1, 2, 3, and 4 and their corresponding probability densities j\u142n(x)j2 are shown in Figure 2.2. The wave function \u1421(x) corre- sponding to the lowest energy level E1 is called the ground state. The other wave functions are called excited states. If we integrate the product of two different wave functions \u142 l(x) and \u142n(x), we find that a 0 \u142 l(x)\u142n(x) dx 2 a a 0 sin lðx a \ufffd \ufffd sin nðx a \ufffd \ufffd dx 2 ð ð 0 sin l\u141 sin n\u141 d\u141 0 (2:41) where equation (A.15) has been introduced. This result may be combined with the normalization relation to give a 0 \u142 l(x)\u142n(x) dx ä ln (2:42) where ä ln is the Kronecker delta, ä ln 1, l n 0, l 6 n (2:43) Functions that obey equation (2.41) are called orthogonal functions. If the orthogonal functions are also normalized, as in equation (2.42), then they are 2.5 Particle in a one-dimensional box 51 said to be orthonormal. The orthogonal property of wave functions in quantum mechanics is discussed in a more general context in Section 3.3. The stationary states Ø(x, t) for the particle in a one-dimensional box are given by substitution of equations (2.39) and (2.40) into (2.31), Ø(x, t) 2 a r sin nðx a \ufffd \ufffd eÿi(n 2ð2"=2ma2) t (2:44) The most general solution (2.33) is, then, Ø(x, t) 2 a r X n cn sin nðx a \ufffd \ufffd eÿi(n 2ð2"=2ma2) t (2:45) Figure 2.2 Wave functions \u142i and probability densities j\u142ij2 for a particle in a one- dimensional box of length a. \u1421 0 a x |\u1421|2 0 a \u1422 0 a x |\u1422|2 0 a \u1423 0 a x |\u1423|2 0 a \u1424 0 a x |\u1424|2 0 a 52 Schro¨dinger wave mechanics 2.6 Tunneling As a second example of the application of the Schro¨dinger equation, we consider the behavior of a particle in the presence of a potential barrier. The specific form that we choose for the potential energy V (x) is given by V (x) V0, 0 < x < a 0, x , 0, x . a and is shown in Figure 2.3. The region where x , 0 is labeled I, where 0 < x < a is labeled II, and where x . a is labeled III. Suppose a particle of mass m and energy E coming from the left approaches the potential barrier. According to classical mechanics, if E is less than the barrier height V0, the particle will be reflected by the barrier; it cannot pass through the barrier and appear in region III. In quantum theory, as we shall see, the particle can penetrate the barrier and appear on the other side. This effect is called tunneling. In regions I and III, where V (x) is zero, the Schro¨dinger equation (2.30) is d2\u142(x) dx2 ÿ 2mE "2 \u142(x) (2:46) The general solutions to equation (2.46) for these regions are \u142I AeiÆx BeÿiÆx (2:47 a) \u142III FeiÆx GeÿiÆx (2:47 b) where A, B, F, and G are arbitrary constants of integration and Æ is the abbreviation Æ 2mE p " (2:48) In region II, where V (x) V0 . E, the Schro¨dinger equation (2.30) becomes V(x) x 0 a V0 I II III Figure 2.3 Potential energy barrier of height V0 and width a. 2.6 Tunneling 53 d2\u142(x) dx2 2m "2 (V0 ÿ E)\u142(x) (2:49) for which the general solution is \u142II Ceâx Deÿâx (2:50) where C and D are integration constants and â is the abbreviation â 2m(V0 ÿ E) p " (2:51) The term exp[iÆx] in equations (2.47) indicates travel in the positive x- direction, while exp[ÿiÆx] refers to travel in the opposite direction. The coefficient A is, then, the amplitude of the incident wave, B is the amplitude of the reflected wave, and F is the amplitude of the transmitted wave. In region III, the particle moves in the positive x-direction, so that G is zero. The relative probability of tunneling is given by the transmission coefficient T T jFj 2 jAj2 (2:52) and the relative probability of reflection is given by the reflection coefficient R R jBj 2 jAj2 (2:53) The wave function for the particle is obtained by joining the three parts \u142I, \u142II, and \u142III such that the resulting wave function \u142(x) and its first derivative \u1429(x) are continuous. Thus, the following boundary conditions apply \u142I(0) \u142II(0), \u1429I(0) \u1429II(0) (2:54) \u142II(a) \u142III(a), \u1429II(a) \u1429III(a) (2:55) These four relations are sufficient to determine any four of the constants A, B, C, D, F in terms of the fifth. If the particle were confined to a finite region of space, then its wave function could be normalized, thereby determining the fifth and final constant. However, in this example, the position of the particle may range from ÿ1 to 1. Accordingly, the wave function cannot be normalized, the remaining constant cannot be evaluated, and only relative probabilities such as the transmission and reflection coefficients can be determined. We first evaluate the transmission coefficient T in equation (2.52). Applying equations (2.55) to (2.47 b) and (2.50), we obtain Ceâa Deÿâa FeiÆa â(Ceâa ÿ Deÿâa) iÆFeiÆa from which it follows that 54 Schro¨dinger wave mechanics C F â iÆ 2â \ufffd \ufffd e(iÆÿâ)a D F âÿ iÆ 2â \ufffd \ufffd e(iÆâ)a (2:56) Application of equation (2.54) to (2.47 a) and (2.50) gives A B C D iÆ(Aÿ B) â(C ÿ D) (2:57) Elimination of B from the pair of equations (2.57) and substitution of equations (2.56) for C and for D yield A 1 2iÆ [(â iÆ)C ÿ (âÿ iÆ)D] F e iÆa 4iÆâ [(â iÆ)2eÿâa ÿ (âÿ iÆ)2eâa] At this point it is easier to form jAj2 before any further algebraic simplifica- tions jAj2 A\ufffdA jFj2 1 16Æ2â2 [(â2 Æ2)2eÿ2âa (â2 Æ2)2e2âa ÿ (âÿ iÆ)4 ÿ (â iÆ)4] jFj2 1 16Æ2â2 [(Æ2 â2)2(eâa ÿ eÿâa)2 16Æ2â2] jFj2 1 (Æ 2 â2)2 4Æ2â2 sinh2 âa " # where equation (A.46) has been used. Combining this result with equations (2.48), (2.51), and (2.52), we obtain T 1 V 2 0 4E(V0 ÿ E) sinh 2( 2m(V0 ÿ E) p a=") " #ÿ1 (2:58) To find the reflection coefficient R, we eliminate A from the pair of equations (2.57) and substitute equations (2.56) for C and for D to obtain B 1 2iÆ [ÿ(âÿ iÆ)C (â iÆ)D] F e iÆa 4iÆâ [(Æ2 â2)eâa ÿ (Æ2 â2)eÿâa] F e iÆa 2iÆâ (Æ2 â2)sinh âa 2.6 Tunneling 55 where again equation (A.46) has been used. Combining this result with equations (2.52) and (2.53), we find that R T jBj 2 jFj2 T (Æ2 â2)2 4Æ2â2 sinh2 âa Substitution of equations (2.48), (2.51), and (2.58) yields R V 20 4E(V0 ÿ E) sinh 2( 2m(V0 ÿ E) p a=") 1 V 2 0 4E(V0 ÿ E) sinh 2( 2m(V0 ÿ E) p a=") (2:59) The transmission coefficient T in equation (2.58) is the relative probability that a particle impinging on the potential barrier tunnels through the barrier. The reflection coefficient R in equation (2.59) is the relative probability that the particle bounces off the barrier and moves in the negative x-direction. Since the particle must do one or the other of these two possibilities, the sum of T and R should equal unity T R 1 which we observe from equations (2.58) and (2.59) to be the case. We also note that the (relative) probability for the particle being in the region 0 < x < a is not zero. In this region, the potential energy is greater than the total particle energy, making the kinetic energy of the particle negative. This concept is contrary to classical theory and does not have a physical signifi- cance. For this reason we cannot observe the particle experimentally within the potential barrier. Further, we note that because the particle is not confined to a finite region, the boundary conditions on the wave function have not imposed any restrictions